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[ "We consider the most basic form of the summation problem, i.e., the summation of Boolean functions.", "Let B N denote the set of Boolean functions f : {0, . . . , N − 1} → {0, 1}. Let", "denote the arithmetic mean of all values of f . Clearly, a f ∈ [0, 1].", "Problem: For f ∈ B N , compute an ε-approximationā f of the sum a f such that" ]

[ "It is known that in the worst case setting, we need roughly N(1− ε) evaluations of the function f .", "In the randomized setting, we assume thatā f is a random variable and require that (2) holds for the expected value of |ā f − a f | for any function f .", "It is known, see e.g., #OTHEREFR , that in the randomized setting we need roughly min{N, ε −1/2 } function evaluations.", "In the quantum setting, we want to compute a random variableā f such that (2) holds with a high probability (greater than 1 2 ) either for all Boolean functions or on the average with respect to a probability measure defined on the set B N .", "These two error criteria in the quantum setting will be precisely defined in Section 3." ]

[ "minimal number" ]

{ "title": "Sharp Error Bounds on Quantum Boolean Summation in Various Settings", "abstract": "We study the quantum summation (QS) algorithm of Brassard, Høyer, Mosca and Tapp, see [1] , that approximates the arithmetic mean of a Boolean function defined on N elements. We improve error bounds presented in [1] in the worst-probabilistic setting, and present new error bounds in the average-probabilistic setting. In particular, in the worst-probabilistic setting, we prove that the error of the QS algorithm using M − 1 quantum queries is 3 4 πM −1 with probability 8 π 2 , which improves the error bound πM −1 + π 2 M −2 of [1] . We also present error bounds with probabilities p ∈ ( In the average-probabilistic setting, we prove that the QS algorithm has error of order min{M −1 , N −1/2 } iff M is divisible by 4. This bound is optimal, as recently shown in [10] . For M not divisible by 4, the QS algorithm is far from being optimal if M ≪ N 1/2 since its error is proportional to M −1 . The quantum summation (QS) algorithm (also known as the amplitude estimation algorithm) of Brassard, Høyer, Mosca and Tapp, see [1] , computes an approximation to the arithmetic *" }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "FM algorithm and QC algorithm are used to find a good threshold index, and AA is used to find all k indices whose values are less than the value of the threshold index.", "In addition, this paper contributes to the following two.", "First, we explicitly distinguish gate complexity and query complexity by defining new symbols.", "Second, we re-formulate FM algorithm and finding k-minima algorithm following the manner of AA.", "Therefore, all of them can be compared more easily and clearly." ]

[ "From N indices, AA searches one of k * miyamotokohei@protonmail.com † masa@cs.osakafu-u.ac.jp ‡ kise@cs.osakafu-u.ac.jp indices that satisfy some certain condition with the query complexity of O( N/k).", "Many quantum algorithms do not directly return multiple answers because the measurement of quantum states collapses the state of superposition.", "Therefore, many trials are required to obtain all the results.", "This is the disadvantage of quantum algorithms and such trials sometimes increase linearly for the number of results. Hence, O(k) trials are required for k results.", "However, our algorithm solves the problem that returns k results with O( √ k)." ]

[ "quantum database search" ]

{ "title": "A Quantum Algorithm for Finding $k$-Minima", "abstract": "We propose a new finding k-minima algorithm and prove that its query complexity is O( √ kN ), where N is the number of data indices. Though the complexity is equivalent to that of an existing method, the proposed is simpler. The main idea of the proposed algorithm is to search a good threshold that is near the k-th smallest data. Then, by using the generalization of amplitude amplification, all k data are found out of order and the query complexity is O( √ kN ). This generalization of amplitude amplification is also not well discussed and we briefly prove the query complexity. Our algorithm can be directly adapted to distance-related problems like k-nearest neighbor search and clustering and classification. There are few quantum algorithms that return multiple answers and they are not well discussed." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "To simplify notation, we set n = (1, s) and we denote the identity oracle Id n by Id s and the line H n by H s . Also, we refer to s as the secret.", "As it turns out, solving DLOG(G p,1 ) or CDH(G p,1 ) is essentially as hard as finding the secret, therefore we define formally this problem as", "What is the complexity of finding s, that is how many calls to the identity oracle are needed for that task? To answer this question, we consider US, the well studied unstructured search problem. Suppose that the size of C is N .", "It is easily seen that probabilistic query complexity of US(C) is linear in N .", "The quantum query complexity of the problem is also well studied." ]

[ "#OTHEREFR have shown that Ω( √ N ) queries are also necessary.", "The relationship between US and the problem SECRET is given by the fact that the identity oracle Id s and the Grover oracle ∆ s can simulate each other with a single query. Corollary 3.3.", "The randomized query complexity of SECRET(G p,1 ) is Θ(p) and its quantum query complexity is Θ( √ p).", "We will now consider the reductions of SECRET(G p,1 ) to DLOG(G p,1 ) and CDH(G p,1 ).", "The case of DLOG(G p,1 ) in fact follows from the case of CDH(G p,1 ), but it is so simple that it is worth to describe it explicitely." ]

[ "queries", "Grover" ]

{ "title": "Discrete logarithm and Diffie-Hellman problems in identity black-box groups", "abstract": "We investigate the computational complexity of the discrete logarithm, the computational Diffie-Hellman and the decisional Diffie-Hellman problems in some identity black-box groups G p,t , where p is a prime number and t is a positive integer. These are defined as quotient groups of vector space Z t+1 p by a hyperplane H given through an identity oracle. While in general black-box groups with unique encoding these computational problems are classically all hard and quantumly all easy, we find that in the groups G p,t the situation is more contrasted. We prove that while there is a polynomial time probabilistic algorithm to solve the decisional Diffie-Hellman problem in G p,1 , the probabilistic query complexity of all the other problems is Ω(p), and their quantum query complexity is Ω( √ p). Our results therefore provide a new example of a group where the computational and the decisional Diffie-Hellman problems have widely different complexity." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "In case b) the hidden shift can be discovered in one query using an algorithm that was found by one of the co-authors #OTHEREFR , provided that the dual of the function can be computed efficiently, where the definition of the dual is via the Fourier spectrum of the function which in this case can be shown to be flat in absolute value.", "If no efficient implementation of the dual is known then still a quantum algorithm exists that can identify the hidden shift in O(n) queries.", "The present paper can be thought of as a generalization of this latter algorithm to the case of Boolean functions other than those having a flat spectrum.", "This is motivated by the quite natural question of what happens when the extremal conditions leading to the family of bent functions are relaxed.", "In this paper we address the question of whether there is a broader class of functions for which hidden shifts of a function can be identified." ]

[ "In this case the corresponding class of Boolean functions are the delta functions, i.e., f, g : {0, 1} n → {0, 1}, where g(x) = f (x + s) and f (x) is the function that takes value 1 on input (0, . . .", ", 0) and 0 elsewhere and g(x) is the function that takes the value 1 on input s and 0 elsewhere.", "Grover's algorithm #OTHEREFR allows to find s in time O( √ 2 n ) on a quantum computer (which is also the fastest possible #OTHEREFR ).", "Thus, the following situation emerges for the quantum and the classical query complexities of these two extremal cases: for bent functions the classical query complexity 1 is Ω( √ 2 n ) and the quantum query complexity 2 is O(n).", "For delta functions the classical query complexity is Θ(2 n ) and the quantum query complexity is Θ( √ 2 n )." ]

[ "hidden shift problem", "Grover's search problem" ]

{ "title": "Quantum algorithm for the Boolean hidden shift problem", "abstract": "Abstract. The hidden shift problem is a natural place to look for new separations between classical and quantum models of computation. One advantage of this problem is its flexibility, since it can be defined for a whole range of functions and a whole range of underlying groups. In a way, this distinguishes it from the hidden subgroup problem where more stringent requirements about the existence of a periodic subgroup have to be made. And yet, the hidden shift problem proves to be rich enough to capture interesting features of problems of algebraic, geometric, and combinatorial flavor. We present a quantum algorithm to identify the hidden shift for any Boolean function. Using Fourier analysis for Boolean functions we relate the time and query complexity of the algorithm to an intrinsic property of the function, namely its minimum influence. We show that for randomly chosen functions the time complexity of the algorithm is polynomial. Based on this we show an average case exponential separation between classical and quantum time complexity. A perhaps interesting aspect of this work is that, while the extremal case of the Boolean hidden shift problem over so-called bent functions can be reduced to a hidden subgroup problem over an abelian group, the more general case studied here does not seem to allow such a reduction." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "What can a quantum computer do with a large classical data set? At first glance it would seem that the costs of loading the data into the quantum computer would overwhelm any possible quantum speedup." ]

[ "Similar questions apply to other quantum optimization algorithms, such as the adiabatic algorithm #OTHEREFR , quantum walks #OTHEREFR , and the QAOA #OTHEREFR .", "Existing quantum algorithms for optimization and machine learning are often less complete than their classical counterparts because they do not use realistic models of their input data #OTHEREFR .", "One way to view this is that they are meant to be subroutines in larger \"end-to-end\" algorithms that will provide the data in the needed format.", "However, there has been relatively little research on these more complete algorithms and their development has often been nontrivial.", "Indeed, the trivial methods of turning large classical datasets into either quantum oracles or quantum states are so expensive as to negate any possible quantum advantage." ]

[ "fast quantum" ]

{ "title": "Small quantum computers and large classical data sets", "abstract": "We introduce hybrid classical-quantum algorithms for problems involving a large classical data set X and a space of models Y such that a quantum computer has superposition access to Y but not X. These algorithms use data reduction techniques to construct a weighted subset of X called a coreset that yields approximately the same loss for each model. The coreset can be constructed by the classical computer alone, or via an interactive protocol in which the outputs of the quantum computer are used to help decide which elements of X to use. By using the quantum computer to perform Grover search or rejection sampling, this yields quantum speedups for maximum likelihood estimation, Bayesian inference and saddle-point optimization. Concrete applications include k-means clustering, logistical regression, zero-sum games and boosting." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Inversion around the mean In order to strengthen amplitude with negative sign we use a procedure called inversion around the mean.", "The operation is fairly simple and implies using 2A − I ⊗2 gate, where A is defined as:", "and the inversion around mean is represented by:", "The gate has following action on the |ψ 2 :", "The searched input has probability amplitude of one, however this is not a general case for number of input qubits greater than two." ]

[ "As a quick example you can check that for 3-qubit inputs (with the second element being searched) the two iterations in the algorithm would follow:", "so that the absolute value of amplitude for the second element is clearly higher.", "The equivalent representation of inversion around mean gate that is implemented in quantum circuit implies using gates:" ]

[ "quantum oracle gate" ]

{ "title": "Quantum machine learning for data scientists", "abstract": "This text aims to present and explain quantum machine learning algorithms to a data scientist in an accessible and consistent way. The algorithms and equations presented are not written in rigorous mathematical fashion, instead, the pressure is put on examples and step by step explanation of difficult topics. This contribution gives an overview of selected quantum machine learning algorithms, however there is also a method of scores extraction for quantum PCA algorithm proposed as well as a new cost function in feed-forward quantum neural networks is introduced. The text is divided into four parts: the first part explains the basic quantum theory, then quantum computation and quantum computer architecture are explained in section two. The third part presents quantum algorithms which will be used as subroutines in quantum machine learning algorithms. Finally, the fourth section describes quantum machine learning algorithms with the use of knowledge accumulated in previous parts. Contents" }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Our next benchmark set consists of circuits that, in addition to preparing encoded quantum states, implement procedures for performing FT quantum operations #OTHEREFR , #OTHEREFR , #OTHEREFR .", "FT operations limit the propagation errors from one qubit in a QECC-register (the block of qubits that encodes a logical qubit) to another qubit in the same register, and a single faulty gate damages at most one qubit in each register.", "One constructs FT stabilizer circuits by executing each Clifford gate transversally 12 across QECC-registers as shown in Figure 14 .", "Non-Clifford gates need to be implemented using a FT architecture that often requires ancilla qubits, measurements and correction procedures conditioned on measurement outcomes.", "Figure 15 shows a circuit that implements a FT-Toffoli operation #OTHEREFR ." ]

[ "The state |cat = ( 0 ⊗5 + 1 ⊗5 )/ √ 2 is obtained using a stabilizer subcircuit (not shown).", "The arrows point to the set of gates that is applied if the measurement outcome is 1; no action is taken otherwise.", "Controlled-Z gates are implemented as H j CN OT i,j H j with control i and target j. Z gates are implemented as P 2 .", "We implemented FT benchmarks for the half-adder and full-adder circuits ( Figure 16 ) as well as for computing f (x) = b x mod 15.", "Each circuit from Figure 17 implements f (x) with a particular coprime base value #OTHEREFR ." ]

[ "5-qubit register" ]

{ "title": "Simulation of Quantum Circuits via Stabilizer Frames", "abstract": "Abstract-Generic quantum-circuit simulation appears intractable for conventional computers and may be unnecessary because useful quantum circuits exhibit significant structure that can be exploited during simulation. For example, Gottesman and Knill identified an important subclass, called stabilizer circuits, which can be simulated efficiently using group-theory techniques and insights from quantum physics. Realistic circuits enriched with quantum error-correcting codes and fault-tolerant procedures are dominated by stabilizer subcircuits and contain a relatively small number of non-Clifford components. Therefore, we develop new data structures and algorithms that facilitate parallel simulation of such circuits. Stabilizer frames offer more compact storage than previous approaches but require more sophisticated bookkeeping. Our implementation, called Quipu, simulates certain quantum arithmetic circuits (e.g., reversible ripple-carry adders) in polynomial time and space for equal superpositions of n-qubits. On such instances, known linear-algebraic simulation techniques, such as the (state-ofthe-art) BDD-based simulator QuIDDPro, take exponential time. We simulate quantum Fourier transform and quantum fault-tolerant circuits using Quipu, and the results demonstrate that our stabilizer-based technique empirically outperforms QuIDDPro in all cases. While previous high-performance, structure-aware simulations of quantum circuits were difficult to parallelize, we demonstrate that Quipu can be parallelized with a nontrivial computational speedup." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Space-time trade-offs are a widely observed phenomenon in data structure complexity.", "In this work, we are interested in space-time trade-offs in inverting random functions, namely, the trade-off between the size (in the number of bits) of pre-computed data structure (or advice) about the function and the algorithm's running time for computing the inverse of a certain image.", "Such trade-offs give lower bound for algorithms that inverts cryptographic functions without taking the specific structure of that family of functions.", "Without pre-computed advice ( = 0), classical computers requires = Ω( ) for inverting a fraction of the input for a random function", ": [ ] ↦ → [ ], and quantum computers requires = Ω( √ ) #OTHEREFR ." ]

[ "However, if we allow some pre-computed advice, classical computers can do much better.", "Hellman #OTHEREFR showed that every permutation can be inverted with = = ( 2/3 ).", "However, it is not known whether we can do better than Grover's algorithm or Hellman's algorithm, even if we allow quantum computers to come into play.", "Therefore motivated by postquantum cryptanalysis, it is natural to ask whether these two algorithms are indeed the best that we can do. For classical computers, De et al.", "#OTHEREFR (going back to ideas of Yao #OTHEREFR ) showed that =Ω( ) is required, and Corrigan-Gibbs and Kogan #OTHEREFR gave some evidence that improving this lower bound seems to be difficult, by connecting function inversion problem to several other hard problems in complexity theory, communication complexity, etc. For quantum computers, Nayebi et al." ]

[ "Grover's algorithm" ]

{ "title": "Lower Bounds for Function Inversion with Quantum Advice", "abstract": "Function inversion is that given a random function : [ ] → [ ], we want to compute some auxiliary information of size that we can find pre-image of any image with a few queries to the function given as a black box in time . It is a well-studied problem in the classical settings, however, it is not clear how a quantum adversary can do better at this task besides invoking Grover's algorithm [Gro96] . Nayebi et al. [NABT15] proved a lower bound for adversaries inverting permutations leveraging only quantum queries to the black box. We give a matching lower bound for functions and permutations where = ( ), and allowing adversaries to be fully quantum, and thus resolving the open question positively raised by Nayebi et al. of whether such lower bound is achievable for inverters with quantum advice. In order to prove these bounds, we also proved a lower bound for a generalized version of quantum random access code (originally introduced by Ambainis et al. [ANTSV99]), i.e. under the setting where the encoding length is variable and each element can be arbitrarily correlated, which may be of independent interest." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Here we discuss four examples of n-bit instances of satisfiability.", "In three of the examples the problems are classically computationally simple to solve.", "These problems also have structure that we exploit to calculate g min in the corresponding quantum version.", "In each case g min goes like 1/n p , so these problems can be solved in polynomial time by adiabatic quantum evolution." ]

[ "If we assume that we treat the clause as an oracle, which may be queried but not analyzed, it takes 2 n classical queries to find the satisfying assignment.", "Our quantum version has g min of order 2 −n/2 , so the time required for quantum adiabatic evolution scales like 2 n , which means that there is no quantum speedup.", "Nonetheless, it is instructive to see how it is possible to evaluate g min for the Grover problem." ]

[ "\"Grover problem" ]

{ "title": "Quantum Computation by Adiabatic Evolution", "abstract": "We give a quantum algorithm for solving instances of the satisfiability problem, based on adiabatic evolution. The evolution of the quantum state is governed by a time-dependent Hamiltonian that interpolates between an initial Hamiltonian, whose ground state is easy to construct, and a final Hamiltonian, whose ground state encodes the satisfying assignment. To ensure that the system evolves to the desired final ground state, the evolution time must be big enough. The time required depends on the minimum energy difference between the two lowest states of the interpolating Hamiltonian. We are unable to estimate this gap in general. We give some special symmetric cases of the satisfiability problem where the symmetry allows us to estimate the gap and we show that, in these cases, our algorithm runs in polynomial time." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Here, the first equality follows from the fact that the random walk can return to location 0 only after an even number of steps.", "The second equality follows by partitioning the paths that return to 0 after 2m steps according to the total number of times the path returns to 0 (including the final return to 0).", "If the path returns to 0 t times, then it also leaves 0 t times.", "Hence, there are t steps in which we move right with probability 1 (the steps which start at location 0).", "There are also m − t other steps when the path moves right (each of those steps is taken with probability p) and m steps when the path moves left (each of those steps is taken with probability 1 − p)." ]

[ "Let S = 2m − t − 1 and A = m − t.", "We can rewrite the sum (16) in the following way:", "where the last equality follows from", "This completes the proof of the claim." ]

[ "path" ]

{ "title": "Quantum strategies are better than classical in almost any XOR game", "abstract": "We initiate a study of random instances of nonlocal games. We show that quantum strategies are better than classical for almost any 2-player XOR game. More precisely, for large n, the entangled value of a random 2-player XOR game with n questions to every player is at least 1.21... times the classical value, for 1 − o(1) fraction of all 2-player XOR games." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "If any one state is true and all others false, the result is just a binary count of n items, a given combination of n computational basis qubits.", "Readout is probabilistic.", "For example, if copies of 0011, 0000 are read a few times, there is a very good chance that entries |0>, |1>, |0> and |1), |1>, |1> will result, permitting reconstruction of the stored word 0011, 0000.", "(Simply note that the entries corresponding to combinations of computational basis qubits count up in binary from the left).", "The H10 structure has the advantage that it can be produced without complex quantum gates." ]

[ "Stored words in the proposed system cannot be searched before reading.", "Ideally Grover's algorithm may be used someday to address, or quickly locate state vectors prior to reading, and then to read them without destroying them." ]

[ "state vector" ]

{ "title": "Note on Needle in a Haystack", "abstract": "Introduced below is a quantum database method, not only for retrieval but also for creation. It uses a particular structure of true's and false's in a state vector of n qubits, permitting up to 2**2**n words, vastly more than for classical bits. Several copies are produced so that later they can be destructively observed and a word determined with high probability. Grover's algorithm is proposed below to read out, nondestructively the unknown contents of a given stored state vector using only one state vector. Holding data in a quantum system is clearly problematic, since such systems quickly become incoherent, although systems of the future might be designed to be more stable. This paper assumes a quantum system with m identical sets of n qubits, where each set is assumed to form a state vector. Then each state vector is loaded with 1's and 0's with an appropriate normalization factor. For n qubits a state vector has dimension 2 n which theoretically holds n 2 2 binary codes. An interesting and practical subset of a general state vector is to initialize qubits to combinations of H |0> = (1 1)'/√2, |1> = (0 1)', and |0> = (1 0)'. Each state vector will then have a particular structure termed here the H10 structure: For 2 n entries, either the pattern repeats in the other half (2 n-1 ) entries, or one of the halves is filled with 0s. For example, one may have 0011, 0000/√2 for n = 3, this being |0> |1> H |0>. Note that anything similar to 0111, 0000 is impossible under H10 symmetry, since this rule also applies to each quarter (block of 2 n-2 ) and so on down to blocks of 2 1 states. If any one state is true and all others false, the result is just a binary count of n items, a given combination of n computational basis qubits. Readout is probabilistic. For example, if copies of 0011, 0000 are read a few times, there is a very good chance that entries |0>, |1>, |0> and |1), |1>, |1> will result, permitting reconstruction of the stored word 0011, 0000. (Simply note that the entries corresponding to combinations of computational basis qubits count up in binary from the left). The H10 structure has the advantage that it can be produced without complex quantum gates. It should be mentioned that the number of codes using H10 symmetry is far more than 2 n but still far less than the n 2 2 available by arbitrarily placing true's" }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Many computational problems involve queries to an oracle (calls to a subroutine) that evaluates some function f at the argument x passed to it and returns the result f (x).", "Typically, the task is to use the oracle to determine some property of the unknown function.", "An important example for quantum computation is PERIOD FINDING #OTHEREFR (the ABELIAN HIDDEN SUBGROUP PROBLEM #OTHEREFR ), where the function is invariant under addition of some constant to its argument and the task is to find that constant.", "Another example is CONCEPT LEARNING, where there is some set (the concept class) of Boolean-valued functions and the task is to identify exactly which one (the concept) the oracle is evaluating #OTHEREFR ." ]

[ "A natural goal is to minimize the number of queries to the oracle needed to solve the problem; this minimum is the query complexity of the problem.", "An alternative goal is to maximize the probability of determining the desired property of f using no more than some fixed number of queries, k.", "Although this probability is clearly non-decreasing in k, when it does not increase with additional queries, we might say that these queries provide no information, or describe them as useless.", "For example, consider Deutsch's problem, in which f : {1, 2} → Z 2 and the property to be determined is f (1) + f (2) #OTHEREFR .", "If f is chosen uniformly at random, then the prior probabilities for the value of this sum are each 1/2." ]

[ "Grover's UNSTRUCTURED SEARCH" ]

{ "title": "On the uselessness of quantum queries", "abstract": "Given a prior probability distribution over a set of possible oracle functions, we define a number of queries to be useless for determining some property of the function if the probability that the function has the property is unchanged after the oracle responds to the queries. A familiar example is the parity of a uniformly random Boolean-valued function over {1, 2, . . . , N }, for which N − 1 classical queries are useless. We prove that if 2k classical queries are useless for some oracle problem, then k quantum queries are also useless. For such problems, which include classical threshold secret sharing schemes, our result also gives a new way to obtain a lower bound on the quantum query complexity, even in cases where neither the function nor the property to be determined is Boolean." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "The generalized Grover or amplitude amplification algorithm #OTHEREFR , can rotate a given source state |s close to a target state |t using any unitary transformation V in no more than 1/ |V ts | steps, where t|V |s = V ts = 0." ]

[ "We have in our numerical study simulated a computer which, instead of the change of sign on the target state has been programmed to implement an arbitrary phase-shift i.e.", "|t → e iφ |t instead of |t → −|t , and we watch the adaptive modification of the state χ(φ) representing an initially unknown phase shift towards a state with a well defined, optimum, phase shift.", "Figure 2 shows the efficiency of the adaptive learning applied to the Grover algorithm, defined as the average success-probability obtained by an ensemble of .", "10 % and 25 % quantiles of the number of iterations needed to obtain a probability of success of 95 % of the theoretical maximum.", "The data show that with the asymmetric push feedback, a near-optimal Grover search on databases with up to 10000 elements can be taught in less than 20 iterations in the best 10 % runs, while 10-12 iterations suffice for smaller databases where success is achieved in 25 % of the runs." ]

[ "original Grover algorithm", "computational states" ]

{ "title": "Quantum learning by measurement and feedback", "abstract": "Abstract. We investigate an approach to quantum computing in which quantum gate strengths are parametrized by quantum degrees of freedom. The capability of the quantum computer to perform desired tasks is monitored by measurements of the output and gradually improved by successive feedback modifications of the coupling strength parameters. Our proposal only uses information available in an experimental implementation, and is demonstrated with simulations on search and factoring algorithms." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "For both reasons, Shor's discovery sparked a great deal of interest in the design of quantum algorithms and computers that endures today.", "In addition to Shor's algorithm, there is a wealth of other interesting and important algorithms that have been developed for quantum computers.", "Two of those algorithms will be described in detail in this tutorial in order to better elucidate the study of quantum computing theory and quantum algorithm design.", "These two algorithms are good models for our current understanding of quantum computation as many other quantum algorithms use similar techniques to achieve their results, whether they be algorithms to solve linear systems of equations #OTHEREFR , or quickly compute discrete logarithms.", "Shor, for example, credits one of these algorithms as the inspiration for his own #OTHEREFR ." ]

[ "Grover's algorithm searches for a specified entry in an unordered database, em-An Introduction to Quantum Algorithms" ]

[ "Lov Grover's quantum" ]

{ "title": "An Introduction to Quantum Algorithms", "abstract": "" }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "If ε = 0, we says that the quantum algorithm is exact.", "For more details on quantum query complexity, we may refer to #OTHEREFR .", "Quantum query models are one of most important computing models in quantum computing.", "In this complexity models #OTHEREFR , an algorithm is charged for \"queries\" to the input bits, while any intermediate computation is considered as free.", "For many functions one can obtain large quantum speed-ups in the case algorithms are allowed a constant small probability of error (bounded error)." ]

[ "The model of exact quantum query, where the algorithms must output the correct answer with certainty for every possible input, seems to be more intriguing #OTHEREFR .", "It is much more difficult to come up with exact quantum algorithms that outperform classical deterministic algorithms.", "In the exact quantum query complexity, it was recognized that the best quantum speed-up for computing total functions was by a factor of 2 for many years #OTHEREFR .", "In a breakthrough result, Ambainis has presented the first example of a Boolean function f : {0, 1} n → {0, 1} for which exact quantum algorithms have superlinear advantage over classical deterministic algorithms #OTHEREFR . The result was improved in 2016 #OTHEREFR .", "Based on the results in #OTHEREFR , Ambainis, Gruska, and Zheng #OTHEREFR have verified that exact quantum algorithms have certain advantage for most of Boolean functions." ]

[ "n-bit", "classical algorithm" ]

{ "title": "Characterizations of symmetrically partial Boolean functions with exact quantum query complexity", "abstract": "We give and prove an optimal exact quantum query algorithm with complexity k + 1 for computing the promise problem (i.e., symmetric and partial Boolean function) DJ k n defined as: DJ k n (x) = 1 for |x| = n/2, DJ k n (x) = 0 for |x| in the set {0, 1, . . . , k, n − k, n − k + 1, . . . , n}, and it is undefined for the rest cases, where n is even, |x| is the Hamming weight of x. The case of k = 0 is the well-known Deutsch-Jozsa problem. We outline all symmetric (and partial) Boolean functions with degrees 1 and 2, and prove their exact quantum query complexity. Then we prove that any symmetrical (and partial) Boolean function f has exact quantum 1-query complexity if and only if f can be computed by the Deutsch-Jozsa algorithm. We also discover the optimal exact quantum 2-query complexity for distinguishing between inputs of Hamming weight {⌊n/2⌋, ⌈n/2⌉} and Hamming weight in the set {0, n} for all odd n. In addition, a method is provided to determine the degree of any symmetrical (and partial) Boolean function." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Even if quantum computers are unlikely to become widely available in the next couple of years, the cryptographic community has decided to start worrying about this threat and to study its impact.", "One compelling reason for taking action is that even current pre-quantum long-term secrets are at risk as it seems feasible for a malicious organization to simply store all encrypted data until it has access to a quantum computer.", "This explains why post-quantum cryptosystems, based for instance on lattices or codes, have become a very hot topic in cryptology, and researchers are now concentrating their efforts in order to provide efficient alternatives that would resist quantum adversaries.", "In this paper, we focus on symmetric cryptography, the other main branch of cryptography.", "Symmetric primitives also suffer from a reduced ideal security in the quantum world, but this security reduction turns out to be much less drastic than for many asymmetric primitives." ]

[ "It can be applied to any generic exhaustive key search, but merely offers a quadratic speedup compared to a classical attack.", "Therefore, the current consensus is that key lengths should be doubled in order to offer the same security against quantum algorithms.", "This was one of the motivations to require a version of AES with a 256-bit key, that appears in the initial recommendations of the European PQCRYPTO project [ABB + 15]:", "\"Symmetric systems are usually not affected by Shor's algorithm, but they are affected by Grover's algorithm.", "Under Grover's attack, the best security a key of length n can offer is 2 n/2 , so AES-128 offers only 2 64 post-quantum security." ]

[ "main quantum attack", "Grover's algorithm" ]

{ "title": "Quantum Differential and Linear Cryptanalysis", "abstract": "Abstract. Quantum computers, that may become available one day, would impact many scientific fields, most notably cryptography since many asymmetric primitives are insecure against an adversary with quantum capabilities. Cryptographers are already anticipating this threat by proposing and studying a number of potentially quantum-safe alternatives for those primitives. On the other hand, symmetric primitives seem less vulnerable against quantum computing: the main known applicable result is Grover's algorithm that gives a quadratic speed-up for exhaustive search. In this work, we examine more closely the security of symmetric ciphers against quantum attacks. Since our trust in symmetric ciphers relies mostly on their ability to resist cryptanalysis techniques, we investigate quantum cryptanalysis techniques. More specifically, we consider quantum versions of differential and linear cryptanalysis. We show that it is usually possible to use quantum computations to obtain a quadratic speed-up for these attack techniques, but the situation must be nuanced: we don't get a quadratic speed-up for all variants of the attacks. This allows us to demonstrate the following non-intuitive result: the best attack in the classical world does not necessarily lead to the best quantum one. We give some examples of application on ciphers LAC and KLEIN. We also discuss the important difference between an adversary that can only perform quantum computations, and an adversary that can also make quantum queries to a keyed primitive." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "The most significant discovery in quantum computation would be Shor's polynomial-time quantum algorithms for factoring integers and computing discrete logarithms #OTHEREFR , both of which are believed to be hard to solve in classical settings and are thus used in arguments for the security of the widely used cryptosystems." ]

[ ", N − 1} such that f (i) = 1 for a hidden Boolean function f ; it has yielded a variety of generalizations #OTHEREFR .", "Grover's algorithm and its generalizations assume the oracle computation model, in which a problem instance is given as a black box (called an oracle) and any algorithm needs to make queries to the black box to get sufficient information on the instance.", "In the case of searching an unstructured set, any algorithm needs to make queries of the form \"what is the value of function f for input i ?\" to the given oracle.", "In the oracle computation model, the efficiency of an algorithm is usually measured by the number of queries the algorithm needs to make, i.e., the query complexity of the algorithm.", "The query complexity of a problem means the query complexity of the algorithm that solves the problem with fewest queries." ]

[ "Grover's quantum algorithm" ]

{ "title": "Claw Finding Algorithms Using Quantum Walk", "abstract": "The claw finding problem has been studied in terms of query complexity as one of the problems closely connected to cryptography. For given two functions, f and g, as an oracle which have domains of size N and M (N ≤ M), respectively, and the same range, the goal of the problem is to find x and y such that f (x) = g(y). This problem has been considered in both quantum and classical settings in terms of query complexity. This paper describes an optimal algorithm using quantum walk that solves this problem. Our algorithm can be slightly modified to solve a more general problem of finding a tuple consisting of elements in the two function domains that has prespecified property. Our algorithm can also be generalized to find a claw of k functions for any constant integer k > 1, where the domains of the functions may have different size." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Various quantum computational frameworks, such as Quantum Circuit Model #OTHEREFR , Topological Quantum Computation #OTHEREFR , Adiabatic Quantum Computation (AQC) #OTHEREFR , Quantum Walk (QW) #OTHEREFR , Resonant Transition Based Quantum Computation (RTBQC) #OTHEREFR and Measurement Based Quantum Computation (MBQC) #OTHEREFR have been proposed to attack problems that are considered extremely difficult for classical computers.", "Notable successes include the inventions of Shor's factoring algorithm and Grover's search algorithm, which manifest indisputable enhancement over all known classical algorithms designed for the same purpose.", "Among the proposed quantum computational frameworks above, quantum walk models are certainly among the most heavily supported." ]

[ "In addition, they are central to quantum algorithms #OTHEREFR created to tackle other computationally hard problems, such as graph isomorphism #OTHEREFR , network analysis and navigation #OTHEREFR , and quantum simulation #OTHEREFR , even including certain aspects of complex biological processes #OTHEREFR .", "Furthermore, due to the simple physics principle behind quantum walk models, various efforts have been made to establish a better understanding of quantum walk models by relating to other major quantum computational frameworks or explore novel approaches to exploit quantum walks to perform a greater variety of tasks #OTHEREFR .", "Quantum walks can be formulated in both discrete time #OTHEREFR and continuous time #OTHEREFR versions.", "In this work, we focus on the study of continuous-time quantum walk (CTQW), not only because it offers a simpler physical picture but also it is less challenging to perform CTQW experiments in comparison to their discrete-time counterparts.", "Furthermore, if implementing CTQW in a quantum circuit model, robust quantum computations could be attained due to the availability of fault tolerance and error corrections." ]

[ "Grover's search algorithm" ]

{ "title": "Optimizing Quantum Walk Search on a Reduced Uniform Complete Multi-Partite Graph", "abstract": "In a recent work by Novo et al. (Sci. Rep. 5, 13304, 2015), the invariant subspace method was applied to the study of continuous-time quantum walk (CTQW). The method helps to reduce a graph into a simpler version that allows more transparent analyses of the quantum walk model. In this work, we adopt the aforementioned method to investigate the optimality of a quantum walk search of a marked element on a uniform complete multi-partite graph. We formulate the eigenbasis that would facilitate the transport between the two lowest energy eigenstates and demonstrate how to set the appropriate coupling factor to preserve the optimality." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "The prospect of a large-scale, cryptographically relevant quantum computer has prompted increased scrutiny of the post-quantum security of our cryptographic primitives.", "Shor's algorithm for factoring and computing discrete logarithms introduced in [Sho94] and #OTHEREFR will completely break public-key schemes such as RSA, ECDSA and ECDH.", "But symmetric schemes such as block ciphers and hash functions are widely considered post-quantum secure." ]

[ "As Grover's algorithm only provides at most a square root speedup, the rule of thumb is to simply double the cipher's key size to make it post-quantum secure.", "Such conventional wisdom reflects the asymptotic behavior and only gives a rough idea of the security penalties that quantum computers inflict on symmetric primitives.", "In particular, the cost of evaluating the Grover oracle is often ignored.", "In their call for proposals to the standardization of post-quantum cryptography [NIS16], the National Institute of Standards and Technology (NIST) proposes security categories for post-quantum public-key schemes such as key encapsulation and digital signatures.", "The categories are defined by the cost of quantum algorithms for exhaustive key search on the block cipher AES and collision search for the hash function SHA-3, and measure the attack cost in the number of quantum gates." ]

[ "Grover's algorithm" ]

{ "title": "Implementing Grover oracles for quantum key search on AES and LowMC", "abstract": "Grover's search algorithm gives a quantum attack against block ciphers by searching for a key that matches a small number of plaintext-ciphertext pairs. This attack uses O( √ N ) calls to the cipher to search a key space of size N . Previous work in the specific case of AES derived the full gate cost by analyzing quantum circuits for the cipher, but focused on minimizing the number of qubits. In contrast, we study the cost of quantum key search attacks under a depth restriction and introduce techniques that reduce the oracle depth, even if it requires more qubits. As cases in point, we design quantum circuits for the block ciphers AES and LowMC. Our circuits give a lower overall attack cost in both the gate count and depth-times-width cost models. In NIST's post-quantum cryptography standardization process, security categories are defined based on the concrete cost of quantum key search against AES. We present new, lower cost estimates for each category, so our work has immediate implications for the security assessment of post-quantum cryptography. As part of this work, we release Q# implementations of the full Grover oracle for and for the three LowMC instantiations used in Picnic, including unit tests and code to reproduce our quantum resource estimates. To the best of our knowledge, these are the first two such full implementations and automatic resource estimations." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "The inefficiency of the classical algorithm arises only in determining a physical quantity of interest.", "In the case of Monte-Carlo algorithms, the \"measurement\" of a physical quantity suffers from the so-called \"sign problem\", often resulting in exponentially large, random errors that can be reduced only by repeating the computation extremely many times.", "In contrast, the quantum algorithms for emulation can determine many (but not all) of the interesting physical quantities with polynomially bounded statistical errors.", "How to efficiently implement measurements of these quantities has been the topic of more recent work in this area, much of which is based on variants of the phase estimation algorithm #OTHEREFR .", "Although several researchers have suggested that there are interesting quantum physics simulations that can be implemented with well below 100 qubits, one of the interesting problems in this area of research is to come up with a specific simulation algorithm that uses small numbers of qubits and quantum gates, and that computes an interesting physical quantity not easily obtainable using available classical computers." ]

[ "Given is a black box that computes a binary function f on inputs x with 0 ≤ x < N.", "The function f has the property that there is a unique input a for which f (a) = 1.", "The standard quantum version of this black box implements the transformationf| | |x | | |b = | | |x | | |b ⊕ f (x) , where b is a bit and b ⊕ f (x) is computed modulo 2.", "Unstructured quantum search finds a quadratically faster, that is, in time of order N 1/2 , than the best classical black-box search, which requires time of order N.", "The context for this algorithm is the famous P = NP conjecture, which is captured by the following algorithmic problem: Given is a classical circuit C that computes an output." ]

[ "quantum computers", "unstructured quantum search" ]

{ "title": "0 Ju l 2 00 2 Introduction to Quantum Information Processing", "abstract": "" }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Q# is scalable in that it allows to write programs to target machines of various sizes, ranging from small machines with only a few hundred qubits to large machines with millions of qubits.", "Finally, being a bona-fide stand-alone language, Q# allows a programmer to code complex quantum algorithms, offers rich and informative error reporting, and allows to perform various tasks such as debugging, profiling, resource estimation, and certain special-purpose simulations.", "RevKit is an open source C++ framework and library that implements a large set of reversible synthesis, optimization, and mapping algorithms.", "By default, RevKit is executed as a command-based shell application, which allows to perform synthesis scripts by combining a variety of different commands.", "As an example, the command sequence revgen --hwb 4; tbs; revsimp; rptm; tpar; ps -c #OTHEREFR generates a reversible function describing the 4-input reversible hidden-weighted bit function, synthesizes it into a reversible circuit using transformation-based synthesis #OTHEREFR , performs simplification of the resulting circuit, maps it into Clifford+T gates using the mapping described in #OTHEREFR , optimizes the T count using the T-par algorithm presented in #OTHEREFR , and finally prints statistics about the final quantum circuit." ]

[ "Using the Python interface, RevKit can be executed from within ProjectQ using the projectq.libs.revkit module." ]

[ "RevKit commands", "first command" ]

{ "title": "Programming quantum computers using design automation", "abstract": "Recent developments in quantum hardware indicate that systems featuring more than 50 physical qubits are within reach. At this scale, classical simulation will no longer be feasible and there is a possibility that such quantum devices may outperform even classical supercomputers at certain tasks. With the rapid growth of qubit numbers and coherence times comes the increasingly difficult challenge of quantum program compilation. This entails the translation of a high-level description of a quantum algorithm to hardware-specific low-level operations which can be carried out by the quantum device. Some parts of the calculation may still be performed manually due to the lack of efficient methods. This, in turn, may lead to a design gap, which will prevent the programming of a quantum computer. In this paper, we discuss the challenges in fully-automatic quantum compilation. We motivate directions for future research to tackle these challenges. Yet, with the algorithms and approaches that exist today, we demonstrate how to automatically perform the quantum programming flow from algorithm to a physical quantum computer for a simple algorithmic benchmark, namely the hidden shift problem. We present and use two tool flows which invoke RevKit. One which is based on ProjectQ and which targets the IBM Quantum Experience or a local simulator, and one which is based on Microsoft's quantum programming language Q#." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "In this approach #OTHEREFR , we first decompose the Hamiltonian into leading-and higher-order terms:", "for large M .", "From this, we next find the eigenvalues and eigenvectors of H (0) , some of which may be degenerate.", "Finally, adding the perturbation H #OTHEREFR , certain linear combinations of the degenerate eigenvectors of H (0) are eigenvectors of H (0) + H #OTHEREFR , and this \"lifts\" the degeneracy.", "This mixing drives evolution between degenerate eigenvectors of H (0) , and the energy or eigenvalue gap dictates the rate of evolution." ]

[ "So, H #OTHEREFR should include terms Θ(M 3/2 ), and H (0) should include anything of higher order, i.e., ω(M 3/2 ):", "With this decomposition, we next need to find the eigenvectors and eigenvalues of H (0) , but unfortunately, this is prohibitively complicated.", "To circumvent this obstacle, we can try changing the basis, as in #OTHEREFR .", "Besides |a , we choose the uniform superposition of unmarked vertices to be another basis state:", "A state that is obviously orthogonal to this, which we use as a third basis state, is" ]

[ "perturbation H", "marked vertex |a" ]

{ "title": "Quantum Walk Search on Kronecker Graphs", "abstract": "Kronecker graphs, obtained by repeatedly performing the Kronecker product of the adjacency matrix of an \"initiator\" graph with itself, have risen in popularity in network science due to their ability to generate complex networks with real-world properties. In this paper, we explore spatial search by continuous-time quantum walk on Kronecker graphs. Specifically, we give analytical proofs for quantum search on first-, second-, and third-order Kronecker graphs with the complete graph as the initiator, showing that search takes Grover's O( √ N ) time. Numerical simulations indicate that higher-order Kronecker graphs with the complete initiator also support optimal quantum search." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "In the last example, we consider programming a quantum system composed of 16 qubits, as for example the ibmqx5 quantum computer from IBM #OTHEREFR ." ]

[ "The function can be represented using the DAG in Fig. 6(a) .", "Our first attempt is to use the Bennett strategy for qubits clean-up. This method leads to the circuit shown in Fig. 6(b) .", "This 17-qubits design cannot be mapped on the chosen 16-bits hardware.", "A second possibility is to apply the well known decomposition method proposed by Barenco in #OTHEREFR to the 9-controls Toffoli gates, as in Fig. 6(d) .", "In this case only one extra ancilla is required (11 qubits in total)." ]

[ "Grover's algorithm" ]

{ "title": "A pr 2 01 9 Reversible Pebbling Game for Quantum Memory Management", "abstract": "Abstract-Quantum memory management is becoming a pressing problem, especially given the recent research effort to develop new and more complex quantum algorithms. The only existing automatic method for quantum states clean-up relies on the availability of many extra resources. In this work, we propose an automatic tool for quantum memory management. We show how this problem exactly matches the reversible pebbling game. Based on that, we develop a SAT-based algorithm that returns a valid clean-up strategy, taking the limitations of the quantum hardware into account. The developed tool empowers the designer with the flexibility required to explore the trade-off between memory resources and number of operations. We present three show-cases to prove the validity of our approach. First, we apply the algorithm to straight-line programs, widely used in cryptographic applications. Second, we perform a comparison with the existing approach, showing an average improvement of 52.77%. Finally, we show the advantage of using the tool when synthesizing a quantum circuit on a constrained near-term quantum device. The prospective of experimenting with a practical quantum computer is closing up thanks to the recent developments in hardware technology [1], [2], [3] . Driven by the revolutionary potential capabilities of quantum computing, research is extremely active both in academic and in industrial environments [4] . The race is on to develop quantum algorithms capable of proving quantum supremacy, which is the ability to solve problems that cannot be solved classically [5] , [6] , [7] , [8] . A large part of the design of quantum algorithms is still performed manually, despite the emergence of several automatic methods for both synthesis [9], [10] and optimization [11] , [12] , [13] of quantum circuits. Most manual and automatic approaches for quantum circuit synthesis decompose large functionality into smaller parts in order to deal with complexity. Each part requires some resources in terms of qubits and quantum operations. The components can be connected together in order to obtain the desired computation for the overall circuit. Most of the parts of a large function are used to compute intermediate values, which are stored on qubits. However, the final composed circuit must not emit any of those values. Otherwise, the computed results may entangle with intermediate values and compromise the overall quantum algorithm. Since quantum operations are reversible, intermediate results can be \"uncomputed\" by performing the same operations that computed them, in reverse order. Fig. 1 illustrates a small example. The composition of the two functions f and g generates an unknown state that can be uncomputed by performing f in reverse order. There are many possible ways to combine the small parts of a decomposition, each of which resulting in different accumulated costs for number of qubits and number of quantum" }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "The first ideas of using quantum devices for computation were expressed at the beginning of the eighties by Manin #OTHEREFR , see also #OTHEREFR , and Feynman #OTHEREFR .", "They observed that simulating quantum mechanics on a classical computer is extremly hard, probably infeasible, since it leads to differential equations whose dimensions are exponential in the number of system components.", "To overcome this, they suggested the idea to simulate quantum mechanics using quantum devices itself.", "In 1985, Deutsch #OTHEREFR developed the rigorous theoretical basis of quantum computation -the model of a quantum Turing machine, which became the so far most serious and still standing challenge to the Turing-Church Thesis (the latter stating that, roughly, every reasonable physical computing device can be simulated with only polynomial increase of resources on a classical Turing machine).", "A breakthrough for quantum computing happened in 1994, when Shor #OTHEREFR showed that efficient factorization of integers would be possible on a quantum computer, which in turn, would mean the possibility of breaking the foremost public key codes like the RSA cryptosystem." ]

[ "Since then we witness an explosion of efforts, broad research on quantum algorithms for all kinds of (mostly discrete) problems, on quantum cryptography, and quantum information theory.", "Physicists are intensively working on how to construct quantum computers, that means, finding quantum mechanical systems that can be manipulated to fulfill the abstractly proposed requirements.", "Systems with a few qubits are already successfully realized in laboratories.", "Important forerunners for the development of quantum algorithms for Monte Carlo related, numerical problems were the work of Boyer, Brassard, Høyer, Mosca, and Tapp #OTHEREFR , #OTHEREFR on quantum counting and the results of Beals, Buhrman, Cleve, Mosca and de Wolf #OTHEREFR and Nayak and Wu #OTHEREFR on lower bounds." ]

[ "efficient quantum search" ]

{ "title": "From Monte Carlo to quantum computation", "abstract": "Quantum computing was so far mainly concerned with discrete problems. Recently, E. Novak and the author studied quantum algorithms for high dimensional integration and dealt with the question, which advantages quantum computing can bring over classical deterministic or randomized methods for this type of problem. In this paper we give a short introduction to the basic ideas of quantum computing and survey recent results on high dimensional integration. We discuss connections to the Monte Carlo methology and compare the optimal error rates of quantum algorithms to those of classical deterministic and randomized algorithms." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Examples include MicroMint #OTHEREFR , RMAC #OTHEREFR , chopMD #OTHEREFR , Leamnta-LW [HIK + 11], PHO-TON and Parazoa #OTHEREFR , the Keyed-Sponge #OTHEREFR , all of which assume the multi-collision resistance of a certain function.", "Multi-collisions algorithms have also been used in attacks, such as the MDC-2 #OTHEREFR , HMAC #OTHEREFR , Even-Mansour #OTHEREFR , and LED #OTHEREFR .", "Multicollision resistance for polynomial k has also recently emerged as a theoretical way to avoid keyed hash functions #OTHEREFR , or as a useful cryptographic primitives, for example, to build statistically hiding commitment schemes with succinct interaction #OTHEREFR .", "Quantum.", "Quantum computing stands to fundamentally change the field of cryptography." ]

[ "In turn, Grover's algorithm can be used to find ordinary collisions (k = 2) in time O(2 n/3 ), speeding up the classical \"birthday\" attack which requires O(2 n/2 ) time.", "It is also known that, in some sense (discussed below), these speedups are optimal #OTHEREFR .", "These attacks require updated symmetric primitives with longer keys in order to make such attacks intractable." ]

[ "hash functions", "Grover's algorithm" ]

{ "title": "On Finding Quantum Multi-collisions", "abstract": "A k-collision for a compressing hash function H is a set of k distinct inputs that all map to the same output. In this work, we show that for any constant k, Θ N 1 2 (1− 1 2 k −1 ) quantum queries are both necessary and sufficient to achieve a k-collision with constant probability. This improves on both the best prior upper bound (Hosoyamada et al., ASIACRYPT 2017) and provides the first non-trivial lower bound, completely resolving the problem." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Hence, a quantum network, namely a multipartite system, can be represented by a tensor network.", "Indeed, quantum circuits are a special class of tensor networks, where the arrangement of the tensors and their types are restricted #OTHEREFR , #OTHEREFR , #OTHEREFR .", "Quantum computers are devices that perform calculations by utilizing quantum mechanical features including superposition and entanglement.", "Although large-scale quantum computers are not built yet, theoretical research on quantum algorithms has been conducted for several years.", "In 1994, Shor's algorithm #OTHEREFR , is proved to be able to solve integerfactorization problem with polymonial time on a quantum computer, while it is NP in classical computing." ]

[ "In 2009, Harrow, Hassidim and Lloyd put forward a quantum algorithm for solving linear systems of equations, which is famous as the HHL Algorithm #OTHEREFR .", "Base on this algorithm, many quantum version of classical machine learning methods are designed, such as quantum least-squares linear regression #OTHEREFR and support vector machines #OTHEREFR .", "The runtimes of such algorithms are polylogarithmic in the dimensions of the matrix, so that they provide exponential speedups over their classical counterparts.", "There are several types of tensor decompositions, such as canonical polyadic (CP) decomposition #OTHEREFR , #OTHEREFR , tensor-train (TT) decomposition #OTHEREFR , Tucker decompostion #OTHEREFR , and etc. However, currently there are no quantum tensor decomposition algorithms.", "In this paper, we propose a quantum higher order singular value decomposition (Q-HOSVD)." ]

[ "corresponding classical algorithm", "Grover's search algorithm" ]

{ "title": "Quantum Higher Order Singular Value Decomposition", "abstract": "Higher order singular value decomposition (HOSVD) is an important tool for analyzing big data in multilinear algebra and machine learning. In this paper, we present a quantum algorithm for higher order singular value decomposition. Our method allows one to decompose a tensor into a core tensor containing tensor singular values and some unitary matrices by quantum computers. Compared to the classical HOSVD algorithm, our quantum algorithm provides an exponential speedup." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Taking ǫ = 1 2 now leads to the stated result.", "Corollary 1 which we now present shows that, for quantum noise levels as large as p = p(N ) = 1 log(N ) and advice distribution following a power law, the best possible classical algorithm for SEARCH WITH ADVICE in this case has expected query complexity growing no more slowly than a rate of order log(N ) as N → ∞, whereas the query complexity of our geometric quantum search algorithm, Algorithm 2, is in fact bounded by a fixed constant (not depending on N ) for all positive integers N . .", "Suppose that the ambient quantum noise level as in (8) is described by p = p(N ) =", "for constants c 3 , c 4 > 0, where we can take c 3 = In Corollary 2, we extend Corollary 1 by allowing increased levels of quantum noise, at the expense of obtaining less dramatic reductions in query complexity by means of quantum search.", "Corollary 2." ]

[ "Suppose that the ambient quantum noise level as in (8) is described by p = p(N ) = 1 (log(N )) q , for some q, 0 < q ≤ 1.", "Then, D(µ δ ) ≥ Ω(log(N )), but T AQS(p(N )) (µ δ ) ≤ O((log(N )) 1−q ).", "In fact, for all N ≥ 100, D(µ δ ) ≥ c 3 log(N ), and T AQS (p(N )) (µ δ ) ≤ c 4 (log(N )) 1−q ,", "for constants c 3 , c 4 > 0, where we can take c 3 = Proof.", "The proof is the same as that for Corollary 1 except that we now take p = p(N ) = 1 (log(N )) q ." ]

[ "advice distribution", "n ∈" ]

{ "title": "Super-exponential query complexity reduction via noise-resistant quantum search", "abstract": "In the SEARCH WITH ADVICE problem, a single entry of interest within a database of N entries is to be found assuming that an ordering of the entries, from that with the highest probability of being the entry of interest (as determined by a so-called advice distribution) to that with the lowest, is provided. We present a quantum algorithm that, in the presence of significant levels of quantum noise, solves SEARCH WITH ADVICE for a power law advice distribution with average-case query complexity O(1) as N → ∞. Since as we also show the best classical algorithms for this problem exhibit average-case query complexity of order no better than log(N ), our quantum algorithm provides a super-exponential reduction in query complexity." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Here we consider the case where we don't know better than to repeatedly calculate the function without looking at the algorithm, e.g.", "because the function is calculated in a black box subroutine into which we are not allowed to look. In computer science this is called an oracle.", "Here I consider only oracles which give 1 for exactly one input.", "Quantum searching for the case with several inputs which give 1 and even with an unknown number of such inputs is treated in #OTHEREFR .", "Obviously on a classical computer we have to query the oracle O(N ) times to find the answer." ]

[ "This is an interesting improvement even though in computer science terms this is still an exponential (and not polynomial) time algorithm. Bennett et al.", "#OTHEREFR have shown that no quantum algorithm can solve the problem in less than O( √ N ) steps. Boyer et al. #OTHEREFR have improved this result to show that e.g.", "for a 50% success probability no quantum algorithm can do better than only a few percent faster than Grover's algorithm.", "I improve the proof in #OTHEREFR , showing that for any success probability Grover's algorithm is optimal.", "The abovementioned proofs have shown that O( √ N ) steps are necessary for quantum searching." ]

[ "quantum" ]

{ "title": "Grover’s Quantum Searching Algorithm is Optimal” Phys", "abstract": "I improve the tight bound on quantum searching [4] to a matching bound, thus showing that for any probability of success Grover's quantum searching algorithm is optimal. E.g. for near certain success we have to query the oracle π/4 √ N times, where N is the size of the search space. I also show that unfortunately quantum searching cannot be parallelized better than by assigning different parts of the search space to independent quantum computers. Earlier results left open the possibility of a more efficient parallelization." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "One easily shows (using sin x ≤ x for x ≥ 0, which implies that x(1 − sin x) ≥ x(1 − x)) that the maximum value of this product is greater than a quarter:", "D.", "Estimating a completely unknown Hamiltonian acting in a d dimensional space Equation (31) gives a lower bound on the product of the precision with which a completely unkown Hamiltonian is measured and the time taken to estimate it.", "We beleive that this lower bound is not tight and that in general a stronger lower bound should hold.", "We do not know at present what form this stronger lower bound will take, but we beleive that it should depend on the dimensionality of the Hilbert space on which the Hamiltonian acts." ]

[ "In this example one must distinguish between d Hamiltonians of the form H k = E|k k| where the d states |k form an orthonormal basis.", "Farhi and Gutmann show that in order to distinguish these Hamiltonians perfectly, a minimum time of ∆t ≥ cd 1/2 /E is necessary (where c is some positive constant).", "This example shows that there are situations where estimating an unkown Hamiltonian becomes increasingly difficult as the dimension d of the Hilbert space on which it acts increases.", "However in the Fahri-Gutmann example, the unkown Hamiltonian has a very specific form which is known before hand.", "We have obtained preliminary indications that when the Hamiltonian is completely unkown, estimating it should take substantially more time than sugested by the Fahri-Gutmann example." ]

[ "Grover's search algorithm" ]

{ "title": "Measuring energy, estimating Hamiltonians, and the time-energy uncertainty relation", "abstract": "Suppose that the Hamiltonian acting on a quantum system is unknown and one wants to determine what is the Hamiltonian. We show that in general this requires a time ∆t which obeys the uncertainty relation ∆t∆H 1 where ∆H is a measure of how accurately the unknown Hamiltonian must be estimated. We apply this result to the problem of measuring the energy of an unknown quantum state. It has been previously shown that if the Hamiltonian is known, then the energy can in principle be measured with arbitrarily large precision in an arbitrarily short time. On the other hand we show that if the Hamiltonian is not known then an energy measurement necessarily takes a minimum time ∆t which obeys the uncertainty relation ∆t∆E 1 where ∆E is the precision of the energy measurement. Several examples are studied to address the question of whether it is possible to saturate these uncertainty relations. Their interpretation is discussed in detail." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Therefore, by means of a classical computer, the oracle must be queried on average N 2 times ( O (N ) classical steps).", "However, by using the same amount of hardware as in the classical case but by having the input and output in superpositions of states, Grover has developed a quantum mechanical algorithm capable of solving this search problem in about #OTHEREFR .", "Although this is not dramatic as the exponential quantum advantage achieved by Shor's algorithm for factoring, the extremely wide applicability of searching problems makes Grover's algorithm interesting and important.", "In particular, Grover's algorithm gives a quadratic speed-up in the solution of N P -complete problems, which account for many of the important hard problems in computer science.", "Indeed, it was shown in #OTHEREFR that relative to an oracle chosen uniformly at random, with probability 1, the class NP cannot be solved on a quantum Turing machine in time O √ N ." ]

[ "A detailed proof of Grover's statement about the optimality of his quantum searching algorithm appears in #OTHEREFR .", "Finally, in #OTHEREFR , it is shown that for any number of oracle lookups, Grover's algorithm is exactly (and not just asymptotically) optimal.", "What makes a quantum search algorithm more efficient than another? An algorithm is an abstract mathematical concept, whereas it is useful to consider how efficiently we can run an algorithm on a computer.", "Computer scientists associate a cost with each step of the algorithm and with the amount of memory required, embodying the idea that physical computers have a finite size (memory) and work at a finite rate of elementary calculation steps per unit time.", "This gives us a way to determine if one algorithm is intrinsically faster than another." ]

[ "possible quantum algorithms" ]

{ "title": "On Grover's Search Algorithm from a Quantum Information Geometry Viewpoint", "abstract": "We present an information geometric characterization of Grover's quantum search algorithm. First, we quantify the notion of quantum distinguishability between parametric density operators by means of the Wigner-Yanase quantum information metric. We then show that the quantum searching problem can be recast in an information geometric framework where Grover's dynamics is characterized by a geodesic on the manifold of the parametric density operators of pure quantum states constructed from the continuous approximation of the parametric quantum output state in Grover's algorithm. We also discuss possible deviations from Grover's algorithm within this quantum information geometric setting." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "More convincing as an argument against a connection, however, is probably the observation that 3, 4, 20, 21 also appear, say, in the sequence of numbers which appear the same #OTHEREFR when written in base 5 and base 10/2.", "This is easily revealed by using the On-Line Encyclopedia of Integer Sequences of AT&T Research #OTHEREFR .", "It is an interesting and educational pastime to see how essentially every finite sequence of integer numbers that one can possibly come up with appears in, for example, the \"number of isolated-pentagon fullerenes with a certain number of vertices\", or the \"decimal expansion of Buffon's constant\".", "The sequence 2, 4, 6, 9 in this order, to consider a different random example, appears in no fewer than 165 (!) listed integer sequences, each of which is equipped with a different construction or operational meaning.", "The lesson to learn is that one should probably be not too surprised about coincidences of small tuples of integers." ]

[ "It is difficult to conceive how such a hard-wired coherent oracle would be realized at the genome level.", "The optimal improvement in the sampling efficiency, in turn, would be of the order of the square root of N .", "It does seem unlikely that the overhead needed in a reliable quantum computation, possibly even enhanced by error correction requiring again an enormous overhead, would by any figure of merit be more economical than, say, a simple doubling of the waiting time in case of N = 4." ]

[ "quantum mechanics", "Grover's search" ]

{ "title": "Nontrivial quantum effects in biology: A skeptical physicists' view", "abstract": "Invited contribution to\"Quantum Aspects of Life\", D. Abbott Ed. (World Scientific, Singapore, 2007)." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Since each concept in C is defined uniquely by 2 k many (n − k)-bit strings a 0 , . . . , a", "It is easy to see that in fact R(C) = Θ(2 k (n − k)): For each of the 2 k parities which one must learn (corresponding to the 2 k possible prefixes of an input), one can learn the (n − k)-bit parity with n − k classical queries.", "It is also easy to see that by running the Bernstein-Vazirani algorithm 2 k times (once for each different k-bit prefix), a quantum algorithm can learn an unknown concept from C exactly using 2 k queries, and thus", ", and the lemma is proved.", "Based on these observations, we pose the following question:" ]

[]

[ "Grover search" ]

{ "title": "Improved Bounds on Quantum Learning Algorithms", "abstract": "In this article we give several new results on the complexity of algorithms that learn Boolean functions from quantum queries and quantum examples. • Hunziker et al. [17] conjectured that for any class C of Boolean functions, the number of quantum black-box queries which are required to exactly identify an unknown function from C is O( C is a combinatorial parameter of the class C. We essentially resolve this conjecture in the affirmative by giving a quantum algorithm that, for any class C, identifies any unknown function from C using O( log |C| log log |C| √γ C ) quantum black-box queries. • We consider a range of natural problems intermediate between the exact learning problem (in which the learner must obtain all bits of information about the black-box function) and the usual problem of computing a predicate (in which the learner must obtain only one bit of information about the black-box function). We give positive and negative results on when the quantum and classical query complexities of these intermediate problems are polynomially related to each other. • Finally, we improve the known lower bounds on the number of quantum examples (as opposed to quantum black-box queries) required for (ǫ, δ)-PAC learning any concept class of VapnikChervonenkis dimension d over the domain {0, 1} n from Ω( ). This new lower bound comes closer to matching known upper bounds for classical PAC learning." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Of course, quadratically faster spreading on a line is not an algorithm yet, but it is a good start, and #OTHEREFR", "(2003) showed that a quantum walk could search an unsorted database with a quadratic speed up." ]

[ "starting with a phone number and searching a telephone directory to find the corresponding name), potentially has to check all N entries in the database, and on an average has to check at least half.", "A quantum search only needs to make ffiffiffiffi ffi N p queries, though the queries ask for many answers in superposition.", "The quantum walk search algorithm sort of works backwards, starting in a uniform superposition over the whole database, and converging on the answer as the quantum walk proceeds.", "As already noted, quantum walks are reversible; a quantum walk running backwards is also a quantum walk. #OTHEREFR", "(2003) proved that a quantum walk could perform exponentially faster than any classical algorithm when finding a route across a particular sort of network (figure 2)." ]

[ "first quantum algorithm" ]

{ "title": "A random walk approach to quantum algorithms", "abstract": "The development of quantum algorithms based on quantum versions of random walks is placed in the context of the emerging field of quantum computing. Constructing a suitable quantum version of a random walk is not trivial; pure quantum dynamics is deterministic, so randomness only enters during the measurement phase, i.e. when converting the quantum information into classical information. The outcome of a quantum random walk is very different from the corresponding classical random walk owing to the interference between the different possible paths. The upshot is that quantum walkers find themselves further from their starting point than a classical walker on average, and this forms the basis of a quantum speed up, which can be exploited to solve problems faster. Surprisingly, the effect of making the walk slightly less than perfectly quantum can optimize the properties of the quantum walk for algorithmic applications. Looking to the future, even with a small quantum computer available, the development of quantum walk algorithms might proceed more rapidly than it has, especially for solving real problems." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Moreover, A.Berthiaume and G.Brassard in [BB] showed, that Q-M computations even can beat the nondeterministic ones in computations with oracles.", "But as for absolute (without oracles) computations, advantages of Q-M computers over the probabilistic classical machines were not so obvious.", "Situation has been changed in 1994 when P.Shor in his work [Sh] suggested polynomial time quantum algorithms solving problems: of factorization and of finding discrete logarithms.", "For both problems classical probabilistic algorithms with polynomial time complexity are not known.", "In Shor's results advantage was taken of discrete Fourier transformation." ]

[ "He was able to construct the quantum algorithm which for the given function F : {0, 1} n −→ {0, 1} finds such x that F (x) = 1, after O( √ N ) quantum evaluations of F provided such x is unique, in opposite to classical probabilistic computers which require Ω(N ) evaluations in average, N = 2 n . This is known as the problem of searching.", "In Grover's algorithm advantage was taken of the so-called Walsh-Hadamard transformation.", "Later M.Boyer, G.Brassard, P.Hoyer and A.Tapp in [BBHT] extended this method to the case of arbitrary number t of x such that F (x) = 1 and provided tight lower bounds for this algorithm depending on t.", "In particular, they showed, that if t = N/4 then a solution is found with certainty after single iteration of algorithm! Note, that the general number O( √ N ) of evaluations in quantum searching can not be reduced in view of result of C.Bennett, G.Brassard, E.Bernstein and U.Vazirani [BBBV] who proved, that relative to an oracle chosen at random with probability 1 the class NP cannot be solved in time o(2 n/2 ).", "Having Grover's algorithm as a good precedent it is interesting to elusidate is it possible to accelerate sufficiently complicated classical algorithms on quantum computers by some regular way of conversion of a classical program to a quantum one." ]

[ "L.Grover" ]

{ "title": "About the quantum mechanical speeding up of classical algorithms Y.Ozhigov", "abstract": "This work introduces a relative diffusion transformation (RDT) -a simple unitary transformation which acts in a subspace, localized by an oracle. Such a transformation can not be fulfilled on quantum Turing machines with this oracle in polynomial time in general case. It is proved, that every function f : ω n −→ ω n , card(ω) = 4, computable in time T (n) and space S(n) on classical 1-dimensional cellular automaton, can be computed with certainty in time O(T 1/2 S) on quantum computer with RDTs over the parts of intermediate products of classical computation. This requires multiprocessor, which consists of Quantum mechanical computations (QC) are distinct in nature from the classical ones. The point is that a quantum system can be in different classical states simultaneously with the corresponding amplitudes. The vector, composed of these amplitudes completely determines the quantum state of system. The module squared of every amplitude is the probability of detecting the system in the corresponding classical state after observation. Such an observation is the only way to obtain a result of QC. An evolution of such a system is represented by the application of unitary transformation to its vector of amplitudes. Quantum computers became one of the most popular areas of investigations in theoretical computer science as well as in quantum physics because of that in the past 3-4 years considerable progress has been made in the theory of QC. Since that time when R.Feunmann in the work [Fe] proposed quantum mechanical (Q-M) computer, D.Deutsch in the work [De] gave the first formal model of computations on quantum Turing machine (QTM), and S.Lloyd in the work [Ll] presented the physical scheme of Q-M computational device, the advantages of quantum computations over the classical ones in a variety of particular problems became apparent from the sequence of results (look, for example, at [Be], [Sh], [BE] ). Moreover, A.Berthiaume and G.Brassard in [BB] showed, that Q-M computations even can beat the nondeterministic ones in computations with oracles. But as for absolute (without oracles) computations, advantages of Q-M computers over the probabilistic classical machines were not so obvious. Situation has been changed in 1994 when P.Shor in his work [Sh] suggested polynomial time quantum algorithms solving problems: of factorization and of finding discrete logarithms. For both problems classical probabilistic algorithms with polynomial time complexity are not known. In Shor's results advantage was taken of discrete Fourier transformation. The following bright result which is closely connected to the present work was obtained by L.Grover in [Gr1] . He was able to construct the quantum algorithm which for the 1" }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Generally speaking, an idea of efficient quantum algorithms is very simple. Given the input", "µ p e p , we already have the result e s = T (A) in this linear combination, but it can be observed only with probability |µ s | 2 which may be even equal to zero.", "One multiplication on appropriate unitary matrix U should raise absolute value of amplitude of correct answer e s on some fairly big constant.", "Thus after sufficient number of iterations T (n) the required state e s will have amplitude large enough for observation with probability at least 2/3.", "The exact value of T (n) may be of importance, for example in quantum searching the required amplitude grows during √ N transformations and after this instant fells down to zero (look at [BBHT] )." ]

[ "This transform is remarkable for the following reason: been applied to the state x = p µ p e p , µ p ∈ R, it raises by some constant an absolute value of amplitude µ s , opposite to average amplitude.", "Diffusion transform D is defined by it's matrix D:", "Unitarity of D can be easily verified.", "Note that D = W RW , where R is the rotation matrix, defined by Proposition 1 (Grover , #OTHEREFR ).", "For every state x e p |x − x av = x av − e p |D(x) ." ]

[ "significant diffusion transform", "L.Grover" ]

{ "title": "About the quantum mechanical speeding up of classical algorithms Y.Ozhigov", "abstract": "This work introduces a relative diffusion transformation (RDT) -a simple unitary transformation which acts in a subspace, localized by an oracle. Such a transformation can not be fulfilled on quantum Turing machines with this oracle in polynomial time in general case. It is proved, that every function f : ω n −→ ω n , card(ω) = 4, computable in time T (n) and space S(n) on classical 1-dimensional cellular automaton, can be computed with certainty in time O(T 1/2 S) on quantum computer with RDTs over the parts of intermediate products of classical computation. This requires multiprocessor, which consists of Quantum mechanical computations (QC) are distinct in nature from the classical ones. The point is that a quantum system can be in different classical states simultaneously with the corresponding amplitudes. The vector, composed of these amplitudes completely determines the quantum state of system. The module squared of every amplitude is the probability of detecting the system in the corresponding classical state after observation. Such an observation is the only way to obtain a result of QC. An evolution of such a system is represented by the application of unitary transformation to its vector of amplitudes. Quantum computers became one of the most popular areas of investigations in theoretical computer science as well as in quantum physics because of that in the past 3-4 years considerable progress has been made in the theory of QC. Since that time when R.Feunmann in the work [Fe] proposed quantum mechanical (Q-M) computer, D.Deutsch in the work [De] gave the first formal model of computations on quantum Turing machine (QTM), and S.Lloyd in the work [Ll] presented the physical scheme of Q-M computational device, the advantages of quantum computations over the classical ones in a variety of particular problems became apparent from the sequence of results (look, for example, at [Be], [Sh], [BE] ). Moreover, A.Berthiaume and G.Brassard in [BB] showed, that Q-M computations even can beat the nondeterministic ones in computations with oracles. But as for absolute (without oracles) computations, advantages of Q-M computers over the probabilistic classical machines were not so obvious. Situation has been changed in 1994 when P.Shor in his work [Sh] suggested polynomial time quantum algorithms solving problems: of factorization and of finding discrete logarithms. For both problems classical probabilistic algorithms with polynomial time complexity are not known. In Shor's results advantage was taken of discrete Fourier transformation. The following bright result which is closely connected to the present work was obtained by L.Grover in [Gr1] . He was able to construct the quantum algorithm which for the 1" }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "A quantum computer program, or algorithm, is a definite sequence of such unitary transformations.", "For a given initial spin state, the output of the program is the spin state after the sequence of transformations has acted.", "The length of the algorithm is equal to the number of elementary unitary transformations which make up the algorithm.", "This framework for quantum computation is general enough that any ordinary digital computer program can be turned into a quantum computer algorithm.", "(It is required that the ordinary program be reversible; however any ordinary computer program can be written in reversible code.) Quantum computers can go beyond ordinary computers when they act on superpositions of states and take advantage of interference effects." ]

[ "Here we are given a function f (a) defined on the integers a from 1 to N.", "The function has the property that it takes the value 1 on just a single element of its domain, w, and it has the value 0 for all a = w.", "With only the ability to call the function f , the task is to find w.", "On a classical computer this requires, on average, N/2 calls of the function f .", "However Grover showed that with a quantum computer w can be found with of order N 1/2 function calls. This remarkable speed-up illustrates the power of quantum computation." ]

[ "Grover algorithm", "quantum" ]

{ "title": "An analog analogue of a digital quantum computation", "abstract": "We solve a problem, which while not fitting into the usual paradigm, can be viewed as a quantum computation. Suppose we are given a quantum system described by an N dimensional Hilbert space with a Hamiltonian of the form E|w w| where |w is an unknown (normalized) state. We show how to discover |w by adding a Hamiltonian (independent of |w ) and evolving for a time proportional to N 1/2 /E. We show that this time is optimally short. This process is an analog analogue to Grover's algorithm, a computation on a conventional (!) quantum computer which locates a marked item from an unsorted list of N items in a number of steps proportional to N 1/2 ." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Diffie and Hellman 1 lay the foundation of modern cryptography by presenting the concept of public-key encryption.", "Contrary to symmetric-key encryption which uses the same key to encrypt and to decrypt, publickey encryption uses a pair of different keys, public-key and private-key, to encrypt and to decrypt.", "Classical public-key encryption has been widely used in many fields." ]

[ "The research on quantum public-key encryption can pre-empt such potential threats.", "Analogous to the analysis of quantum symmetric-encryption protocols, 4 a quantum public-key encryption consists of six elements: plaintext, ciphertext, public-key, private-key, encryption algorithm and decryption algorithm.", "If an encryption protocol involved quantum part, it can be taken as the counterpart of classical publickey encryption.", "With the advent of quantum key distribution protocol, BB84 5 , quantum public-key encryption has attained great development.", "Okamoto's protocol 6 is considered as the first public-key encryption protocol which is based on the Knapsack-set problem." ]

[ "classical publickey encryption", "Shor's algorithm" ]

{ "title": "A complete classification of quantum public-key encryption protocols", "abstract": "We present a classification of quantum public-key encryption protocols. There are six elements in quantum public-key encryption: plaintext, ciphertext, public-key, private-key, encryption algorithm and decryption algorithm. According to the property of each element which is either quantum or classical, the quantum public-key encryption protocols can be divided into 64 kinds. Among 64 kinds of protocols, 8 kinds have already been constructed, 52 kinds can be proved to be impossible to construct and the remaining 4 kinds have not been presented effectively yet. This indicates that the research on quantum public-key encryption protocol should be focus on the existed kinds and the unproposed kinds." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[]

[ "(i) Initializing a set of qubits | S〉 , which represent the solutions and an output qubit.", "(ii) Apply quantum rotation gate, G, 'r' times on the qubits to amplify the probability of finding target element.", "(iii) Measure solution vector | S〉 .", "(iv) If | S〉 contain the correct information, then stop.", "(v) Else, go to step (i)." ]

[ "canonical quantum search" ]

{ "title": "A Fast fixed-point Quantum Search Algorithm by using Disentanglement and Measurement", "abstract": "Abstract-Generic quantum search algorithm searches for target entity in an unsorted database by repeatedly applying canonical Grover's quantum rotation transform to reach near the vicinity of the target entity. Thus, upon measurement, there is a high probability of finding the target entity. However, the number of times quantum rotation transform is to be applied for reaching near the vicinity of the target is a function of the number of target entities present in an unsorted database, which is generally unknown. A wrong estimate of the number of target entities can lead to overshooting or undershooting the targets, thus reducing the success probability. Some proposals have been made to overcome this limitation. These proposals either employ quantum counting to estimate the number of solutions or fixed-point schemes. This paper proposes a new scheme for stopping the application of quantum rotation transformation on reaching near the targets by disentanglement, measurement and subsequent processing to estimate the distance of the state vector from the target states. It ensures a success probability, which is greater than half for all practically significant ratios of the number of target entities to the total number of entities in a database. The search problem is trivial for remaining possible ratios. The proposed scheme is simpler than quantum counting and more efficient than the known fixed-point schemes. It has same order of computational complexity as canonical Grover`s search algorithm but is slow by a factor of two and requires two additional ancilla qubits." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "(1) W is a unitary transformation over a n-dimensional Hilbert space with basis vectors {|1 , . . . , |n }.", "It is a unitary transformation described as follows:", "(2) U denotes a unitary transformation over a n + n 2 -dimensional Hilbert space with basis vectors {|k , |i, j | i, j, k ∈ [n], i < j}. It is a unitary completion of the following transformation:", "It is easy to verify that both W and U are unitary transformations.", "For convenience, we further define two unitary transformations G and R such that G := W O x and R := U O x ." ]

[ "After applying the operators G and R respectively, the initial state |Ψ 0 becomes G|Ψ 0 = cos(3θ)|α ⊥ + sin(3θ)|α , R|Ψ 0 = cos(2θ)|β ⊥ + sin(2θ)|β ,", "|i and |β := 1 √ (n−|x|)|x| i,j:x i =0,x j =1 |i, j .", "Next, we investigate the special cases of k = κn, l = λn when 1−2κ and 1−2λ are two consecutive extrema of a Chebyshev polynomial (i.e. γ − χ = 1).", "After applying Grover operators G for d − 1 times on |Ψ 0 , the initial state becomes", "Without loss of generality, we may assume γ is odd and continue the discussion for δ = 0 and δ = 1." ]

[ "Grover operator" ]

{ "title": "Exact quantum query complexity of weight decision problems via Chebyshev polynomials", "abstract": "Abstract. The weight decision problem, which requires to determine the Hamming weight of a given binary string, is a natural and important problem with lots of applications in cryptanalysis, coding theory, fault-tolerant circuit design and so on. In this work, we investigate the exact quantum query complexity of weight decision problems, where the exact quantum algorithm must always output the correct answer within finite steps. More specifically we consider a partial Boolean function which distinguishes the Hamming weight of the length-n input between k and l. In particular, both Deutsch-Jozsa problem and Grover search problem can be interpreted as special cases of this problem. Our contributions include both upper bounds and lower bounds for the precise number of queries. For most choices of (k, l)(or ( k n , l n )) and sufficiently large n, the gap between our upper and lower bounds is no more than one query. To get the results, we first build the connection between Chebyshev polynomials and our problem, then determine all the boundary cases of ( k n , l n ) with matching upper and lower bounds, and finally we generalize to other cases via a new quantum padding technique. This quantum padding technique can be of independent interest in designing other quantum algorithms." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "A classical description of P 2 n can be given by describing its effect on a given vector.", "If Z is a 2 n -dimensional vector and Y = P 2 n Z, then Y i = Z j , for i = 0, 1, · · · , 2 n − 1, wherein j is obtained by reversing the bits in the binary representation of index i.", "Therefore, a description of the matrix P 2 n , in terms of its elements P ij , for i and j = 0, 1, · · · , 2 n − 1, is given as", "A factorization of P 2 n in terms of Π 2 i is given as #OTHEREFR" ]

[ "This quantum description can be seen from the factorization of P 2 n , given by (5), and quantum description of permutation matrices Π 2 i .", "It is interesting to note that for classical computation the term \"bit-reversal\" refers to reversing the bits in the binary representation of index of the elements of a vector while, for quantum computation, the matrix P 2 n literally performs a reversal of the order of qubits.", "Note that, P 2 n is symmetric, i.e., P 2 n = P t 2 n #OTHEREFR .", "This can be also easily proved based on the quantum description of P 2 n since if the qubits are reversed twice then the original ordering of the qubits is restored.", "This implies that, P 2 n P 2 n = I 2 n and since P 2 n is orthogonal, i.e., P 2 n P t 2 n = I 2 n , it then follows that P 2 n = P t 2 n ." ]

[ "n qubits", "quantum description" ]

{ "title": "Quantum wavelet transforms: fast algorithms and complete circuits", "abstract": "The quantum Fourier transform (QFT), a quantum analog of the classical Fourier transform, has been shown to be a powerful tool in developing quantum algorithms. However, in classical computing there is another class of unitary transforms, the wavelet transforms, which are every bit as useful as the Fourier transform. Wavelet transforms are used to expose the multi-scale structure of a signal and are likely to be useful for quantum image processing and quantum data compression. In this paper, we derive efficient, complete, quantum circuits for two representative quantum wavelet transforms, the quantum Haar and quantum Daubechies D (4) transforms. Our approach is to factor the classical operators for these transforms into direct sums, direct products and dot products of unitary matrices. In so doing, we find that permutation matrices, a particular class of unitary matrices, play a pivotal role. Surprisingly, we find that operations that are easy and inexpensive to implement classically are not always easy and inexpensive to implement quantum mechanically, and vice versa. In particular, the computational cost of performing certain permutation matrices is ignored classically because they can be avoided explicitly. However, quantum mechanically, these permutation operations must be performed explicitly and hence their cost enters into the full complexity measure of the quantum transform. We consider the particular set of permutation matrices arising in quantum wavelet transforms and develop efficient quantum circuits that implement them. This allows us to design efficient, complete quantum circuits for the quantum wavelet transform." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "The observation that arbitrary computations can be carried out by a computational device in such a way that in principle each time-step can be reversed-allowing to recover the input from the output of a computation that has been orchestrated in such a fashion-goes back to Bennett's landmark paper #OTHEREFR .", "While the original motivation for reversible computing was to demonstrate that the amount of heat generated by irreversible gates-as implied by Landauer's principle-can in principle be avoided by making computations that never erase any information, it transpires that compared to the actual energy dissipation of modern integrated chips, this saving in energy is quite small, see, e.g., the survey #OTHEREFR .", "For modern chips, the amount of energy savings due to avoiding erasure of information would be more than 10 orders of magnitude smaller than the amount of energy savings that arise from other dissipative processes that heat up the chip.", "Aside from an adiabatic regime where chips would operate at ultra-low power, yet ultra-slow, therefore arguably the main application of reversible computing is therefore in quantum computing, namely as a vehicle that allows a quantum computer to carry out any function that a classical computer might carry out.", "It should be noted that the ability to compute classical functions is at the core of many interesting quantum algorithms, including Shor's algorithm for discrete log and factoring #OTHEREFR where the reversible computations are arithmetic operations in suitable algebraic data structures such as rings and fields." ]

[ "Many variations of this general theme exist including quantum walk algorithms that allow to traverse graphs faster than classical algorithms can, in some cases even exponentially faster, as well as some algorithms for simulation of Hamiltonians, where reversible computations may be needed for the efficient accessing of the matrix elements of the underlying Hamiltonian #OTHEREFR .", "While this may illustrate the need for techniques to turn classical computations into quantum circuits, it also may serve as an illustration of the difficulties that such a translation will present to an compiler system that aims at supporting this translation from a classical computation.", "Such a classical description could be, say, a program expressed in a higher-level programming language such as C or Haskell.", "Among the difficulties are the following issues: (i) qubits that are used as intermediate scratch space during the computation have to be cleaned up at the end of the computation or else the interference effects, on which quantum computations typically rely heavily, would disappear which would render the computation useless, (ii) the number of qubits that are needed for scratch space grows linearly with the number of classical instructions if a simple method for turning the irreversible computation into a reversible one is used such as the original Bennett method #OTHEREFR .", "What is more, the simple methods for making circuits reversible are extremely inefficient regarding the load-factor of the computation, namely they lead to circuit that only manipulate a tiny subset of the qubits at a given time and leave the bulk of the qubits idle." ]

[ "reversible computations", "Grover's algorithm" ]

{ "title": "Reversible circuit compilation with space constraints", "abstract": "We develop a framework for resource efficient compilation of higher-level programs into lower-level reversible circuits. Our main focus is on optimizing the memory footprint of the resulting reversible networks. This is motivated by the limited availability of qubits for the foreseeable future. We apply three main techniques to keep the number of required qubits small when computing classical, irreversible computations by means of reversible networks: first, wherever possible we allow the compiler to make use of in-place functions to modify some of the variables. Second, an intermediate representation is introduced that allows to trace data dependencies within the program, allowing to clean up qubits early. This realizes an analog to \"garbage collection\" for reversible circuits. Third, we use the concept of so-called pebble games to transform irreversible programs into reversible programs under space constraints, allowing for data to be erased and recomputed if needed. We introduce REVS, a compiler for reversible circuits that can translate a subset of the functional programming language F# into Toffoli networks which can then be further interpreted for instance in LIQui| , a domain-specific language for quantum computing and which is also embedded into F#. We discuss a number of test cases that illustrate the advantages of our approach including reversible implementations of SHA-2 and other cryptographic hash-functions, reversible integer arithmetic, as well as a test-bench of combinational circuits used in classical circuit synthesis. Compared to Bennett's method, REVS can reduce space complexity by a factor of 4 or more, while having an only moderate increase in circuit size as well as in the time it takes to compile the reversible networks." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Simon's algorithm gives an exponential speedup with respect to a classical algorithm, but it solves a very narrow problem that does not have practical applications.", "We now describe an algorithm that gives only a polynomial -more specifically, quadratic -speedup with respect to classical, but it applies to a very large class of problems.", "The algorithm is known as Grover's search #OTHEREFR .", "The problem solved by the algorithm can be described as black-box search: we are given a circuit that computes an unknown function on binary variables, and we want to determine for which value(s) of the input the function gives output 1.", "In other words, we are trying to determine the unique binary string that satisfies a property encoded by a circuit." ]

[ "Such an algorithm can be applied whenever we are searching for a specific element in a set, and we do not have enough information to do anything smarter than a brute force search, i.e., testing all elements in the set.", "As mentioned earlier, the basic idea of the algorithm is to start with the uniform superposition of all basis states, and iteratively increase the coefficients of basis states that correspond to binary strings for which the unknown function gives output 1.", "We need some definitions. Let f : {0, . . .", ", 2 n − 1} → {0, 1}, and assume that there exists a unique ∈ {0, . . .", ", 2 n − 1} : f ( ) = 1, i.e., there is a unique element in the domain of the function that yields output 1. We want to determine ." ]

[ "original paper", "database" ]

{ "title": "An Introduction to Quantum Computing, Without the Physics", "abstract": "This paper is a gentle but rigorous introduction to quantum computing intended for discrete mathematicians. Starting from a small set of assumptions on the behavior of quantum computing devices, we analyze their main characteristics, stressing the differences with classical computers, and finally describe two well-known algorithms (Simon's algorithm and Grover's algorithm) using the formalism developed in previous sections. This paper does not touch on the physics of the devices, and therefore does not require any notion of quantum mechanics. The quantum computing device is, in abstract terms, similar to a classical computing device: it has a state, and the state of the device evolves according to certain operations. More specifically, the model of computation that we consider is inspired by the current generation of quantum computing hardware, and it works as follows: 1. The state of the quantum computer is contained in a quantum register, which is initialized in a predefined way. arXiv:1708.03684v2 [cs.DM] 1 Oct 2017 2. The state evolves according to operations specified in advance according to an algorithm. 3. At the end of the computation, some information on the state of the quantum register is obtained by means of a special operation, called a measurement. All terms in italics will be the subject of the assumptions mentioned earlier, upon which our exposition will build. Note that this type of computing device is similar to a Turing machine, except for the presence of a tape. It is possible to assume the presence of a tape and be more formal in defining a device that is the quantum equivalent of a Turing machine, see [Deutsch, 1985] , but there is no need to do so for the purposes of this work. A discussion on quantum computers requires notational devices to switch back and forth from the decimal and the binary representation of integers, some bit operations, and familiarity with the properties of the tensor product. We describe here the necessary concepts and the notation, so that the reader can come back to this section at any time to clarify symbols. Definition 1. Given two vector spaces V and W over a field K with bases e 1 , . . . , e m and f 1 , . . . , f n respectively, the tensor product V ⊗ W is another vector space over K of dimension mn. The tensor product space is equipped with a bilinear operation ⊗ : V × W → V ⊗ W . The vector space V ⊗ W has basis e i ⊗ f j ∀i = 1, . . . , m, j = 1, . . . , n." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[]

[ "To be able to define the target problem of this paper, we have to organize the efforts done by others in that field.", "The unstructured search problem targeted by Grover's original algorithm is deviated in the literature to the following four major problems:", "• Unstructured list with a unique match.", "• Unstructured list with one or more matches, where the number of matches is known", "• Unstructured list with one or more matches, where the number of matches is unknown." ]

[ "quantum" ]

{ "title": "Fixed Phase Quantum Search Algorithm", "abstract": "Building quantum devices using fixed operators is a must to simplify the hardware construction. Quantum search engine is not an exception. In this paper, a fixed phase quantum search algorithm that searches for M matches in an unstructured search space of size N will be presented. Selecting phase shifts of 1.91684π in the standard amplitude amplification will make the technique perform better so as to get probability of success at least 99.58% in O N /M better than any know fixed operator quantum search algorithms. The algorithm will be able to handle either a single match or multiple matches in the search space. The algorithm will find a match in O N /M whether the number of matches is known or not in advance. In 1996, Lov Grover [10] presented an algorithm that quantum mechanically searches an unstructured list assuming that a unique match exists in the list with quadratic speed-up over classical algorithms. To be able to define the target problem of this paper, we have to organize the efforts done by others in that field. The unstructured search problem targeted by Grover's original algorithm is deviated in the literature to the following four major problems: • Unstructured list with a unique match. • Unstructured list with one or more matches, where the number of matches is known • Unstructured list with one or more matches, where the number of matches is unknown. • Unstructured list with strictly multiple matches. The efforts done in all the above cases, similar to Grover's original work, used quantum parallelism by preparing superposition that represents all the items in the list. The superposition could be uniform or arbitrary. The techniques used in most of the cases to amplify the amplitude(s) of the required state(s) have been generalized to an amplitude amplification technique that iterates * ayounes2@yahoo.com" }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ ".", "With this choice, we guarantee that there is no algorithm that can solve the SDP more efficiently than ISD, thus we consider the ISD work factor (WF) to compute the security level of LEDAsig against decoding attacks.", "The ISD approach, which was pioneered by Prange in #OTHEREFR , attempts at performing the decoding of a general linear code more efficiently than an exhaustive search approach.", "Subsequent improvements of Prange's algorithm were presented by Lee and Brickell #OTHEREFR , Leon #OTHEREFR and Stern #OTHEREFR .", "Among these variants, Stern's algorithm #OTHEREFR is currently the one best exploiting the speedups provided by quantum computers, as shown in #OTHEREFR ." ]

[ "By contrast, when execution on classical computers is considered, the most efficient ISD turns out to be the Becker-Joux-May-Meurer (BJMM) algorithm proposed in #OTHEREFR , which is part of a family of results on the subject #OTHEREFR .", "All the aforementioned approaches have a running time growing exponentially in the effective key size of the scheme (a function of the number of errors, code size and rate), regardless of the availability of a quantum computer.", "As a consequence, the security levels against attackers performing a decoding attack (DA) with classical computers have been estimated by considering the WF of the BJMM algorithm, while the security levels against quantum computer-equipped attackers were computed taking into account Stern's algorithm.", "We defend LEDAsig from DAs employing parameters which prevent the syndrome decoding from succeeding given a computational power bounded by the desired security level.", "To this end, we take into account the fact that the nature of the QC codes employed in LEDAsig provides a speedup by a factor √ p with respect to the running time of the ISD algorithm employed to perform decoding of a general linear code #OTHEREFR ." ]

[ "Grover's algorithm" ]

{ "title": "Design and Implementation of a Digital Signature Scheme Based on Low-density Generator Matrix Codes", "abstract": "In this paper we consider a post-quantum digital signature scheme based on low-density generator matrix codes and propose efficient algorithmic solutions for its implementation. We also review all known attacks against this scheme and derive closed-form estimates of their complexity when running over both classical and quantum computers. Based on these estimates, we propose new parametrization for the considered system to achieve given pre-quantum and post-quantum security levels. Finally, we provide and discuss performance benchmarks obtained through a suitably developed and publicly available reference implementation of the considered system." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Shor' algorithm #OTHEREFR indicates that once scalable quantum computers are available, many widely used asymmetric cryptosystems, such as RSA, will be broken.", "This has sparked a upsurge of research on post-quantum cryptography, which studies classical systems that are secure against quantum adversaries.", "In response to the threat of quantum computing, NIST has initiated the process of standardizing post-quantum public-key algorithms #OTHEREFR .", "On the other hand, although less attention is paid than public-key cryptography, symmetric cryptosystems are also suffering the threat from quantum attacks." ]

[ "More strikingly, some symmetric systems that have been proved to be secure against classical adversaries have been broken by polynomial-time quantum algorithms.", "Kuwakado and Morii made use of Simon's algorithm #OTHEREFR to distinguish the three-round Feistel construction #OTHEREFR and recover the key in Even-Mansour cipher #OTHEREFR . Santoli et al. #OTHEREFR and Kaplan et al.", "#OTHEREFR subsequently extended their results independently and applied Simon's algorithm to other symmetric primitives.", "All these attacks are executed in the model of quantum chosen-plaintext attack #OTHEREFR , where the attacker can query the encryption oracle with superpositions.", "When quantum chosen-plaintext attack has been widely studied, quantum related-key attack has also started to draw attention." ]

[ "Grover's algorithm general" ]

{ "title": "A quantum related-key attack based on Bernstein-Vazirani algorithm", "abstract": "Due to the powerful computing capability of quantum computers, cryptographic researchers have applied quantum algorithms to cryptanalysis and obtained many interesting results in recent years. In this paper, we study related-key attack in the quantum setting, and proposed a specific relatedkey attack which can recover the key of block ciphers efficiently, as long as the attacked block ciphers satisfy certain conditions. The attack algorithm employs Bernstein-Vazirani algorithm as a subroutine and requires the attacker to query the encryption oracle with quantum superpositions. Afterwards, we rigorously demonstrate the validity of the attack and analyze its complexity. Our work shows that related-key attack is quite powerful when combined with quantum algorithms, and provides some guidance for the design of block ciphers that are secure against quantum adversaries." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "The complexity measurement is the number of queries made in order to compute the desired task.", "For sorting by comparisons, the oracle is just the comparison matrix of the numbers being sorted.", "Decision tree model has been studied intensively for classical deterministic and probabilistic algorithms, and lately, for quantum decision tree algorithms.", "Several problems have been identified such that quantum computers can make asymptotically fewer queries than any classical one.", "One of the most well known problems is finding the location of the only 1 in a n bit binary string." ]

[ "His algorithm has been adapted for other tasks that admit quantum speeding-up: for example, finding the minimum in O( √ n) #OTHEREFR , counting the number of 1's #OTHEREFR , and, Element Distinctness Problem in O(n 3/4 log n) #OTHEREFR .", "Advancing shoulder to shoulder with fast quantum algorithms are works on lower bounds that unveil the limit of quantum computational power.", "For example, Grover's algorithm was soon shown to be optimal by several authors #OTHEREFR , tight bounds for all symmetric Boolean functions shown by #OTHEREFR , Ω(log n) for ordered searching by #OTHEREFR , Ω( √ n) for AND OF OR by #OTHEREFR .", "Sorting and the problem of searching in a sorted list are closely related.", "[10] finds a 0.526 log 2 n algorithm and #OTHEREFR a 0.631 log 2 n algorithm for the latter problem." ]

[ "striking quantum" ]

{ "title": "Quantum lower bound for sorting", "abstract": "We prove that Ω(n log n) comparisons are necessary for any quantum algorithm that sorts n numbers with high success probability and uses only comparisons. If no error is allowed, at least 0.110n log 2 n − 0.067n + O(1) comparisons must be made. Previous known lower bound is Ω(n)." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[]

[ "His search algorithm can be used to compute the logical OR of N bits in the same time; simply search for a one in the input string.", "Since Grover search can be used as a subroutine, even within another instance of Grover search, one can speed up the computation of more general logical formulas.", "For example, a two-level AND-OR tree (with one AND gate of fan-in √ N and √ N OR gates of the same fan-in as its children) can be computed in time O( √ N log N ).", "Here the log factor comes from amplifying the success probability of the inner quantum search to be polynomially close to one, so that the total error is at most constant.", "By iterating the same argument, regular AND-OR trees of depth d can be evaluated with constant error in time" ]

[ "quantum computer" ]

{ "title": "Every NAND formula of size N can be evaluated in time O ( N 1 2 + ε ) on a quantum computer", "abstract": "For every NAND formula of size N , there is a bounded-error O(N 1 2 +ε )-time quantum algorithm that evaluates this formula on a black-box input, for ε > 0 an arbitrarily small constant. It follows that the (2 − ε)-th power of the quantum query complexity is a lower bound on the formula size, almost solving in the positive an open problem posed" }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "In contrast, we keep the protocols the same, with only classical communication and only change adversary's internal power (by allowing it to be a polynomial-time quantum computer which may access quantum oracles). We believe that this is the first such separation.", "Boneh, Dagdelen, Fischlin, Lehmann, Schaffner, and Zhandry #OTHEREFR first showed how to correctly define the random oracle in the quantum setting (namely, the adversary has to have superposition access to it).", "For the Fiat-Shamir construction (using random oracles as modeled by #OTHEREFR ), an impossibility result was given by Dagdelen, Fischlin, and Gagliardoni #OTHEREFR .", "However, their impossibility only shows that security of Fiat-Shamir cannot be shown using extractors that do not perform quantum rewinding; 1 but such quantum rewinding is possible and used in the existing positive results from #OTHEREFR , #OTHEREFR which would also not work in a model without quantum rewinding.", "A variant of Fiat-Shamir has been shown to be a quantum secure signature scheme #OTHEREFR ." ]

[ "These are a much stronger assumption that usual sigma-protocols: as shown in [21, Appendix A], sigma-protocols with oblivious commitments are by themselves already non-interactive zeroknowledge proofs in the CRS model (albeit single-theorem, non-adaptive ones).", "#OTHEREFR presents a non-interactive quantum zero-knowledge proof of knowledge in the random oracle model, based on arbitrary sigma-protocols (it does not even need strict soundness).", "That protocol uses ideas different from both Fiat-Shamir and Fischlin's scheme to avoid rewinding.", "It was known for a long time that it is difficult to use classical definitions for computational binding in the quantum setting ([22] is the first reference we are aware of), but none showed so far that the computational definition was truly insufficient. Our technique.", "The schemes we analyze are all based on sigma-protocols which have the special soundness property: In a proof of a statement s, given two accepting conversations (com, ch, resp) and (com, ch , resp ), one can efficiently ex- #OTHEREFR They do allow extractors that restart the adversary with the same classical randomness from the very beginning." ]

[ "quantum zero-knowledge proof" ]

{ "title": "Quantum Attacks on Classical Proof Systems: The Hardness of Quantum Rewinding", "abstract": "Abstract-Quantum zero-knowledge proofs and quantum proofs of knowledge are inherently difficult to analyze because their security analysis uses rewinding. Certain cases of quantum rewinding are handled by the results by Watrous (SIAM J Comput, 2009) and Unruh (Eurocrypt 2012), yet in general the problem remains elusive. We show that this is not only due to a lack of proof techniques: relative to an oracle, we show that classically secure proofs and proofs of knowledge are insecure in the quantum setting. More specifically, sigma-protocols, the Fiat-Shamir construction, and Fischlin's proof system are quantum insecure under assumptions that are sufficient for classical security. Additionally, we show that for similar reasons, computationally binding commitments provide almost no security guarantees in a quantum setting. To show these results, we develop the \"pick-one trick\", a general technique that allows an adversary to find one value satisfying a given predicate, but not two." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "NP-hard and NP-complete optimization problems can be mapped on Ising spin Hamiltonians, whose ground states encode the solution of the given problem #OTHEREFR .", "In adiabatic quantum computers #OTHEREFR , the target ground state is reached via a so-called quantum annealing procedure #OTHEREFR .", "This technique exploits quantum tunneling as an attempt to explore the system phase space more efficiently than classical analogues, such as simulated thermal annealing.", "In the last decade, many efforts have focused on studying quantitatively how the efficiency of quantum annealing is affected by dissipation, as technological advancements in quantum computation have to take into account the unavoidable interaction of quantum processors with their surroundings." ]

[ "The environment surrounding the qubit system is modeled as a collection of harmonic oscillators.", "The approximate dynamics of the spin system is obtained by tracing away the degrees of freedom of the reservoir and using a Markovian quantum master equation in Lindblad form, preserving the complete positivity of the reduced density matrix.", "Details on the model Hamiltonian, annealing procedure and Lindblad equation are provided in the next section." ]

[ "dissipative quantum annealing", "Grover's search task" ]

{ "title": "May a dissipative environment be beneficial for quantum annealing?", "abstract": "We discuss the quantum annealing of the fully-connected ferromagnetic p-spin model in a dissipative environment at low temperature. This model, in the large p limit, encodes in its ground state the solution to the Grover's problem of searching in unsorted databases. In the framework of the quantum circuit model, a quantum algorithm is known for this task, providing a quadratic speed-up with respect to its best classical counterpart. This improvement is not recovered in adiabatic quantum computation for an isolated quantum processor. We analyze the same problem in the presence of a low-temperature reservoir, using a Markovian quantum master equation in Lindblad form, and we show that a thermal enhancement is achieved in the presence of a zero temperature environment moderately coupled to the quantum annealer." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Quadratic improvement by Grover's algorithm #OTHEREFR is achieved in several applications.", "We point out here how that can be exploited in our algorithm.", "Although we can construct the target Dicke state by measuring the intermediate quantum state, we may increase the efficiency further by using the amplitude amplification method.", "Based on this, an adiabatic evolution has been used towards amplitude amplification of the desired states in #OTHEREFR , but no complexity analysis was shown." ]

[ "Instead of equal superposition |ψ = H ⊗n |0 ⊗n = 1 2 n 2 x∈{0,1} n |x in Grover algorithm, we will use the symmetric state of the form", "2 n |x exploiting the properly chosen Boolean function f (x), as explained in the previous sections.", "Further, towards inverting the phase, we will use another symmetric Boolean function g(x), different from f (x), where g(x) = 1, when wt(x) = w, and g(x) = 0, otherwise.", "Based on g(x), we implement the inversion operator as O g , that inverts the phase of the states |x where {x ∈ {0, 1} n |wt(x) = w}. Thus, we consider the operator", "Consider the n-qubit state |Ψ = s∈S u s |s + s∈{0,1} n \\S v s |s , where u s , v s are real and s∈S u t operator on |Ψ produces |Ψ t , in which the probability amplitude of |X is sin(2t + 1)θ." ]

[ "conventional Grover algorithm" ]

{ "title": "Efficient quantum algorithm to construct arbitrary Dicke states", "abstract": "In this paper, we study efficient algorithms towards the construction of any arbitrary Dicke state. Our contribution is to use proper symmetric Boolean functions that involve manipulations with Krawtchouk polynomials. Deutsch-Jozsa algorithm, Grover algorithm and the parity measurement technique are stitched together to devise the complete algorithm. Further, motivated by the work of Childs et al (2002), we explore how one can plug the biased Hadamard transformation in our strategy. Our work compares fairly with the results of Childs et al (2002)." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "This interaction destroys the quantum parallelism by turning a quantum calculation into a classical one.", "The other type of error, inaccuracies in the implementation of gate operations, accumulates over time and destroys the results of the calculation.", "In this paper we show results of simulations of a quantum computer which is subject to both decoherence and inaccuracies.", "These simulations assume the trapped ion model of a quantum computer proposed by Cirac and Zoller #OTHEREFR .", "We study Shor's factorization algorithm by simulating circuits which factor the numbers 15, 21, 35, and 57 #OTHEREFR ." ]

[ "The rest of this section gives a brief overview of quantum computers." ]

[ "Grover's database search" ]

{ "title": "Simulating the Effect of Decoherence and Inaccuracies on a Quantum Computer", "abstract": "A Quantum Computer is a new type of computer which can solve problems such as factoring and database search very efficiently. The usefulness of a quantum computer is limited by the effect of two different types of errors, decoherence and inaccuracies. In this paper we show the results of simulations of a quantum computer which consider both decoherence and inaccuracies. We simulate circuits which factor the numbers 15, 21, 35, and 57 as well as circuits which use database search to solve the circuit satisfaction problem. Our simulations show that the error rate per gate is on the order of 10 −6 for a trapped ion quantum computer whose noise is kept below π/4096 per gate and with a decoherence rate of 10 −6 . This is an important bound because previous studies have shown that a quantum computer can factor more efficiently than a classical computer if the error rate is of order 10 −6 ." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Firstly, in the exact model we exhibit a P with Q E P n n,1, so the evasiveness conjecture fails in the case of quantum computers.", "However, we also prove Q E P 2 n 2 for all P, so evasiveness does hold up to a constant factor for exact quantum computers.", "Secondly, we give a nontrivial monotone graph property for which the evasiveness conjecture is violated by a zero-error quantum algorithm: let STAR be the property that the graph has a vertex which is adjacent to all other vertices.", "Any classical (zero-error or bounded-error) algorithm for STAR requires n 2 queries.", "We give a zero-error quantum algorithm that determines STAR with only On 3=2 queries." ]

[]

[ "boundederror quantum algorithms" ]

{ "title": "Bounds for small-error and zero-error quantum algorithms", "abstract": "We present a number of results related to quantum algorithms with small error probability and quantum algorithms that are zero-error. First A general goal in the design of randomized algorithms is to obtain fast algorithms with small error probabilities. Along these lines is also the goal of obtaining fast algorithms that are zero-error (a.k.a. Las Vegas), as opposed to bounded-error (a.k.a. Monte Carlo). We examine these themes in the context of quantum algorithms, and present a number of new upper and lower bounds that contrast with those that arise in the classical case. The error probabilities of many classical probabilistic algorithms can be reduced by techniques that are commonly is known, then an error probability bounded above by an arbitrarily small \" 0 can be obtained by running A independently log1=\" times and taking the majority value of the outcomes. This amplification procedure increases the running time of the algorithm by a multiplicative factor of log1=\" and is optimal (assuming that A is only used as a black-box). We first consider the question of whether or not it is possible to perform amplification more efficiently on a quantum computer. A classical probabilistic algorithm A is said to p; qcompute a function f : f0; 1g ! f 0; 1g if Pr Ax = 1 p if fx = 0 q if fx = 1 . Algorithm A can be regarded as a deterministic algorithm with an auxiliary input r, which is uniformly distributed over some underlying sample space S (usually S is of the form f0; 1g ljxj ). We will focus our attention on the onesided-error case (i.e. when p = 0) and prove bounds on quantum amplification by translating them to bounds on quantum search. In this case, for any x 2 f 0; 1g n , fx = 1 iff 9r 2 SAx; r = 1 . Grover's quantum search algorithm [15] (and some refinements of it [6, 7, 8, 29, 16] ) can be cast as a quantum amplification method that is provably more efficient than any classical method. It amplifies a 0; q -algorithm to a 0; 1 2 -quantum-computer with O1= p q executions of A, whereas classically 1=q executions of A would be required to achieve this. It is natural to consider other amplification problems, such as amplifying 0; q -computers to 0; 1 , \"-quantum-computers (0 q 1 , \" 1). We give a tight analysis of this. Theorem 1 Let A : f0; 1g n S ! f0; 1g be a classical probabilistic algorithm that 0; q -computes some function f, and let N = jSj and \" 2 ,N . Then, given a blackbox for A, the number of calls to A that are necessary and sufficient to 0; 1 , \"-quantum-compute f is p N p log1=\" + qN, p qN :" }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "In this section, we prove Theorem 1, stated in Section 1.", "The search problem is the following: for a given black-box x, find a j such that x j = 1 using as few queries to x as possible." ]

[ "We address the question of how large the number of queries should be in order to be able to achieve a very small error \".", "We will prove that if T N, then T 2 p N log1=\" : This result will actually be a special case of a more general theorem that involves a promise on the number of solutions.", "Suppose we want to search a space of N items with error \", and we are promised that there are at least some number t N solutions.", "The higher t is, the fewer queries we will need.", "In the appendix we give the following lower bound on \" in terms of T, using tools from #OTHEREFR ." ]

[ "quantum computer" ]

{ "title": "Bounds for small-error and zero-error quantum algorithms", "abstract": "We present a number of results related to quantum algorithms with small error probability and quantum algorithms that are zero-error. First A general goal in the design of randomized algorithms is to obtain fast algorithms with small error probabilities. Along these lines is also the goal of obtaining fast algorithms that are zero-error (a.k.a. Las Vegas), as opposed to bounded-error (a.k.a. Monte Carlo). We examine these themes in the context of quantum algorithms, and present a number of new upper and lower bounds that contrast with those that arise in the classical case. The error probabilities of many classical probabilistic algorithms can be reduced by techniques that are commonly is known, then an error probability bounded above by an arbitrarily small \" 0 can be obtained by running A independently log1=\" times and taking the majority value of the outcomes. This amplification procedure increases the running time of the algorithm by a multiplicative factor of log1=\" and is optimal (assuming that A is only used as a black-box). We first consider the question of whether or not it is possible to perform amplification more efficiently on a quantum computer. A classical probabilistic algorithm A is said to p; qcompute a function f : f0; 1g ! f 0; 1g if Pr Ax = 1 p if fx = 0 q if fx = 1 . Algorithm A can be regarded as a deterministic algorithm with an auxiliary input r, which is uniformly distributed over some underlying sample space S (usually S is of the form f0; 1g ljxj ). We will focus our attention on the onesided-error case (i.e. when p = 0) and prove bounds on quantum amplification by translating them to bounds on quantum search. In this case, for any x 2 f 0; 1g n , fx = 1 iff 9r 2 SAx; r = 1 . Grover's quantum search algorithm [15] (and some refinements of it [6, 7, 8, 29, 16] ) can be cast as a quantum amplification method that is provably more efficient than any classical method. It amplifies a 0; q -algorithm to a 0; 1 2 -quantum-computer with O1= p q executions of A, whereas classically 1=q executions of A would be required to achieve this. It is natural to consider other amplification problems, such as amplifying 0; q -computers to 0; 1 , \"-quantum-computers (0 q 1 , \" 1). We give a tight analysis of this. Theorem 1 Let A : f0; 1g n S ! f0; 1g be a classical probabilistic algorithm that 0; q -computes some function f, and let N = jSj and \" 2 ,N . Then, given a blackbox for A, the number of calls to A that are necessary and sufficient to 0; 1 , \"-quantum-compute f is p N p log1=\" + qN, p qN :" }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Google, IBM, and Intel all announced their quantum chips of 72, 50, and 49 superconducting qubits (quantum bits), respectively, in 2018.", "Modern computing and communication rest on the digital abstraction of information, measured in bits. A bit has a state either 0 or 1.", "Quantum mechanics allows a quantum bit (qubit) to be in a superposition of both states 0 and 1.", "In addition, the dimension of the state space grows exponentially with the number of qubits.", "These properties endow a quantum computer the power to achieve tasks that are beyond the capability of classical computers." ]

[ "Even more impressive is Shor's factoring algorithm #OTHEREFR , which provides an exponential speedup over the best known classical factoring approach.", "It is anticipated that \"quantum supremacy\"-the superiority of quantum computing over classical devices for a well-defined computational problem -will be achieved by NISQ (noisy intermediate scale quantum) devices in the near future #OTHEREFR .", "All these phenomena indicate that we are in the transitions from studing quantum theory to engineering quantum information-the second quantum revolution #OTHEREFR .", "An extraordinary quantum effect is entanglement-a strong quantum correlation between qubits that are even far apart.", "With preshared quantum entanglement between different parties, communication of quantum information can be done by so-called quantum teleportation #OTHEREFR . Thus establishing entanglement attracts lots of attention #OTHEREFR ." ]

[ "quantum computer" ]

{ "title": "Protocols for PacketQuantum Network Intercommunication", "abstract": "A quantum network, which involves multiple parties pinging each other with quantum messages, could revolutionize communication, computing and basic sciences. A global system of various packet switching quantum and classical networks is called quantum internet, the internet in the future. To pave the way to the future quantum internet, unified protocols that support the distribution of quantum messages within the quantum internet are necessary. Classical network functionalities, ranging from error-control mechanisms to overhead-control strategies, assume that classical information can be correctly read and copied. However, developing quantum internet protocols is more challenging since some classical techniques are forbidden by quantum effects, such as entanglement, measurement, and no-cloning. In this paper, we investigate and propose protocols for packet quantum network intercommunication: quantum User Datagram Protocol (qUDP) and quantum Transmission Control Protocol (qTCP). To protect the fragile quantum information in the quantum internet, qTCP employs techniques of quantum error-correcting codes as well as classical techniques of stack design. In particular, the creation of the logical process-to-process connections of qTCP is accomplished by a quantum version of the three-way handshake protocol." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "If ε = 0, we says that the quantum algorithm is exact.", "For more details on quantum query complexity, we may refer to #OTHEREFR .", "Quantum query models are one of most important computing model in quantum computing.", "In this complexity models #OTHEREFR , an algorithm is charged for \"queries\" to the input bits, while any intermediate computation is considered as free.", "For many functions one can obtain large quantum speedups in the case algorithms are allowed a constant small probability of error (bounded error)." ]

[ "The model of exact quantum query, where the algorithms must output the correct answer with certainty for every possible input, seems to be more intriguing #OTHEREFR .", "It is much more difficult to come up with exact quantum algorithms that outperform classical determistic algorithms.", "In the exact quantum query complexity, it was recognized that the best quantum speed-up for computing total functions was by a factor of 2 for many years #OTHEREFR .", "In a breakthrough result, Ambainis has presented the first example of a Boolean function f : {0, 1} n → {0, 1} for which exact quantum algorithms have superlinear advantage over classical deterministic algorithms #OTHEREFR .", "Based on the results in #OTHEREFR , Ambainis, Gruska, and Zheng #OTHEREFR have verified that exact quantum algorithms have certain advantage for most of Boolean functions." ]

[ "n-bit", "classical algorithm" ]

{ "title": "Characterizations of promise problems with exact quantum query complexity", "abstract": "We give and prove an optimal exact quantum query algorithm with complexity k + 1 for computing the promise problem (i.e., symmetric and partial Boolean function) DJ k n defined as: DJ k n (x) = 1 for |x| = n/2, DJ k n (x) = 0 for |x| in the set {0, 1, . . . , k, n − k, n − k + 1, . . . , n}, and it is undefined for the rest cases, where n is even, |x| is the Hamming weight of x. The case of k = 0 is the well-known Deutsch-Jozsa problem. We outline all symmetric (and partial) Boolean functions with degrees 1 and 2, and prove their exact quantum query complexity. Then we prove that any symmetrical (and partial) Boolean function f has exact quantum 1-query complexity if and only if f can be computed by the Deutsch-Jozsa algorithm. We also discover the optimal exact quantum 2-query complexity for distinguishing between inputs of Hamming weight {⌊n/2⌋, ⌈n/2⌉} and Hamming weight in the set {0, n} for all odd n. In addition, a method is provided to determine the degree of any symmetrical (and partial) Boolean function." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "One-wayness is exactly 1 st -preimage resistance; it is implied by 2 nd -preimage resistance which is in turn implied by collision resistance; see #OTHEREFR .", "There is no reverse implication, however a collision resistant, length-reducing function can be constructed from a one-way function #OTHEREFR .", "Hence 2 nd -preimage resistant or collision resistant functions exist iff one-way functions exist.", "It is expected that one-way functions do exist, although this conjecture, implying the famous conjecture P = N P , is not likely to be proven in the near future 5 .", "What happens in a world with quantum computers? Although some number-theory-based constructions used in asymmetric cryptography collapse, one-way functions may very well still exist." ]

[ "Of course, the practical situation is more complex, since even if one-way functions and secure symmetric cryptography exist, it is not known whether the symmetric primitives used today are good approximations of their idealized counterparts.", "In fact, symmetric ciphers and cryptographic hash functions like the SHA family #OTHEREFR , do not seem to rely on a small family of well-identified hypotheses of hardness of simple mathematical problems, unlike asymmetric algorithms 6 .", "This lack of structure has two consequences: there is no provable security reduction between symmetric algorithms, but conversely their security is not likely to collapse because of some sudden theoretical advance.", "In fact, the last 30 years of cryptanalytic progress showed that the security of symmetric primitives of early designs like DES #OTHEREFR or hashing functions like SHA1 tend to erode slowly rather than abruptly, and that more mature designs (the AES competition contenders, the SHA2 family) exhibit a very good resistance to cryptanalysis.", "It turns out that a family of signature algorithms, Lamport signatures #OTHEREFR , and their derivatives, only require a function f with 1 st -preimage resistance (i.e." ]

[ "quantum computer" ]

{ "title": "Using Hash-Based Signatures to Bootstrap Quantum Key Distribution", "abstract": "Quantum Key Distribution is a secret distribution technique that requires an authenticated channel. This channel is usually created on top of an un-authenticated communication medium using unconditionally secure Message Authentication Codes (MAC) and an initial common secret. We examine the consequences of replacing this MAC algorithm by a cryptographic hash-based signature algorithm, like the Lamport algorithm, and show that in practical settings it results in an increase of the security of QKD and ease its deployment. Quantum Key Distribution (QKD) is a way to create shared and secret random values at both ends of a communication link, with a security guaranteed without computational hardness assumptions [SBpC + 09]. It requires however a classical authenticated channel, together with an untrusted 'quantum' channel (usually realized with an optical fiber or an free space optical transmission). This authenticated channel can be realized on top of an un-authenticated network connection using cryptographic primitives. The natural choice for these primitives is to use symmetric, unconditionally secure Message Authentication Codes like Wegman-Carter [WC81], Evaluation Hash [MV84] or LFSR-based Toeplitz [Kra94] . Being symmetric, these primitives require a common secret; this is not a problem as soon as enough secret is created by the QKD link, but it is an undesirable constraint for the first run, as it forces the user to dispatch securely a common secret at both ends of the link. A very common argument against QKD is that, instead of exchanging a short common secret and using QKD to amplify it, one may as well exchange initially a very large secret and use it in place of the QKD output; the latter solution is easily realized thanks to the very low current price of storage. While this argument is not entirely correct, 1 it is desirable to have alternatives to the pre-sharing of a common secret. Another argument against methods based on a common secret is that they are very hard to operate securely in practice. Indeed, the right way to implement them would be to store the secret on a device providing hardware security like a smart card (acting as a safe for the secret), but for this to be of interest the whole authentication tag computation needs to be performed inside the secure device. Unfortunately, the complete computation 1 Indeed, QKD is forward secure, which means that each key produced is completely independent of past values; as a consequence, even an attacker having at some point in time a complete knowledge of the equipment state including its secrets, does not learn anything about future keys in a passive attack scenario. Contrary to the hard disk scenario where a one-time compromise is enough to obtain all the keys, QKD forces the attacker to perform a persistent active attack in order to obtain new keys, with a much higher risk of being detected." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Even if we make the time-steps of the discrete walk smaller and smaller, the \"coin\" register remains.", "Therefore, the limit cannot be the continous walk without the \"coin\" register.", "This means that one variant of quantum walks could be more powerful than the other in some context, but so far all known examples have given similar behavior of the two walks (see e.g. [CFG02, Kem03b, CCD + 03]).", "In this paper, we present the first example where the discrete walk (with \"coin\") outperforms the continous walk (with no \"coin\").", "Our example is the spatial search #OTHEREFR variant of Grover's search problem." ]

[ "Then, we can find the marked item in O( √ N ) quantum steps, with one quantum step querying a superposition of items. In contrast, classically Ω(N ) queries are required.", "In the \"spatial search\" variant, we have the extra constraint that the N items are stored in N different memory locations and we need time to move betwen locations.", "This may increase the running time of a quantum algorithm.", "The first \"spatial\" version of Grover's algorithms with optimal performance was given by #OTHEREFR who showed how to search N items arranged on the n-dimensional hypercube, using a discrete quantum walk.", "In this paper, we consider the 2-dimensional arrangement where N memory locations are arranged in an √ N × √ N grid." ]

[ "usual Grover's search" ]

{ "title": "Coins Make Quantum Walks Faster", "abstract": "We show how to search N items arranged on a √ N × √ N grid in time O( √ N log N ), using a discrete time quantum walk. This result for the first time exhibits a significant difference between discrete time and continuous time walks without coin degrees of freedom, since it has been shown recently that such a continuous time walk needs time Ω(N ) to perform the same task. Our result furthermore improves on a previous bound for quantum local search by Aaronson and Ambainis. We generalize our result to 3 and more dimensions where the walk yields the optimal performance of O( √ N ) and give several extensions of quantum walk search algorithms for general graphs. The coin-flip operation needs to be chosen judiciously: we show that another \"natural\" choice of coin gives a walk that takes Ω(N ) steps. We also show that in 2 dimensions it is sufficient to have a two-dimensional coin-space to achieve the time O( √ N log N )." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "3. Measure the position register.", "4. Check if the measured vertex is the marked item.", "An item on a vertex of the graph could be marked by setting an auxiliary qubit to |1 , whereas the unmarked items could have this qubit set to |0 .", "Then this auxiliary qubit can control the coin to be C for the unmarked items and C ′ for the marked item.", "We will analyse this algorithm to obtain upper bounds on the query complexity of search by random walks." ]

[ "Each vertex has N edges (we will include a self-loop for each vertex). Both vertices and edges are labelled with 1, . . .", ", N ; the coin space and the vertex Hilbert space are both N -dimensional and we will write states as |i ⊗ |j , where the first register is the coin-register.", "The shift operation S is defined as S : |i ⊗ |j −→ |j ⊗ |i .", "The marked coin in this case is chosen to be C 1 = −C 0 , which gives C 1 − C 0 = −2C 0 and C ′ = C 0 ⊗ (I − 2|v v|), where |v is the marked state.", "Note that C 0 = 2|s s| − 1 N is the reflection around the mean operator of Grover's (\"standard\") algorithms and I − 2|v v|) =: R v the phase flip of the oracle." ]

[ "random walk search", "Grover's algorithm" ]

{ "title": "Coins Make Quantum Walks Faster", "abstract": "We show how to search N items arranged on a √ N × √ N grid in time O( √ N log N ), using a discrete time quantum walk. This result for the first time exhibits a significant difference between discrete time and continuous time walks without coin degrees of freedom, since it has been shown recently that such a continuous time walk needs time Ω(N ) to perform the same task. Our result furthermore improves on a previous bound for quantum local search by Aaronson and Ambainis. We generalize our result to 3 and more dimensions where the walk yields the optimal performance of O( √ N ) and give several extensions of quantum walk search algorithms for general graphs. The coin-flip operation needs to be chosen judiciously: we show that another \"natural\" choice of coin gives a walk that takes Ω(N ) steps. We also show that in 2 dimensions it is sufficient to have a two-dimensional coin-space to achieve the time O( √ N log N )." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "The field is still in its infancy and very theoretical but offers exciting possibilities for the field of computer science --the most important quantum algorithms discovered to date all perform tasks for which there are no classical equivalents.", "For example, Deutsch's algorithm #OTHEREFR is designed to solve the problem of identifying whether a binary function is constant (function values are either all 1 or all 0) or balanced (the function takes an equal number of 0 and 1 values).", "Deutsch's algorithm accomplishes the task in order O(n) time, while classical methods require O(2 n ) time.", "Simon's algorithm #OTHEREFR is constructed for finding the periodicity in a 2-1 binary function that is guaranteed to possess a periodic element.", "Here again an exponential speedup is achieved; however, admittedly, both these algorithms have been designed for artificial, somewhat contrived problems." ]

[ "Here is a real-world problem for which quantum computation provides performance that is classically impossible (though the speedup is less dramatic than exponential).", "Finally, the most well-known and perhaps the most important quantum algorithm discovered so far is Shor's algorithm for prime factorization #OTHEREFR .", "This algorithm find the factors of very large prime numbers in polynomial time, whereas the best known classical algorithms require exponential time.", "Obviously, the implications for the field of cryptography are profound.", "These quantum algorithms take advantage of the unique features of quantum systems to provide exponential speedup over classical approaches." ]

[ "unordered quantum database" ]

{ "title": "Quantum Harmonic Sieve: Learning DNF with a Classical Example Oracle", "abstract": "This paper combines quantum computation with classical computational learning theory to produce a quantum computational learning algorithm. The result is a fourier-based inductive learning algorithm that performs a learning task for which there is no known classical equivalent --that of learning DNF using only an example oracle. The main result is a quantum algorithm for finding the large fourier coefficients of a function approximated by sampling an example oracle. This result is used to improve on Jackson's DNF learning results, which require a membership oracle." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "The query complexity of an algorithm A computing a function F is the number of queries used by A.", "The query complexity of F is the minimum query complexity of any algorithm computing F .", "We are interested in proving lower bounds on the query complexity of specific functions and consider methods for computing such lower bounds.", "An alternative measure of complexity would be to use the time complexity which counts the number of basic operations used by an algorithm.", "The time complexity is always at least as large as the query complexity since each query takes one unit step, and thus a lower bound on the query complexity is also a lower bound on the time complexity." ]

[ "A notorious exception is the so-called Hidden Subgroup Problem which has polynomial query complexity #OTHEREFR , yet polynomial time algorithms are known only for some instances of the problem.", "The oracle model is called decision trees in the classical setting.", "A classical query consists of an index i ∈ [N] , and the answer of the bit x i .", "There is a natural way of modelling a query so that it is reversible. The input is a pair", "There are (at least) two natural ways of generalizing a query to the quantum setting, in which we require all operations to be unitary." ]

[ "existing quantum algorithms" ]

{ "title": "Lower Bounds on Quantum Query Complexity", "abstract": "Shor's and Grover's famous quantum algorithms for factoring and searching show that quantum computers can solve certain computational problems significantly faster than any classical computer. We discuss here what quantum computers cannot do, and specifically how to prove limits on their computational power. We cover the main known techniques for proving lower bounds, and exemplify and compare the methods." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[]

[ "The phase matching method [7] [8] [9] [10] [11] for the Grover quantum search algorithm has been extensively studied and shown to be effective in improving the success probability P k (λ), where λ = M/N is the ratio of the number of target states to the number of database states and k is the number of iterations.", "In spite of the impressive efficacy of this method for most values of λ, it is less so when λ ≪ 1.", "In Ref.", "[7] , we investigated the problem of an exact search with the success probability P k (λ) = 1 for any value of λ, on the basis of the phase-matched search-operator G N (α) ≡ W N (−α)U N (α), where U N (α) is the oracle operator, W N (−α) is the diffusion operator and α is the matching phase.", "The search operator used in the original Grover search is a special case of the above with α = π. We assumed that λ is known preliminarily." ]

[ "Grover-type quantum search" ]

{ "title": "Additive decomposition of iterative quantum search operations in the Grover-type algorithm", "abstract": "In the Grover-type quantum search process a search operator is iteratively applied, say, k times, on the initial database state. We present an additive decomposition scheme such that the iteration process is expressed, in the computational space, as a linear combination of k operators, each of which consists of a single Grover-search followed by an overall phase-rotation. The value of k and the rotation phase are the same as those determined in the framework of the search with certainty. We further show that the final state can be expressed in terms of a single oracle operator of the Grover-search and phase-rotation factors. We discuss how the additive form can be utilized so that it effectively reduces the computational load of the iterative search, and we propose an effective shortcut gate that realizes the same outcome as the iterative search." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "If the left ball is much heavier than the right, then it is harder to slow down and reverse.", "The heavier the left ball, the more collisions are needed, These digits look familiar! In \"Playing Pool with π\" #OTHEREFR , G.", "Galperin proved that this algorithm really is spitting out the digits of π, since for M = 100 N # collisions = π √ M .", "(1.5) I recommend reading the very readable Ref. #OTHEREFR and watching the very watchable Ref. #OTHEREFR .", "Let us make a seemingly abrupt shift and now turn our attention to the field of quantum query complexity." ]

[ "Grover's algorithm provides a way to \"find a needle in a quantum haystack\"or more precisely to determine which out of d mystery functions are being implemented by a black-box quantum oracle.", "For d − 1 = 100 N the runtime is number of oracle calls = 1 4 π √ d − 1 .", "(1.6)", "In this paper I will argue that the similarity between Eq. 1.5 and Eq. 1.6 is not a coincidence.", "It's the same squareroot, and the same π, and for the same reason." ]

[ "quantum mechanics" ]

{ "title": "Playing Pool with $|\\psi \\rangle$: from Bouncing Billiards to Quantum Search", "abstract": "In \"Playing Pool with π\" [1], Galperin invented an extraordinary method to learn the digits of π by counting the collisions of billiard balls. Here I demonstrate an exact isomorphism between Galperin's bouncing billiards and Grover's algorithm for quantum search. This provides an illuminating way to visualize what Grover's algorithm is actually doing." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "In the basis | σ 3 = ±1 , the density operator ρ r (t) will be represented by the matrix", "where", "Equivalently stated, the Bloch representative of the state ρ r (t) is given by the three-dimensional real vector", "We shall assume, in what follows, that the initial state of the cursor is | C(1) and that the initial state of the register is of the form", "namely the eigenstate belonging to the eigenvalue +1 of n(1) · σ, with n(1) = e 1 sin θ + e 3 cos θ." ]

[ "If, indeed, the positive integer µ is the length of the marked binary word to be retrieved, and we set", "and", "then the state (38) correctly describes the initial state | ι of the quantum search as having a component 2 −µ/2 in the direction of the target state, here indicated by | ω = | σ 3 = +1 , and a component √ 1 − 2 −µ in the direction of the flat superposition, here indicated by | σ 3 = −1 , of the 2 µ − 1 basis vectors orthogonal to the target state. In this notations, if", "then the unitary transformation exp(−i α σ 2 /2) corresponds to the product B · A of the oracle step", "and the estimation step" ]

[ "Grover's quantum search" ]

{ "title": "Speed and entropy of an interacting continuous time quantum walk", "abstract": "We present some dynamic and entropic considerations about the evolution of a continuous time quantum walk implementing the clock of an autonomous machine. On a simple model, we study in quite explicit terms the Lindblad evolution of the clocked subsystem, relating the evolution of its entropy to the spreading of the wave packet of the clock. We explore possible ways of reducing the generation of entropy in the clocked subsystem, as it amounts to a deficit in the probability of finding the target state of the computation. We are thus lead to examine the benefits of abandoning some classical prejudice about how a clocking mechanism should operate." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Google, IBM, and Intel all announced their quantum chips of 72, 50, and 49 superconducting qubits (quantum bits), respectively, in 2018.", "Modern computing and communication rest on the digital abstraction of information, measured in bits. A bit has a state either 0 or 1.", "Quantum mechanics allows a quantum bit (qubit) to be in a superposition of both states 0 and 1.", "In addition, the dimension of the state space grows exponentially with the number of qubits.", "These properties endow a quantum computer the power to achieve tasks that are beyond the capability of classical computers." ]

[ "Even more impressive is Shor's factoring algorithm #OTHEREFR , which provides an exponential speedup over the best known classical factoring approach.", "It is anticipated that \"quantum supremacy\"-the superiority of quantum computing over classical devices for a well-defined computational problem -will be achieved by NISQ (noisy intermediate scale quantum) devices in the near future #OTHEREFR .", "All these phenomena indicate that we are in the transitions from studing quantum theory to engineering quantum information-the second quantum revolution #OTHEREFR .", "An extraordinary quantum effect is entanglement-a strong quantum correlation between qubits that are even far apart.", "With preshared quantum entanglement between different parties, communication of quantum information can be done by so-called quantum teleportation #OTHEREFR . Thus establishing entanglement attracts lots of attention #OTHEREFR ." ]

[ "quantum computer" ]

{ "title": "Protocols for Packet Quantum Network Intercommunication", "abstract": "A quantum network, which involves multiple parties pinging each other with quantum messages, could revolutionize communication, computing and basic sciences. A global system of various packet switching quantum and classical networks is called quantum internet, the internet in the future. To pave the way to the future quantum internet, unified protocols that support the distribution of quantum messages within the quantum internet are necessary. Classical network functionalities, ranging from error-control mechanisms to overhead-control strategies, assume that classical information can be correctly read and copied. However, developing quantum internet protocols is more challenging since some classical techniques are forbidden by quantum effects, such as entanglement, measurement, and no-cloning. In this paper, we investigate and propose protocols for packet quantum network intercommunication: quantum User Datagram Protocol (qUDP) and quantum Transmission Control Protocol (qTCP). To protect the fragile quantum information in the quantum internet, qTCP employs techniques of quantum error-correcting codes as well as classical techniques of stack design. In particular, the creation of the logical process-to-process connections of qTCP is accomplished by a quantum version of the three-way handshake protocol." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "The quantum computational speedup is the fact that quantum algorithms solve the respective problems with fewer computation steps (oracle queries in the case of oracle problems) than their best classical counterparts, sometimes fewer than classically possible." ]

[ "Bob, the problem setter, hides a ball in one of four boxes -drawers from now on.", "Alice, the problem solver, is to locate it by opening drawers.", "In the classical case, she needs to open up to three drawers, always one in the quantum case.", "The drawer and ball problem is an example of oracle problem.", "The operation of checking whether the ball is in a drawer is an example of oracle query, or function evaluation." ]

[ "quantum" ]

{ "title": "Completing the physical representation of quantum algorithms provides a quantitative explanation of their computational speedup", "abstract": "The usual representation of quantum algorithms, limited to the process of solving the problem, is physically incomplete. We complete it in three steps: (i) extending the representation to the process of setting the problem, (ii) relativizing the extended representation to the problem solver to whom the problem setting must be concealed, and (iii) symmetrizing the relativized representation for time reversal to represent the reversibility of the underlying physical process. The third steps projects the input state of the relativized representation, where the problem solver is completely ignorant of the setting and thus the solution of the problem, on one where she knows half solution (half of the information specifying it when the solution is an unstructured bit string). Completing the physical representation shows that the number of computation steps (oracle queries) required to solve any oracle problem in an optimal quantum way should be that of a classical algorithm endowed with the advanced knowledge of half solution. This fits the major quantum algorithms known today and would solve the quantum query complexity problem." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Using the laws of quantum mechanics, quantum computers offer novel solutions for resource-intensive problems.", "Quantum computers are theoretically proven to solve certain problems faster than a classical device #OTHEREFR and are well-equipped to handle tasks such as factoring #OTHEREFR , linear systems of equations #OTHEREFR and Monte-Carlo simulations #OTHEREFR .", "In addition, quantum computers could provide practical solutions to optimization problems #OTHEREFR , such as Quantum Unconstrained Binary Optimization (QUBO) problems, which, for instance, find applications in resource allocation, machine learning, and partitioning." ]

[ "It was shown to be optimal, with a quadratic speed-up over the best-known classical algorithms.", "Dürr and Høyer used Grover's algorithm in #OTHEREFR to describe a minimization algorithm, which was later viewed as an example of the algorithmic framework known as Grover Adaptive Search (GAS) #OTHEREFR .", "GAS iteratively applies Grover Search to find the optimum value of an objective function, by using the best-known value as a threshold to flag all values smaller than the threshold in order to find a better solution.", "GAS has been explored as a possible solution for combinatorial optimization problems #OTHEREFR , alongside variational algorithms such as Variational Quantum Eigensolver #OTHEREFR and Quantum Approximate Optimization Algorithm #OTHEREFR .", "One of the challenges inherent in GAS is the creation of efficient oracles, which need to be repeatedly adapted according to new parameters." ]

[ "quantum algorithm" ]

{ "title": "Grover Adaptive Search for Constrained Polynomial Binary Optimization", "abstract": "In this paper we discuss Grover Adaptive Search (GAS) for Constrained Polynomial Binary Optimization (CPBO) problems, and in particular, Quadratic Unconstrained Binary Optimization (QUBO) problems, as a special case. GAS can provide a quadratic speed-up for combinatorial optimization problems compared to brute force search. However, this requires the development of efficient oracles to represent problems and flag states that satisfy certain search criteria. In general, this can be achieved using quantum arithmetic, however, this is expensive in terms of Toffoli gates as well as required ancilla qubits, which can be prohibitive in the near-term. Within this work, we develop a way to construct efficient oracles to solve CPBO problems using GAS algorithms. We demonstrate this approach and the potential speed-up for the portfolio optimization problem, i.e. a QUBO, using simulation. However, our approach applies to higher-degree polynomial objective functions as well as constrained optimization problems. I." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "The first possible reason is that quantum computers operate in a manner so different from classical computers that our techniques for designing algorithms and our intuitions for understanding the process of computation no longer work.", "The second reason is that there really might be relatively few problems for which quantum computers can offer a substantial speed-up over classical computers, and we may have already discovered many or all of the important techniques for constructing quantum algorithms.", "Best quantum algorithms typically provide a quadratic or low-order polynomial speedup #OTHEREFR .", "Furthermore, there are pointers #OTHEREFR suggesting that quantum computers cannot offer more than a (perhaps small) polynomial advantage for NP-complete problems, 7 and such a speedup would struggle to compete with the heuristic approaches commonly used to solve them in practice.", "However, even a polynomial-order speedup could be of significant benefit for problems requiring exact solutions or for problems that can classically be solved in sub-exponential time, like the graph isomorphism problem (see #OTHEREFR )." ]

[ "However, the speedup is not exponential and, more importantly, the problem it solves is far from being realistic: the cost of constructing the quantum database could negate any advantage of the algorithm, and in many classical scenarios one could do much better by simply creating (and maintaining) an ordered database.", "Using Grover's algorithm as a subroutine for solving problems in image processing is more efficient because the cost of preparing the quantum \"database\" can be spread out over several calls #OTHEREFR ; this strategy motivated a new hybrid quantum-classical paradigm for embedded quantum annealing algorithms #OTHEREFR . Other applications are discussed in #OTHEREFR .", "Quantum simulation, quantum-assisted optimisation and quantum sampling are believed to offer near-term quantum solutions to hard problems that may lead even to commercialisation #OTHEREFR .", "3 What is quantum computational supremacy?", "The quantum computational advantage for simulating quantum systems was first stated by Feynman in 1981, in one of the pioneering papers in quantum computing #OTHEREFR (the other one was Manin #OTHEREFR )." ]

[ "Grover's quantum algorithm", "unsorted quantum database" ]

{ "title": "The Road to Quantum Computational Supremacy", "abstract": "Abstract. We present an idiosyncratic view of the race for quantum computational supremacy. Google's approach and IBM challenge are examined. An unexpected side-effect of the race is the significant progress in designing fast classical algorithms. Quantum supremacy, if achieved, won't make classical computing obsolete. A hyper-fast quantum computer is the digital equivalent of a nuclear bomb; whoever possesses one will be able to shred any encryption and break any code in existence. 3 [42] 1 Fairy tales or more cautionary tales? Following the development of Shor's quantum algorithm [61] in 1994 and Grover's quantum algorithm [39] two years later, quantum computing was seen as a bright beacon in computer science, which led to a surge of theoretical and experimental results. The field captured the interest and imagination of the large public and media, and not surprisingly, unfounded claims about the power of quantum computing and its applications proliferated. A certain degree of pessimism began to infiltrate when experimental groups floundered while attempting to control more than a handful of qubits. Recently, a broad wave of ambitious industry-led research programmes in quantum computing-driven by D-Wave Systems, 4 the tech giants Google, IBM, Microsoft, Intel and startups like Rigetti Computing and Quantum Circuits Incorporated-has emerged 5 and bold claims about a future revolutionised by quantum computing are resurfacing. Governments are also involved: phase 1 (2015-2019) £330 million of the UK government programme on quantum technologies [5] is rolling and the European Commission has announced a e1 billion initiative in quantum technology [7] . The European flagship quantum programme, whose explicit goal is to stimulate a \"second quantum revolution\", aims to \"build a universal quantum computer able to demonstrate the resolution of a problem that, with current techniques on a supercomputer, would take longer than the age of the universe\" by 2035, [1]; see also Figure 1 . Undoubtably, these programmes are extremely beneficial to the development of various quantum technologies, but, are the claims about the future of quantum computing realistic? \"We tend to be too optimistic about the short run, too pessimistic about the long run\" said recently J. Preskill [57] ; see also [15, 64] . 3 A typical example of incorrect, largely-spread, sentence quoted from a recent mystery novel. 4 The company relatively steady progress in producing and selling the first series of D-Wave quantum computers has gone from 28 qubits in 2007 to more than 2,000 in their latest 2000Q TM System machine [6]. 5 Of course, the industry work is based and has continued the academic efforts, sometimes using successful experimentalists from academia, like Google." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Given a verifier circuit V for a problem in QMA, what could we do to actually prepare a witness which the circuit accepts? Poulin and Wocjan investigated this #OTHEREFR together with the problem of preparing ground states of local Hamiltonians. This question is not simple." ]

[ "However, this works only when Π 1 commutes with Π 0 (the projector on zeros on the ancillae).", "When [Π 1 , Π 0 ] = 0, the ancilla part of the states we get from Grover searching will very likely be nonzero, and the method does not produce a proper witness of the form |ψ i |0 .", "Poulin and Wocjan found a way for preparing the witness in general.", "First, they run the witness-preserving QMA amplification scheme of Marriott and Watrous #OTHEREFR (see Section 2.2) backwards, and then do Grover search for the part of the state with zero ancillae.", "We simplify their method, showing how to search for QMA witnesses using a reverse of our fast QMA amplification, in a much smaller system with easier initialization." ]

[ "basic Grover search" ]

{ "title": "Fast Amplification of QMA", "abstract": "Given a verifier circuit for a problem in QMA, we show how to exponentially amplify the gap between its acceptance probabilities in the 'yes' and 'no' cases, with a method that is quadratically faster than the procedure given by Marriott and Watrous [1]. Our construction is natively quantum, based on the analogy of a product of two reflections and a quantum walk. Second, in some special cases we show how to amplify the acceptance probability for good witnesses to 1, making a step towards the proof that QMA with one-sided error (QMA 1 ) is equal to QMA. Finally, we simplify the filter-state method to search for QMA witnesses by Poulin and Wocjan [2] ." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "But if the evolution is by U (0,A) with A > 0, say, the observed frequency distribution will be shifted to larger values on the periodic lattice.", "Since some finite, fixed, number of measurements suffices to distinguish these distributions with probability 1−ǫ for a given initial wave packet and external vector potential, the computation takes only constant time, independent of the size of the lattice |L|.", "In fact, for larger lattices the initial wave packet can be broader, i.e., more concentrated around its expected energy, and hence fewer measurements are needed to distinguish the original distribution from the one shifted by the vector potential in case the lattice is periodic.", "In contrast, suppose we try to distinguish a periodic lattice from one with boundaries using a deterministic LGA.", "Since no superpositions of states are possible, all that we can do is to start a single particle off in one direction and see if it ever changes velocity by reflecting from a boundary." ]

[ "In conclusion, we remark that our algorithm exploits the Aharonov-Bohm effect #OTHEREFR which is more usually discussed in two dimensions-we are imagining creating the vector potential by applying a magnetic field which threads a (possibly incomplete, in the nonperiodic case) ring of lattice sites.", "It would be interesting, and potentially useful for pattern recognition #OTHEREFR , to formulate this algorithm for a two dimensional QLGA.", "Preparation of a localized wave packet, and measurement of its energy, each in constant time, is then plausible on physical grounds, provided that the lattice lies within some fixed area.", "For the more realistic situation of a fixed density of lattice sites (and hence an increasing spatial area with increasing lattice size) additional analysis is required.", "As the example of unitary transformations and measurements on the Rydberg states of an atom illustrates, careful attention to the details of the physical implementation of a quantum algorithm is required to correctly quantify its computational complexity #OTHEREFR ." ]

[ "QLGA algorithm", "Grover's algorithm" ]

{ "title": "From Gauge Transformations to Topology Computation in Quantum Lattice Gas Automata", "abstract": "The evolution of a quantum lattice gas automaton (LGA) for a single charged particle is invariant under multiplication of the wave function by a global phase. Requiring invariance under the corresponding local gauge transformations determines the rule for minimal coupling to an arbitrary external electromagnetic field. We develop the Aharonov-Bohm effect in the resulting model into a constant time algorithm to distinguish a one dimensional periodic lattice from one with boundaries; any classical deterministic LGA algorithm distinguishing these two spatial topologies would have expected running time on the order of the cardinality of the lattice." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "In the query model, algorithms access the input only by querying input items and the complexity of the algorithm is measured by the number of queries that it makes.", "Many quantum algorithms can be naturally expressed in this model." ]

[ "In the query setting, one can not only construct efficient quantum algorithms but also prove lower bounds on the number of queries that any quantum algorithm needs.", "For example, it can be shown that any algorithm solving the unordered search problem needs f~(x/N) queries #OTHEREFR .", "(This implies that Grover's algorithm is optimal.) The lower bounds in quantum query model provide insights into the limitations of quantum computing.", "For example, the unordered search problem provides an abstract model for NP-complete problems and the f~(v/-N) lower bound of #OTHEREFR provided evidence of the difficulty of solving these problems on a quantum computer.", "For two related problems -inverting a permutation (often used to model one-way permutation) and AND of ORs only *Supported by Berkeley Fellowship for Graduate Studies and, in part, by NSF grant CCR-9800024. This research was done while visiting Microsoft Research." ]

[ "quantum queries" ]

{ "title": "Quantum lower bounds by quantum arguments", "abstract": "We propose a new method for proving lower bounds on quantum query algorithms. Instead of a classical adversary that runs the algorithm with one input and then modifies the input we use a quantum adversary that runs the input with a superposition of inputs. Using this method, we prove two new ~2(x/-N) lower bounds on AND of ORs and inverting a permutation and also provide more uniform proofs for some known lower bounds which have been previously proven via variety of different techniques." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "the broader fact that no super-linear gap is known between R( f ) and R 0 ( f ) for a total function f ." ]

[]

[ "n bits" ]

{ "title": "Separations in Query Complexity Based on Pointer Functions", "abstract": "In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized query complexity for a total Boolean function is given by the function f on n = 2 k bits defined by a complete binary tree of NAND gates of depth k, which achieves R 0 ( f ) = O (D ( f ) 0.7537... ). We show that this is false by giving an example of a total Boolean function f on n bits whose deterministic query complexity is Ω(n) while its zero-error randomized query complexity is O ( √ n). We further show that the quantum query complexity of the same function is O (n 1/4 ), giving the first example of a total function with a super-quadratic gap between its quantum and deterministic query complexities. We also construct a total Boolean function д on n variables that has zero-error randomized query complexity Ω(n/ log(n)) and bounded-error randomized query complexity . This is the first superlinear separation between these two complexity measures. The exact quantum query complexity of the same function is 2 ) are optimal, up to polylogarithmic factors. Further variations of these functions give additional separations between other query complexity measures: a cubic separation between Q and R 0 , a 3/2-power separation between Q E and R, and a 4th-power separation between approximate degree and bounded-error randomized query complexity. All of these examples are variants of a function recently introduced by Göös, Pitassi, and Watson, which they used to separate the unambiguous 1-certificate complexity from deterministic query complexity and to resolve the famous Clique versus Independent Set problem in communication complexity. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Query complexity has been very useful for understanding the power of different computational models. In the standard version of the query model, we want to compute a Boolean function f : {0, 1} n → {0, 1} on an initially unknown input x ∈ {0, 1} n that can only be accessed by asking queries of the form x i =?. The advantage of query complexity is that we can often prove tight lower bounds and have provable separations between different computational models. This is in contrast to the Turing machine world, in which lower bounds and separations between complexity classes often have to rely on unproven assumptions. At the same time, the model of query complexity is simple and captures the essence of quite a few natural computational processes. We use D( f ), R( f ), and Q ( f ) to denote the minimum number of queries in deterministic, randomized, and quantum query algorithms that compute f . By default, we use R( f ) and Q ( f ) to refer to bounded-error algorithms that compute f (x ) correctly on every input x with probability at least 9/10. It is easy to see that for any function f . For partial functions (i.e., functions whose domain is a strict subset of {0, 1} n ), huge separations are known between all these measures. For example, a randomized algorithm can tell if an n-bit Boolean string has 0 ones or at least n/2 ones with a constant number of queries, while any deterministic algorithm requires Ω(n) queries to do this. Similarly, Aaronson and Ambainis (2015) recently constructed a partial Boolean function f on n variables that can be evaluated using one quantum query but requires Ω( √ n) queries for randomized algorithms. The situation is quite different for total functions, which will be the subject of this work. Here, it is known that D( f ), R( f ), and Q ( f ) are all polynomially related. In fact, et al. 2001). A popular variant of randomized algorithms is the zero-error (Las Vegas) model in which a randomized algorithm always has to output the correct answer, but the number of queries after which it stops can depend on the algorithm's coin flips. The complexity R 0 ( f ) is defined as the expected number of queries, over the randomness of the algorithm, for the worst-case input x. A tighter relation D( f ) ≤ R 0 ( f ) 2 is known for Las Vegas algorithms (this was independently observed by several authors (Hartmanis and Hemachandra 1987; Blum and Impagliazzo 1987; Tardos 1990) ). Nisan (1991) where R 1 ( f ) is the one-sided error randomized complexity of f . Recently, Kulkarni and Tal (2016) , based on a result by Midrijānis (2005) , where the O notation hides polylogarithmic factors. While it has been widely conjectured that these relations are not tight, little progress has been made in the past 20 years on improving these upper bounds or exhibiting functions with separations approaching them. Between D( f ) and R 0 ( f ), the best separation known for a total function is the function NAND k on n = 2 k variables defined by a complete binary NAND tree of depth k. This function satisfies R 0 (NAND k ) = O (D (NAND k ) 0.7537... ) (Snir 1985) . Saks and Wigderson (1986) showed that this upper bound is optimal for NAND k and conjectured that this is the largest gap possible between R 0 ( f ) and D( f ). This function also provides the largest known gap between R( f ) and D( f ), and satisfies R(NAND k ) = Ω(R 0 (NAND k )) (Santha 1995). This situation points to" }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "There are two different cases for the search problem: (S1) one is the case that we know there exists at least one solution x of f (x) = y in X.", "(S2) The other is the case that we do not know the existence of such a solution.", "The second one is more difficult than the first one.", "S1 belongs to a class NP, however S2 does to a class NP-hard #OTHEREFR .", "The search problem has been originally discussed by Levin #OTHEREFR , and Solomonoff #OTHEREFR described an algorithm of it." ]

[ "The computational complexity of Grover's searching algorithm is a square root of the cardinality of X denoted by card{X}.", "In this paper, we studied the new quantum algorithm of the search problem S1 and S2 whose computational complexity is polynomial of card{X}.", "The idea of this quantum algorithm is based on the amplification process of the OMV-SAT algorithm #OTHEREFR ." ]

[ "quantum algorithm" ]

{ "title": "On Quantum Algorithm for Binary Search and Its Computational Complexity", "abstract": "A new quantum algorithm for a search problem and its computational complexity are discussed. It is shown in the search problem containing 2 n objects that our algorithm runs in polynomial time." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "The Boolean variables are mapped by real non-negative variables (coordinates of the machine parts) submitted to idealized physical constraints (perfectly accurate, rigid and reversible) representing the simultaneous system of Boolean equations.", "The solution of the problem is reversibly and nondeterministically produced under the simultaneous influence of all equations -in fact through an idealized many body interaction.", "This machine can be seen as the many body generalization of another perfect machine, the bounching ball model of reversible computation.", "Section 4.", "Simultaneous computation turns out to be a representation of quantum problem solving, an extension of the quantum algorithms that comprises the physical representation of the problem-solution interdependence." ]

[ "The solution of the problem is reversibly and nondeterministically produced under the simultaneous influence of the state before measurement and the quantum principle.", "Section 5.", "The quantum speed up turns out to be \"precognition of the solution\" -the reduction of the initial ignorance of the solution due to backdating, to before running the algorithm, a time-symmetric part of the state vector reduction on the solution; as such, it is bounded by state vector reduction through an entropic inequality.", "Section 6.", "The notion of simultaneous computation is positioned within the development of quantum computation." ]

[ "quantum correlation" ]

{ "title": "Quantum problem solving as simultaneous computation", "abstract": "I provide an alternative way of seeing quantum computation. First, I describe an idealized classical problem solving machine that, thanks to a many body interaction, reversibly and nondeterministically produces the solution of the problem under the simultaneous influence of all the problem constraints. This requires a perfectly accurate, rigid, and reversible relation between the coordinates of the machine parts -the machine can be considered the many body generalization of another perfect machine, the bounching ball model of reversible computation. The mathematical description of the machine, as it is, is applicable to quantum problem solving, an extension of the quantum algorithms that comprises the physical representation of the problem-solution interdependence. The perfect relation between the coordinates of the machine parts is transferred to the populations of the reduced density operators of the parts of the computer register. The solution of the problem is reversibly and nondeterministically produced under the simultaneous influence of the state before measurement and the quantum principle. At the light of the present notion of simultaneous computation, the quantum speed up turns out to be \"precognition\" of the solution, namely the reduction of the initial ignorance of the solution due to backdating, to before running the algorithm, a timesymmetric part of the state vector reduction on the solution; as such, it is bounded by state vector reduction through an entropic inequality." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[]

[ "Let N = 2 n and V = {0, 1, · · · , N − 1}.", "We use |x to denote |j 0 ⊗ · · · ⊗ |j n−1 ∈ (C 2 ) ⊗n where {|j } j=0,1 is the standard basis of C 2 and j i ∈ {0, 1} (i = 0, · · · , n − 1) are the 2-adic digits, i.e., the 2-adic expansion of x is given by n−1 i=0 j i 2 i .", "It is useful to identify (C 2 ) ⊗n with the Hilbert space ℓ 2 (V ) of functions on V , in which case |x is identified with a function δ x , i.e., δ x (y) = 1 if y = x and δ x (y) = 0 otherwise. With this identification, we write", "and consider the ONB {|x ⊗ |⋆ | x ∈ V, ⋆ = ±} of H, where we use |± to denote vectors (|0 ± |1 )/ √ 2 ∈ C 2 .", "We now introduce two operators on H known as the oracle operator and the diffusion operator." ]

[ "n-qubit states", "Grover's searching" ]

{ "title": "Supersymmetric quantum walks with chiral symmetry", "abstract": "Quantum walks have attracted attention as a promising platform realizing topological phenomena and many physicists have introduced various types of indices to characterize topologically protected bound states that are robust against perturbations. In this paper, we introduce an index from a supersymmetric point of view. This allows us to define indices for all chiral symmetric quantum walks such as multidimensional split-step quantum walks and quantum walks on graphs, for which there has been no index theory. Moreover, the index gives a lower bound on the number of bound states robust against compact perturbations. We also calculate the index for several concrete examples including the unitary transformation that appears in Grover's search algorithm." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Positive cost:", "Negative witness: witness w ′ gets extended to the witnessw ′ of P 1 by letting The logarithmic factors in the positive and negative costs arise from encoding c into binary.", "We believe it is possible to avoid this logarithmic overhead by querying c directly, but we have no means to realize such a query in a span program yet.", "The subroutine can be used also in the opposite direction, to replace e c by g. Functionally, the subroutine is a multiplexor then. We use it in this mode in Subroutine 16.", "Proof." ]

[ "For a = 0, 1, . . . , k and ℓ = 0, 1, . . .", ", 2 a − 1, define vector f For each a = 0, . . .", ", k − 1, b ∈ {0, 1} and ℓ from {0, 1, . . . , 2 a − 1}, define an input vector", "that is labeled by value b of variable c a .", "In the positive case, it is easy to see that" ]

[ "resulting subroutine", "n" ]

{ "title": "Span-program-based quantum algorithm for the rank problem", "abstract": "Recently, span programs have been shown to be equivalent to quantum query algorithms. It is an open problem whether this equivalence can be utilized in order to come up with new quantum algorithms. We address this problem by providing span programs for some linear algebra problems. We develop a notion of a high level span program, that abstracts from loading input vectors into a span program. Then we give a high level span program for the rank problem. The last section of the paper deals with reducing a high level span program to an ordinary span program that can be solved using known quantum query algorithms." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "In Section 2, we saw that weak Schur sampling cannot efficiently solve the hidden subgroup problem since such a solution would imply an efficient solution of the collision problem.", "In fact, the problem faced by weak Schur sampling is considerably more difficult, since no information is available about the basis in which the collisions occur.", "This motivates quantum generalizations of the usual (i.e., classical) collision problem, which we study in this section.", "The , 11] .", "The classical algorithm is quite simple: after querying the function on O( d/r) randomly chosen inputs, there is a reasonable probability that a collision will appear, provided one exists." ]

[ "In particular, while the classical algorithm queries the black box non-adaptively, it is essential for the quantum algorithm to make adaptive queries.", "We begin in Section 4.1 by considering what one might call the quantum r-collision sampling problem, which is simply the problem of deciding whether one has k copies of the d-dimensional maximally mixed state or of a state that is maximally mixed on an unknown subspace of dimension d/r.", "This is exactly the problem faced by the weak Schur sampling approach to the hidden subgroup problem, so our results on the quantum collision sampling problem allow us to put tight bounds on the effectiveness of weak Schur sampling for the hsp.", "It turns out that k = Θ(d/r) copies are necessary and sufficient to distinguish these two cases with constant advantage.", "Interestingly, this measurement is entangled across all k registers, and we do not know whether its performance can be matched by an adaptive sequence of singleregister measurements, as is the case for asymptotically large k #OTHEREFR ." ]

[ "quantum algorithm" ]

{ "title": "Weak Fourier-Schur Sampling, the Hidden Subgroup Problem, and the Quantum Collision Problem", "abstract": "Schur duality decomposes many copies of a quantum state into subspaces labeled by partitions, a decomposition with applications throughout quantum information theory. Here we consider applying Schur duality to the problem of distinguishing coset states in the standard approach to the hidden subgroup problem. We observe that simply measuring the partition (a procedure we call weak Schur sampling) provides very little information about the hidden subgroup. Furthermore, we show that under quite general assumptions, even a combination of weak Fourier sampling and weak Schur sampling fails to identify the hidden subgroup. We also prove tight bounds on how many coset states are required to solve the hidden subgroup problem by weak Schur sampling, and we relate this question to a quantum version of the collision problem." }

{ "title": "A fast quantum mechanical algorithm for database search", "abstract": "An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm." }

[ "Finally, X 0 is an F 0 -measurable H-valued random variable with finite mean.", "If With G : H → R being a twice Fréchet differentiable function with bounded and continuous first and second derivatives, we study the weak error (1.4) e(T ) = E G(X(T )) − G(X(T )) , whereX(T ) is some approximation of the process X at time T . This paper is a sequel to #OTHEREFR . It consists of three parts.", "In Section 2 we define some central concepts and state important background results used throughout the paper.", "In Section 3 we show that the error formula in #OTHEREFR Theorem 3 .1] holds for a much wider class of approximations of the solution to (1.1) than stated in that paper, which is concerned with spatial semidiscretization by finite elements.", "Finally, in Sections 4 and 5, the usefulness of the general error formula in Theorem 3.1 is demonstrated through the fact that it can be applied to analyze fully discrete schemes for a wide class of stochastic evolution equations: hyperbolic and parabolic alike." ]

[ "The statement of Theorem 3.1 deserves a motivation.", "Consider, for example, the case whenX(T ) is the value of a semidiscretization in space with finite elements, so thatX has the same form as the solution (1.3) of the original problem (1.1).", "Then an error formula may be derived with the aid of Kolmogorov's backward equation and Itô's formula as in #OTHEREFR .", "It turns out that the error analysis is substantially simplified if new processes are constructed by multiplying X(t) andX(t) by suitable integrating factors.", "That is, define new, drift-free processes (1.5) Y (t) = E(T − t)X(t) = E(T ) with {Y (t)} t≥0 being the solution of the equation dY (t) = E(T − t)B dW (t), t > 0; Y (0) = E(T )X 0 (1." ]

[ "stochastic heat equation" ]

{ "title": "Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes", "abstract": "Abstract. We present an abstract framework for analyzing the weak error of fully discrete approximation schemes for linear evolution equations driven by additive Gaussian noise. First, an abstract representation formula is derived for sufficiently smooth test functions. The formula is then applied to the wave equation, where the spatial approximation is done via the standard continuous finite element method and the time discretization via an I-stable rational approximation to the exponential function. It is found that the rate of weak convergence is twice that of strong convergence. Furthermore, in contrast to the parabolic case, higher order schemes in time, such as the Crank-Nicolson scheme, are worthwhile to use if the solution is not very regular. Finally we apply the theory to parabolic equations and detail a weak error estimate for the linearized Cahn-Hilliard-Cook equation as well as comment on the stochastic heat equation." }

{ "title": "Weak order for the discretization of the stochastic heat equation", "abstract": "In this paper we study the approximation of the distribution of X t Hilbert-valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as driven by a Gaussian space time noise whose covariance operator Q is given. We assume that A −α is a finite trace operator for some α > 0 and that Q is bounded from H into D(A β ) for some β ≥ 0. It is not required to be nuclear or to commute with A. The discretization is achieved thanks to finite element methods in space (parameter h > 0) and implicit Euler schemes in time (parameter ∆t = T /N ). We define a discrete solution X n h and for suitable functions ϕ defined on H, we show that where γ < 1 − α + β. Let us note that as in the finite dimensional case the rate of convergence is twice the one for pathwise approximations. MSC classification: 35A40, 60H15, 60H35, 65C30, 65M60" }

[ "That is, define new, drift-free processes (1.5) Y (t) = E(T − t)X(t) = E(T ) with {Y (t)} t≥0 being the solution of the equation dY (t) = E(T − t)B dW (t), t > 0; Y (0) = E(T )X 0 (1.", "#OTHEREFR and {Ỹ (t)} t≥0 solves dỸ (t) =Ẽ(T − t)B dW (t), t > 0;Ỹ (0) =Ẽ(T )X 0 .", "(1.9) However, ifX(T ) is the result of a time-stepping scheme, then a process of the form (1.3) is not immediate.", "We note that in #OTHEREFR the interpolation of the time-stepping operator family is performed in a manner that results in a family of deterministic operators {Ẽ(t)} t≥0 with a weaker type of semigroup property.", "This property is sufficient to mimic the computations in (1.6), but the operator family does not naturally admit deterministic error estimates." ]

[ "This will, as we shall see, yield a drift-free process as in the right hand side of (1.6) such that (1.7) still holds and known deterministic error estimates can be used almost immediately.", "In this case {Ẽ(t)} t≥0 will not be continuous in t and does not have the semigroup property but it turns out that these are not necessary.", "All we need to obtain the fundamental formulas (3.6)-(3.8) for the error is to assume that there exists a well-defined process {Ỹ (t)} t∈[0,T ] of the form (1.10)Ỹ (t) =Ẽ(T )X 0 + t 0Ẽ (T − s)B dW (s) such that (1.7) holds.", "Here {Ẽ(t)} t∈[0,T ] ⊂ B(S, S) andB ∈ B(U, S), where S is a Hilbert subspace of H with the same norm (typically S = H or S is a finitedimensional subspace of H).", "The processỸ is then well defined if (1.11) Tr T" ]

[ "time-stepping operator family" ]

{ "title": "Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes", "abstract": "Abstract. We present an abstract framework for analyzing the weak error of fully discrete approximation schemes for linear evolution equations driven by additive Gaussian noise. First, an abstract representation formula is derived for sufficiently smooth test functions. The formula is then applied to the wave equation, where the spatial approximation is done via the standard continuous finite element method and the time discretization via an I-stable rational approximation to the exponential function. It is found that the rate of weak convergence is twice that of strong convergence. Furthermore, in contrast to the parabolic case, higher order schemes in time, such as the Crank-Nicolson scheme, are worthwhile to use if the solution is not very regular. Finally we apply the theory to parabolic equations and detail a weak error estimate for the linearized Cahn-Hilliard-Cook equation as well as comment on the stochastic heat equation." }

{ "title": "Weak order for the discretization of the stochastic heat equation", "abstract": "In this paper we study the approximation of the distribution of X t Hilbert-valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as driven by a Gaussian space time noise whose covariance operator Q is given. We assume that A −α is a finite trace operator for some α > 0 and that Q is bounded from H into D(A β ) for some β ≥ 0. It is not required to be nuclear or to commute with A. The discretization is achieved thanks to finite element methods in space (parameter h > 0) and implicit Euler schemes in time (parameter ∆t = T /N ). We define a discrete solution X n h and for suitable functions ϕ defined on H, we show that where γ < 1 − α + β. Let us note that as in the finite dimensional case the rate of convergence is twice the one for pathwise approximations. MSC classification: 35A40, 60H15, 60H35, 65C30, 65M60" }

[ "The reason for this is explained in #OTHEREFR with the fact that the Green's function of the leap-frog scheme coincides at the meshpoints with the Green's function for the wave equation, which is not true for the present scheme.", "The latter paper, #OTHEREFR , studies spatial discretization with finite elements in several dimensions with findings in agreement (in one dimension) with the finite difference spatial approximation studied in #OTHEREFR .", "In connection to hyperbolic equations #OTHEREFR should be mentioned, where the authors are concerned with weak as well as strong convergence of a time discretization scheme for a nonlinear stochastic Schrödinger equation.", "Finally, in Section 5, we apply Theorem 3.1 to the backward Euler in time and finite element in space approximation of the linearized Cahn-Hilliard-Cook equation. The reason for doing this is twofold.", "First, we demonstrate that the general error representation formula is useful in the parabolic setting as well." ]

[ "While our general approach would certainly be applicable to the heat equation as well, it would just reprove a known result, maybe with a more transparent proof, see Remark 5.3 .", "Similarly to the wave equation, also here we find that the rate of weak convergence is twice that of the strong convergence #OTHEREFR under essentially the same assumptions.", "The literature on weak convergence for parabolic equations is richer.", "We have already mentioned #OTHEREFR that proves results for fully discrete schemes of the linear stochastic heat equation.", "In #OTHEREFR spatial, finite element schemes are considered for the same equation." ]

[ "stochastic heat equation" ]

{ "title": "Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes", "abstract": "Abstract. We present an abstract framework for analyzing the weak error of fully discrete approximation schemes for linear evolution equations driven by additive Gaussian noise. First, an abstract representation formula is derived for sufficiently smooth test functions. The formula is then applied to the wave equation, where the spatial approximation is done via the standard continuous finite element method and the time discretization via an I-stable rational approximation to the exponential function. It is found that the rate of weak convergence is twice that of strong convergence. Furthermore, in contrast to the parabolic case, higher order schemes in time, such as the Crank-Nicolson scheme, are worthwhile to use if the solution is not very regular. Finally we apply the theory to parabolic equations and detail a weak error estimate for the linearized Cahn-Hilliard-Cook equation as well as comment on the stochastic heat equation." }

{ "title": "Weak order for the discretization of the stochastic heat equation", "abstract": "In this paper we study the approximation of the distribution of X t Hilbert-valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as driven by a Gaussian space time noise whose covariance operator Q is given. We assume that A −α is a finite trace operator for some α > 0 and that Q is bounded from H into D(A β ) for some β ≥ 0. It is not required to be nuclear or to commute with A. The discretization is achieved thanks to finite element methods in space (parameter h > 0) and implicit Euler schemes in time (parameter ∆t = T /N ). We define a discrete solution X n h and for suitable functions ϕ defined on H, we show that where γ < 1 − α + β. Let us note that as in the finite dimensional case the rate of convergence is twice the one for pathwise approximations. MSC classification: 35A40, 60H15, 60H35, 65C30, 65M60" }

[ "In this section, we compare numerically the performances of the new postprocessed method (13) with the standard linearized implicit Euler method #OTHEREFR , and the trapezoidal method (11), both in finite and infinite dimensions." ]

[ "Taking f 1 (x) = Ax and f 2 (x) = f (x) in (13), we consider the nonlinearities f (x) = −x−sin(x) and f (x) = −2x−x 3 , respectively, and we compute the averages over 10 10 independent trajectories with final time T = 1 and compare for many time stepsizes the accuracy for E(exp(−X(T ) 2 )) = +∞ −∞ exp(−x 2 )ρ(x)dx.", "The final time T = 1 is chosen large enough so that the equilibrium is reached and the exponentially decaying term e −λT in (30) is negligible.", "In the left picture of Figure 1 where the nonlinearity f (x) = −x − sin(x) is Lipschitz, we observe as shown in Theorem 2.8 the expected order 2 of convergence for the new method, while the standard methods exhibit order 1 of convergence (see the reference lines with slopes 1, 2).", "Although our analysis in Section 2.2 applies only to globally-Lipschitz vector fields, we observe that the excellent performances of the new method persist also in the example with the non-Lipschitz nonlinearity f (x) = −2x−x 3 and the globally bounded test function φ(x) = exp(−x 2 ) (right picture of Figure 1) .", "We next consider a standard finite-difference approximation U j (t) ≃ u(j∆x, t) of the 1D heat equation (4) with zero Dirichlet boundary conditions on a uniform grid with size ∆x = 1/(N + 1)." ]

[ "scalar nonlinear SDE" ]

{ "title": "High-order integrator for sampling the invariant distribution of a class of parabolic SPDEs with additive space-time noise", "abstract": "We introduce a time-integrator to sample with high order of accuracy the invariant distribution for a class of semilinear SPDEs driven by an additive space-time noise. Combined with a postprocessor, the new method is a modification with negligible overhead of the standard linearized implicit Euler-Maruyama method. We first provide an analysis of the integrator when applied for SDEs (finite dimension), where we prove that the method has order 2 for the approximation of the invariant distribution, instead of 1. We then perform a stability analysis of the integrator in the semilinear SPDE context, and we prove in a linear case that a higher order of convergence is achieved. Numerical experiments, including the semilinear heat equation driven by space-time white noise, confirm the theoretical findings and illustrate the efficiency of the approach." }

{ "title": "Weak order for the discretization of the stochastic heat equation", "abstract": "In this paper we study the approximation of the distribution of X t Hilbert-valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as driven by a Gaussian space time noise whose covariance operator Q is given. We assume that A −α is a finite trace operator for some α > 0 and that Q is bounded from H into D(A β ) for some β ≥ 0. It is not required to be nuclear or to commute with A. The discretization is achieved thanks to finite element methods in space (parameter h > 0) and implicit Euler schemes in time (parameter ∆t = T /N ). We define a discrete solution X n h and for suitable functions ϕ defined on H, we show that where γ < 1 − α + β. Let us note that as in the finite dimensional case the rate of convergence is twice the one for pathwise approximations. MSC classification: 35A40, 60H15, 60H35, 65C30, 65M60" }

[ ", p, with p ≥ 2 (p depends on the model, for instance whether noise is additive or multiplicative):", "For spectral Galerkin discretization of the SPDE (1), in dimension N, with h = 1 N , one may choose r ∈ [0, 1 2 ).", "For linear implicit Euler discretization of (1), with time step size h = ∆t, one may choose r ∈ [0, 1 4 ).", "This article investigates whether (4) holds true if φ p , where p ≥ 2, is replaced with φ 1 or φ 0 .", "This question is motivated by the positive answer for hypoelliptic SDEs, for instance in the additive noise case with constant σ; on the contrary, the contribution of this article shows that the answer is negative for SPDEs, and we exhibit some family of functions which allow us to identify the rate of convergence." ]

[ "Under an appropriate hypoellipticity assumption (which is satisfied in the additive non-degenerate noise case σ(x) = Id), using Malliavin calculus techniques and regularization effect in the associated Kolmogorov equation, the authors in #OTHEREFR , #OTHEREFR (see also #OTHEREFR ), have proved that the standard approach of #OTHEREFR , to prove the weak error estimate for sufficiently regular functions,", "with p ≥ 2, can be extended with φ 0 instead of φ p on the right-hand side.", "In other words, weak convergence is also of order 1 when considering bounded measurable test functions, in particular for bounded continuous test functions.", "Our contribution is to prove that the situation is quite different for SPDEs.", "Note that thanks to (4), the weak error (2) converges to 0 when h → 0, for any given bounded continuous function φ." ]

[ "Euler-Maruyama discretization" ]

{ "title": "Influence of the regularity of the test functions for weak convergence in numerical discretization of SPDEs", "abstract": "This article investigates the role of the regularity of the test function when considering the weak error for standard discretizations of SPDEs of the form dX(t) = AX(t)dt + F (X(t))dt + dW (t), driven by space-time white noise. In previous results, test functions are assumed (at least) of class C 2 with bounded derivatives, and the weak order is twice the strong order. We prove, in the case F = 0, that to quantify the speed of convergence, it is crucial to control some derivatives of the test functions, even when the noise is non-degenerate. First, the supremum of the weak error over all bounded continuous functions, which are bounded by 1, does not converge to 0 as the discretization parameter vanishes. Second, when considering bounded Lipschitz test functions, the weak order of convergence is divided by 2, i.e. it is not better than the strong order. This is in contrast with the finite dimensional case, where the Euler-Maruyama discretization of elliptic SDEs dY (t) = f (Y (t))dt + dB t has weak order of convergence 1 even for bounded continuous functions. 1991 Mathematics Subject Classification. 60H15,60H35,65C30." }

{ "title": "Weak order for the discretization of the stochastic heat equation", "abstract": "In this paper we study the approximation of the distribution of X t Hilbert-valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as driven by a Gaussian space time noise whose covariance operator Q is given. We assume that A −α is a finite trace operator for some α > 0 and that Q is bounded from H into D(A β ) for some β ≥ 0. It is not required to be nuclear or to commute with A. The discretization is achieved thanks to finite element methods in space (parameter h > 0) and implicit Euler schemes in time (parameter ∆t = T /N ). We define a discrete solution X n h and for suitable functions ϕ defined on H, we show that where γ < 1 − α + β. Let us note that as in the finite dimensional case the rate of convergence is twice the one for pathwise approximations. MSC classification: 35A40, 60H15, 60H35, 65C30, 65M60" }

[ "in #OTHEREFR , #OTHEREFR , #OTHEREFR , #OTHEREFR , #OTHEREFR , #OTHEREFR .", "The monograph #OTHEREFR gives a good overview about SPDEs driven by Lévy noise.", "In #OTHEREFR and #OTHEREFR , numerical approximations in time and space of SPDEs driven by Poisson random measures are investigated and the strong error is estimated.", "Of course, this especially implies an estimate for the weak approximation error.", "A difference to our result is that we look at impulsive noise which is white in time and coloured in space whereas in #OTHEREFR and #OTHEREFR a class of SPDEs driven by Poisson random measures which correspond to impulsive space time white noise is considered." ]

[ "The main technical difference between (4) and (1) lies in the fact that the impulsive cylindrical process (Z t ) t∈[0,T ] is a purely discontinuous Hilbert space valued martingale, while the cylindrical Wiener process (W t ) t∈[0,T ] is continuous.", "As a consequence, the main tools for estimating the weak order of convergence for the numerical scheme -the Itô formula and (connected with it) the backward Kolmogorov equations for certain processes associated with the solutions of the SPDEs and their discretizations -are completely different for (4) and (1).", "The main task therefore is to find manageable expression for the approximation error, which allows estimates using techniques similar to those in #OTHEREFR .", "We note that (5) remains true for the solution", "and the corresponding discretization." ]

[ "cylindrical Wiener process" ]

{ "title": "Weak order for the discretization of the stochastic heat equation driven by impulsive noise", "abstract": "Considering a linear parabolic stochastic partial differential equation driven by impulsive space time noise, we approximate the distribution of is an impulsive cylindrical process and Q describes the spatial covariance structure of the noise; Tr(A −α ) < ∞ for some α > 0 and where γ < 1 − α + β." }

{ "title": "Weak order for the discretization of the stochastic heat equation", "abstract": "In this paper we study the approximation of the distribution of X t Hilbert-valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as driven by a Gaussian space time noise whose covariance operator Q is given. We assume that A −α is a finite trace operator for some α > 0 and that Q is bounded from H into D(A β ) for some β ≥ 0. It is not required to be nuclear or to commute with A. The discretization is achieved thanks to finite element methods in space (parameter h > 0) and implicit Euler schemes in time (parameter ∆t = T /N ). We define a discrete solution X n h and for suitable functions ϕ defined on H, we show that where γ < 1 − α + β. Let us note that as in the finite dimensional case the rate of convergence is twice the one for pathwise approximations. MSC classification: 35A40, 60H15, 60H35, 65C30, 65M60" }

[ "This last work is in a similar vein as ours, but in our case, learned messages are used to correct the messages from graphical inference.", "In the experiments we will show that this hybrid approach really improves over running GNNs in isolation.", "The Kalman Filter is a widely used algorithm for inference in Hidden Markov Processes.", "Some works have explored the direction of coupling them with machine learning techniques.", "A method to discriminatively learn the noise parameters of a Kalman Filter was introduced by #OTHEREFR ." ]

[ "Similarly, #OTHEREFR replaces the dynamics defined in the Kalman Filter with a neural network.", "In our hybrid model, instead of replacing the already considered dynamics, we simultaneously train a learnable function for the purpose of inference." ]

[ "Kalman Filter" ]

{ "title": "Combining Generative and Discriminative Models for Hybrid Inference", "abstract": "A graphical model is a structured representation of the data generating process. The traditional method to reason over random variables is to perform inference in this graphical model. However, in many cases the generating process is only a poor approximation of the much more complex true data generating process, leading to suboptimal estimation. The subtleties of the generative process are however captured in the data itself and we can \"learn to infer\", that is, learn a direct mapping from observations to explanatory latent variables. In this work we propose a hybrid model that combines graphical inference with a learned inverse model, which we structure as in a graph neural network, while the iterative algorithm as a whole is formulated as a recurrent neural network. By using cross-validation we can automatically balance the amount of work performed by graphical inference versus learned inference. We apply our ideas to the Kalman filter, a Gaussian hidden Markov model for time sequences, and show, among other things, that our model can estimate the trajectory of a noisy chaotic Lorenz Attractor much more accurately than either the learned or graphical inference run in isolation." }

{ "title": "Backprop KF: Learning Discriminative Deterministic State Estimators", "abstract": "Generative state estimators based on probabilistic filters and smoothers are one of the most popular classes of state estimators for robots and autonomous vehicles. However, generative models have limited capacity to handle rich sensory observations, such as camera images, since they must model the entire distribution over sensor readings. Discriminative models do not suffer from this limitation, but are typically more complex to train as latent variable models for state estimation. We present an alternative approach where the parameters of the latent state distribution are directly optimized as a deterministic computation graph, resulting in a simple and effective gradient descent algorithm for training discriminative state estimators. We show that this procedure can be used to train state estimators that use complex input, such as raw camera images, which must be processed using expressive nonlinear function approximators such as convolutional neural networks. Our model can be viewed as a type of recurrent neural network, and the connection to probabilistic filtering allows us to design a network architecture that is particularly well suited for state estimation. We evaluate our approach on synthetic tracking task with raw image inputs and on the visual odometry task in the KITTI dataset. The results show significant improvement over both standard generative approaches and regular recurrent neural networks." }

[ "Inertial navigation systems have long leveraged virtual and pseudo-measurements from IMU signals, e.g.", "the widespread Zero velocity UPdaTe (ZUPT) #OTHEREFR - #OTHEREFR , as covariance adaptation #OTHEREFR .", "In parallel, deep learning and more generally machine learning are gaining much interest for inertial navigation #OTHEREFR .", "In #OTHEREFR velocity is estimated using support vector regression whereas #OTHEREFR use recurrent neural networks for endto-end inertial navigation.", "Those methods are promising but restricted to pedestrian dead-reckoning since they generally consider slow horizontal planar motion, and must infer velocity directly from a small sequence of IMU measurements, whereas we can afford to use larger sequences." ]

[ "Albeit promising, the method obtains large translational error > 30% in their stereo odometry experiment.", "Finally, #OTHEREFR uses deep learning for estimating covariance of a local odometry algorithm that is fed into a global optimization procedure, and in #OTHEREFR we used Gaussian processes to learn a wheel encoders error.", "Our conference paper #OTHEREFR contains preliminary ideas, albeit not concerned at all with covariance adaptation: a neural network essentially tries to detect when to perform ZUPT.", "Dynamic adaptation of noise parameters in the Kalman filter is standard in the tracking literature #OTHEREFR , however adaptation rules are application dependent and are generally the result of manual \"tweaking\" by engineers.", "Finally, in #OTHEREFR the authors propose to use classical machine learning techniques to to learn static noise parameters (without adaptation) of the Kalman filter, and apply it to the problem of IMU-GNSS fusion." ]

[ "Kalman filter", "deep networks" ]

{ "title": "AI-IMU Dead-Reckoning", "abstract": "Abstract-In this paper we propose a novel accurate method for dead-reckoning of wheeled vehicles based only on an Inertial Measurement Unit (IMU). In the context of intelligent vehicles, robust and accurate dead-reckoning based on the IMU may prove useful to correlate feeds from imaging sensors, to safely navigate through obstructions, or for safe emergency stops in the extreme case of exteroceptive sensors failure. The key components of the method are the Kalman filter and the use of deep neural networks to dynamically adapt the noise parameters of the filter. The method is tested on the KITTI odometry dataset, and our dead-reckoning inertial method based only on the IMU accurately estimates 3D position, velocity, orientation of the vehicle and self-calibrates the IMU biases. We achieve on average a 1.10% translational error and the algorithm competes with top-ranked methods which, by contrast, use LiDAR or stereo vision. We make our implementation open-source at: https://github.com/mbrossar/ai-imu-dr Index Terms-localization, deep learning, invariant extended Kalman filter, KITTI dataset, inertial navigation, inertial measurement unit" }

{ "title": "Backprop KF: Learning Discriminative Deterministic State Estimators", "abstract": "Generative state estimators based on probabilistic filters and smoothers are one of the most popular classes of state estimators for robots and autonomous vehicles. However, generative models have limited capacity to handle rich sensory observations, such as camera images, since they must model the entire distribution over sensor readings. Discriminative models do not suffer from this limitation, but are typically more complex to train as latent variable models for state estimation. We present an alternative approach where the parameters of the latent state distribution are directly optimized as a deterministic computation graph, resulting in a simple and effective gradient descent algorithm for training discriminative state estimators. We show that this procedure can be used to train state estimators that use complex input, such as raw camera images, which must be processed using expressive nonlinear function approximators such as convolutional neural networks. Our model can be viewed as a type of recurrent neural network, and the connection to probabilistic filtering allows us to design a network architecture that is particularly well suited for state estimation. We evaluate our approach on synthetic tracking task with raw image inputs and on the visual odometry task in the KITTI dataset. The results show significant improvement over both standard generative approaches and regular recurrent neural networks." }

[ "A more modern approach is the unscented Kalman smoother (UKS) that approximates the dynamical smoothing distribution by passing a set of selected points (sigma points) through the exact nonlinear functions of the state dynamics and the measurement model #OTHEREFR .", "Unfortunately, these methods may introduce systematic biases in the estimated state and require the availability of both a prior and a likelihood function in closed form.", "In theory these shortcomings can be overcome by using sampling methods such as particle smoothers #OTHEREFR .", "However, these methods require a large number of samples in order to be reliable and are affected by particle degeneration.", "In recent years, several authors pioneered the use of deep neural networks and stochastic optimization on dynamical filtering and smoothing problems." ]

[ "Importantly, all the parameters of this model can be optimized using stochastic gradient descent.", "Improvements in stochastic variational inference led to several applications to dynamical filtering and smoothing problems.", "Using variational methods, Krishnan and collaborators introduced the deep Kalman filters #OTHEREFR .", "In this work, the authors assume that the distribution of the latent states is Gaussian with mean and covariance matrix determined from the previous state through a parameterized deep neural network.", "The parameters of the network are trained using stochastic backpropagation #OTHEREFR ." ]

[ "deep neural network" ]

{ "title": "Estimating Nonlinear Dynamics with the ConvNet Smoother", "abstract": "Estimating the state of a dynamical system from a series of noise-corrupted observations is fundamental in many areas of science and engineering. The most well-known method, the Kalman smoother (and the related Kalman filter), relies on assumptions of linearity and Gaussianity that are rarely met in practice. In this paper, we introduced a new dynamical smoothing method that exploits the remarkable capabilities of convolutional neural networks to approximate complex nonlinear functions. The main idea is to generate a training set composed of both latent states and observations from an ensemble of simulators and to train the deep network to recover the former from the latter. Importantly, this method only requires the availability of the simulators and can therefore be applied in situations in which either the latent dynamical model or the observation model cannot be easily expressed in closed form. In our simulation studies, we show that the resulting ConvNet smoother has almost optimal performance in the Gaussian case even when the parameters are unknown. Furthermore, the method can be successfully applied to extremely nonlinear and non-Gaussian systems. Finally, we empirically validate our approach via the analysis of measured brain signals." }

{ "title": "Backprop KF: Learning Discriminative Deterministic State Estimators", "abstract": "Generative state estimators based on probabilistic filters and smoothers are one of the most popular classes of state estimators for robots and autonomous vehicles. However, generative models have limited capacity to handle rich sensory observations, such as camera images, since they must model the entire distribution over sensor readings. Discriminative models do not suffer from this limitation, but are typically more complex to train as latent variable models for state estimation. We present an alternative approach where the parameters of the latent state distribution are directly optimized as a deterministic computation graph, resulting in a simple and effective gradient descent algorithm for training discriminative state estimators. We show that this procedure can be used to train state estimators that use complex input, such as raw camera images, which must be processed using expressive nonlinear function approximators such as convolutional neural networks. Our model can be viewed as a type of recurrent neural network, and the connection to probabilistic filtering allows us to design a network architecture that is particularly well suited for state estimation. We evaluate our approach on synthetic tracking task with raw image inputs and on the visual odometry task in the KITTI dataset. The results show significant improvement over both standard generative approaches and regular recurrent neural networks." }

[ "Our results rely on two main theoretical contributions, which are interesting in their own right, since their generality naturally extends to other applications.", "Balanced bidirectional breadth-first search.", "By leveraging on recent advanced results, we prove that, on many realistic random models of real-world complex networks, it is possible to sample a random path between two nodes s and t in time m 1 2 +o(1) if the degree distribution has finite second moment, or m 4−β 2 +o(1) if the degree distribution is power law with exponent 2 < β < 3.", "The models considered are the Configuration Model #OTHEREFR , and all Rank-1 Inhomogeneous Random Graph models #OTHEREFR Chapter 3] , such as the Chung-Lu model #OTHEREFR , the Norros-Reittu model #OTHEREFR , and the Generalized Random Graph #OTHEREFR Chapter 3] .", "Our proof techniques have the merit of adopting a unified approach that simultaneously works in all models considered." ]

[ "The algorithm used is simply a balanced bidirectional BFS (bb-BFS): we perform a BFS from each of the two endpoints s and t, in such a way that the two BFSs are likely to explore about the same number of edges, and we stop as soon as the two BFSs \"touch each other\".", "Rather surprisingly, this technique was never implemented to approximate betweenness centrality, and it is rarely used in the experimental algorithm community.", "Our theoretical analysis provides a clear explanation of the reason why this technique improves over the standard BFS: this means that many state-of-the-art algorithm for real-world complex networks can be improved by the bb-BFS.", "Adaptive sampling made rigorous.", "To speed up the estimation of the betweenness centrality, previous work make use of the technique of adaptive sampling, which consists in testing during the execution of the algorithm whether some condition on the sample obtained so far has been met, and terminating the execution of the algorithm as soon as this happens." ]

[ "real-world networks" ]

{ "title": "KADABRA is an ADaptive Algorithm for Betweenness via Random Approximation", "abstract": "We present KADABRA, a new algorithm to approximate betweenness centrality in directed and undirected graphs, which significantly outperforms all previous approaches on real-world complex networks. The efficiency of the new algorithm relies on two new theoretical contributions, of independent interest. The first contribution focuses on sampling shortest paths, a subroutine used by most algorithms that approximate betweenness centrality. We show that, on realistic random graph models, we can perform this task in time |E| 1 2 +o(1) with high probability, obtaining a significant speedup with respect to the Θ(|E|) worst-case performance. We experimentally show that this new technique achieves similar speedups on real-world complex networks, as well. The second contribution is a new rigorous application of the adaptive sampling technique. This approach decreases the total number of shortest paths that need to be sampled to compute all betweenness centralities with a given absolute error, and it also handles more general problems, such as computing the k most central nodes. Furthermore, our analysis is general, and it might be extended to other settings, as well. * This work was done while the authors were visiting the Simons Institute for the Theory of Computing. 1 As explained in see Section 2, to simplify notation we consider the normalized betweenness centrality." }

{ "title": "An Axiomatic and an Average-Case Analysis of Algorithms and Heuristics for Metric Properties of Graphs", "abstract": "In recent years, researchers proposed several algorithms that compute metric quantities of real-world complex networks, and that are very efficient in practice, although there is no worst-case guarantee. In this work, we propose an axiomatic framework to analyze the performances of these algorithms, by proving that they are efficient on the class of graphs satisfying certain properties. Furthermore, we prove that these properties are verified asymptotically almost surely by several probabilistic models that generate power law random graphs, such as the Configuration Model, the Chung-Lu model, and the Norros-Reittu model. Thus, our results imply average-case analyses in these models. For example, in our framework, existing algorithms can compute the diameter and the radius of a graph in subquadratic time, and sometimes even in time n 1+o(1) . Moreover, in some regimes, it is possible to compute the k most central vertices according to closeness centrality in subquadratic time, and to design a distance oracle with sublinear query time and subquadratic space occupancy. In the worst case, it is impossible to obtain comparable results for any of these problems, unless widelybelieved conjectures are false." }

[ "However, the effect of synaptic filtering will induce memory effects so that the input noise is no longer white, which prevents a complete analytic treatment of the system (for fast decay synapses, the firing rate has been derived asymptotically through a singular perturbation method) #OTHEREFR .", "Crucially, the f-I curve, which expresses the relationship between input current and output rate, derived from the first-passage time theory has a non-elementary functional form, meaning that a simplified form of the f-I curve has to be assumed to achieve analytical treatments.", "Self-consistent relation of steady-state firing rate of a network #OTHEREFR can be derived by a presumed f-I curve.", "A more challenging but also more useful issue that has attracted much recent attention is to find macroscopic transient dynamic descriptions of the neuronal networks. Schaffer et al.", "#OTHEREFR derived the complex firing rate equations of IF networks through the eigenfunction expansion of the Fokker-Planck equation under diffusion approximation." ]

[ "#OTHEREFR derived the rate equations for quadratic IF networks using the Lorentzian ansatz.", "This approach has been generalised to including gap junctions #OTHEREFR and synaptic filtering kinetics #OTHEREFR .", "Schwalger et al #OTHEREFR developed a method to derive the stochastic rate equations of adaptive IF networks based on mean-field approximation of the renewal equation.", "In general, no universal method has been described to derive the macroscopic rate dynamics of IF networks.", "Most proposed theories failed to capture the synchronous transitions induced by the synaptic filtering effect studied in our model." ]

[ "Fokker-Planck equation" ]

{ "title": "Hopf Bifurcation in Mean Field Explains Critical Avalanches in Excitation-Inhibition Balanced Neuronal Networks: A Mechanism for Multiscale Variability", "abstract": "Cortical neural circuits display highly irregular spiking in individual neurons but variably sized collective firing, oscillations and critical avalanches at the population level, all of which have functional importance for information processing. Theoretically, the balance of excitation and inhibition inputs is thought to account for spiking irregularity and critical avalanches may originate from an underlying phase transition. However, the theoretical reconciliation of these multilevel dynamic aspects remains an open question. Herein, we show that excitation-inhibition (E-I) balanced network with synaptic kinetics can maintain irregular spiking dynamics with different levels of synchrony and critical avalanches emerge near the synchronous transition point. The mechanism is unveiled by a novel mean-field theory that derives the field equations governing the network macroscopic dynamics. It reveals that the E-I balanced state of the network manifesting irregular individual spiking is characterized by a macroscopic stable state, which can be either a fixed point or a periodic motion and the transition is predicted by a Hopf bifurcation in the macroscopic field. Furthermore, these multiscale variable behaviours can be jointly observed in the spontaneous activities of mouse cortical slice in vitro, indicating universality of the theoretical prediction. Our theory unveils the mechanism that permits complex neural activities in different spatiotemporal scales to coexist and elucidates a possible origin of the criticality of neural systems. It also provides a theoretical framework for analyzing the macroscopic dynamics of E-I balanced networks and its relationship to the microscopic counterparts, which can be useful for large-scale modeling and computation of cortical dynamics. in the slow inhibition case (right). The network dynamic transition induced by a looser E-I balance can be characterised by the dynamics of population firing rate ( Fig. 1 E and F) . However, an effective theory that can predict the population dynamics of IF networks with synaptic kinetics is still lacking. Here, we propose a 15 1 ms for I neurons, modelling the refractory periods in real neurons. Synaptic weights are set as 45 EO J mV  , 72 IO J mV  , 36 EE J mV  , 72 IE J mV  , 81 EI J mV  and 144 II J mV  , which will satisfy the balanced condition. Network dynamics are simulated by a modified second-order Runge-Kutta scheme [73] with a time step of II IE J JJ J J J" }

{ "title": "Low-dimensional spike rate models derived from networks of adaptive integrate-and-fire neurons: Comparison and implementation", "abstract": "The spiking activity of single neurons can be well described by a nonlinear integrate-and-fire model that includes somatic adaptation. When exposed to fluctuating inputs sparsely coupled populations of these model neurons exhibit stochastic collective dynamics that can be effectively characterized using the Fokker-Planck equation. This approach, however, leads to a model with an infinite-dimensional state space and non-standard boundary conditions. Here we derive from that description four simple models for the spike rate dynamics in terms of low-dimensional ordinary differential equations using two different reduction techniques: one uses the spectral decomposition of the Fokker-Planck operator, the other is based on a cascade of two linear filters and a nonlinearity, which are determined from the Fokker-Planck equation and semi-analytically approximated. We evaluate the reduced models for a wide range of biologically plausible input statistics and find that both approximation approaches lead to spike rate models that accurately reproduce the spiking behavior of the underlying adaptive integrate-and-fire population. Particularly the cascade-based models are overall most accurate and robust, especially in the sensitive region of rapidly changing input. For the mean-driven regime, when input fluctuations are not too strong and fast, however, the best performing model is based on the spectral decomposition. The low-dimensional models also well reproduce stable oscillatory spike rate dynamics that are generated either by recurrent synaptic excitation and neuronal adaptation or through delayed inhibitory synaptic feedback. The computational demands of the reduced models are very low but the implementation complexity differs between the different model variants. Therefore we have made available implementations that allow to numerically integrate the low-dimensional spike rate models as well as the Fokker-Planck partial differential equation in efficient ways for arbitrary model parametrizations as open source software. The derived spike rate descriptions retain a direct link to the properties of single neurons, allow for convenient mathematical analyses of network states, and are well suited for application in neural mass/mean-field based brain network models." }

[ "In the previous section, we derived the optimum values of the time sharing model parameters, α * ij and β * ij , for all pairs (i, j) ∈ M × N .", "In this section, we study the problem of partner-selection in the cooperative scenario under consideration.", "Indeed, given the optimum time-sharing model for all possible PU-SU pairs in the network, we are going to find out that when the cooperation is profitable for PUs and what the optimum approach is to assign the SUs to the PUs so as to optimize their utilities.", "Originally stems from economics, matching theory #OTHEREFR is a suitable mathematical framework to analyze and optimize the problem of partner-selection among two groups of players with conflicting interest.", "Merits of the stable matching framework lie in the competitiveness of outcomes, generality of the preferences, efficiency and simplicity of its algorithmic implementations, and most importantly, its overall practicality #OTHEREFR ." ]

[ "In this section, we formulate the cooperative spectrum sharing problem as a one-to-one matching game between the set of PUs and the set of SUs to solve the partner-selection problem in the proposed scenario.", "We analyze the existence of a stable matching and also study its optimality.", "Let's consider two disjoint sets of N and M, the primary and secondary users, respectively.", "Each user has a complete and transitive preference over the users on the other side.", "We use ≻ i to denote the ordering relationship of agent i." ]

[ "optimization" ]

{ "title": "Spectrum Sharing in Cooperative Cognitive Radio Networks: A Matching Game Framework", "abstract": "Abstract-Dynamic spectrum access allows the unlicensed wireless users (secondary users) to dynamically access the licensed bands from legacy spectrum holders (primary users) either on an opportunistic or a cooperative basis. In this paper, we focus on cooperative spectrum sharing in a wireless network consisting of multiple primary and multiple secondary users. In particular, we study the partner-selection and resource-allocation problems within a matching theory framework, in which the primary and secondary users aim at optimizing their utilities in terms of transmission rate and power consumption. We propose a distributed algorithm to find the solution of the developed matching game that results in a stable matching between the sets of the primary and secondary users. Both analytical and numerical results show that the proposed matching model is a promising approach under which the utility functions of both primary and secondary users are maximized." }

{ "title": "A context-aware matching game for user association in wireless small cell networks", "abstract": "Small cell networks are seen as a promising technology for boosting the performance of future wireless networks. In this paper, we propose a novel context-aware user-cell association approach for small cell networks that exploits the information about the velocity and trajectory of the users while also taking into account their quality of service (QoS) requirements. We formulate the problem in the framework of matching theory with externalities in which the agents, namely users and small cell base stations (SCBSs), have strict interdependent preferences over the members of the opposite set. To solve the problem, we propose a novel algorithm that leads to a stable matching among the users and SCBSs. We show that the proposed approach can better balance the traffic among the cells while also satisfying the QoS of the users. Simulation results show that the proposed matching algorithm yields significant performance advantages relative to traditional context-unaware approaches." }

[ "The assumption that the refraction index n is constant near the boundary (Assumption 1) is in fact only needed in Section 3 to establish desired regularity results.", "The arguments of Section 4 and Section 5 are still valid for non constant n if the regularity result holds." ]

[]

[ "Assumption", "(simpler) approach" ]

{ "title": "The Spectral Analysis of the Interior Transmission Eigenvalue Problem for Maxwell's Equations", "abstract": "In this paper we consider the transmission eigenvalue problem for Maxwell's equations corresponding to non-magnetic inhomogeneities with contrast in electric permittivity that has fixed sign (only) in a neighborhood of the boundary. We study this problem in the framework of semiclassical analysis and relate the transmission eigenvalues to the spectrum of a Hilbert-Schmidt operator. Under the additional assumption that the contrast is constant in a neighborhood of the boundary, we prove that the set of transmission eigenvalues is discrete, infinite and without finite accumulation points. A notion of generalized eigenfunctions is introduced and a denseness result is obtained in an appropriate solution space." }

{ "title": "Discreteness of Transmission Eigenvalues via Upper Triangular Compact Operators", "abstract": "Transmission eigenvalues are points in the spectrum of the interior transmission operator, a coupled 2x2 system of elliptic partial differential equations, where one unknown function must satisfy two boundary conditions and the other must satisfy none. We show that the interior transmission eigenvalues are discrete and depend continuously on the contrast by proving that the interior transmission operator has upper triangular compact resolvent, and that the spectrum of these operators share many of the properties of operators with compact resolvent. In particular, the spectrum is discrete and the generalized eigenspaces are finite dimensional. Our main hypothesis is a coercivity condition on the contrast that must hold only in a neighborhood of the boundary." }

[ "4 On the case A 1 = A 2 in a neighborhood of Γ -Proof of Theorem 3", "The condition A 1 − A 2 ≥ cd α Γ I in a neighborhood of Γ with 0 ≤ α < 2 plays a crucial role in establishing the compactness of T , more precisely T 1 and T 2 (see Remarks 4 and 6).", "To be able to deal with case A 1 = A 2 in a neighborhood of Γ, we make some modifications on T .", "The idea is to take into account the fact that u 1 − u 2 ∈ H 1 0 (Ω) which is more regular u 1 and u 2 which are in general not in H 1 (Ω).", "A modification on T was also used in the work of Sylvester's #OTHEREFR ." ]

[ "The motivation for reformulating the problem is as follows.", "Let λ ∈ C be an eigenvalue of the ITE problem and let (u 1 , u 2 ) ∈ [H 1 (Ω)] 2 be a corresponding pair of eigenfunctions. Then", "(4.1)", "Fix λ 0 = 0 and set w = u 1 − u 2 in Ω. From (4.1), we have", "It follows from (4.2) and (4.3) that, in Ω," ]

[ "condition" ]

{ "title": "Discreteness of interior transmission eigenvalues revisited", "abstract": "This paper is devoted to the discreteness of the transmission eigenvalue problems. It is known that this problem is not self-adjoint and a priori estimates are non-standard and do not hold in general. Two approaches are used. The first one is based on the multiplier technique and the second one is based on the Fourier analysis. The key point of the analysis is to establish the compactness and the uniqueness for Cauchy problems under various conditions. Using these approaches, we are able to rediscover quite a few known discreteness results in the literature and obtain various new results for which only the information near the boundary are required and there might be no contrast of the coefficients on the boundary." }

{ "title": "Discreteness of Transmission Eigenvalues via Upper Triangular Compact Operators", "abstract": "Transmission eigenvalues are points in the spectrum of the interior transmission operator, a coupled 2x2 system of elliptic partial differential equations, where one unknown function must satisfy two boundary conditions and the other must satisfy none. We show that the interior transmission eigenvalues are discrete and depend continuously on the contrast by proving that the interior transmission operator has upper triangular compact resolvent, and that the spectrum of these operators share many of the properties of operators with compact resolvent. In particular, the spectrum is discrete and the generalized eigenspaces are finite dimensional. Our main hypothesis is a coercivity condition on the contrast that must hold only in a neighborhood of the boundary." }

[ "If there exisit two positive constants ǫ + and ǫ − , and a neighborhood of ∂Ω, neigh(∂Ω), such that ǫ + < ǫ − < n * , ε(x) ≥ −ǫ − for x ∈ Ω, and either ε(x) ≥ ǫ + or ε(x) ≤ −ǫ + for x ∈ neigh(∂Ω).Then one cannot have that", "for any fixed d ∈ S N −1 , and all (x, k) ∈ S N −1 × R + .", "Proof.", "Assume that (2.5) holds for a fixed d ∈ S N −1 and all (x, k) ∈ S N −1 × R + .", "Then, according to our earlier discussion, we know that every k ∈ R + is an interior transmission eigenvalue to" ]

[ "The proof is completed." ]

[ "interior transmission eigenvalue" ]

{ "title": "Uniqueness in Determining Refractive Indices by Formally-determined Far-field Data", "abstract": "We present two uniqueness results for the inverse problem of determining an index of refraction by the corresponding acoustic far-field measurement encoded into the scattering amplitude. The first one is a local uniqueness in determining a variable index of refraction by the fixed incident-direction scattering amplitude. The inverse problem is formally posed with such measurement data. The second one is a global uniqueness in determining a constant refractive index by a single far-field measurement. The arguments are based on the study of certain nonlinear and non-selfadjoint interior transmission eigenvalue problems." }

{ "title": "Discreteness of Transmission Eigenvalues via Upper Triangular Compact Operators", "abstract": "Transmission eigenvalues are points in the spectrum of the interior transmission operator, a coupled 2x2 system of elliptic partial differential equations, where one unknown function must satisfy two boundary conditions and the other must satisfy none. We show that the interior transmission eigenvalues are discrete and depend continuously on the contrast by proving that the interior transmission operator has upper triangular compact resolvent, and that the spectrum of these operators share many of the properties of operators with compact resolvent. In particular, the spectrum is discrete and the generalized eigenspaces are finite dimensional. Our main hypothesis is a coercivity condition on the contrast that must hold only in a neighborhood of the boundary." }

[ "The two cases ν = 0 and ν = 1 could be called Schrödinger and Helmholtz cases, respectively.", "Perturbations with ν = 0 appear in quantum scattering, whereas potentials with ν = 1 appear in electromagnetic and acoustic scattering.", "Although the proofs are mostly parallel for the two cases, the case ν = 1 is harder as in the end one needs to perform a perturbation argument with large λ which is harder for the rather large perturbation λV .", "The case ν = 1 is more delicate in other ways as well: In the case where W is present, the Helmholtz argument will involve W/λ, and this prevents us from excluding the possibility that the transmission eigenvalues accumulate to zero.", "Furthermore, for ν = 1, we need to invoke unique continuation and this imposes restrictions on the main term." ]

[ "We are planning to consider the other coercivity condition elsewhere." ]

[ "two coercivity conditions" ]

{ "title": "Discreteness of Transmission Eigenvalues for Higher-Order Main Terms and Perturbations", "abstract": "In this paper we extend Sylvester's approach via upper triangular compact operators to establish the discreteness of transmission eigenvalues for higher-order main terms and higher-order perturbations. The coefficients of the perturbations must be sufficiently smooth and the coefficients of the higher-order terms of the perturbation must vanish in a neighbourhood of the boundary of the underlying domain. The zeroeth order term must satisfy a suitable coercivity condition in a neighbourhood of the boundary." }

{ "title": "Discreteness of Transmission Eigenvalues via Upper Triangular Compact Operators", "abstract": "Transmission eigenvalues are points in the spectrum of the interior transmission operator, a coupled 2x2 system of elliptic partial differential equations, where one unknown function must satisfy two boundary conditions and the other must satisfy none. We show that the interior transmission eigenvalues are discrete and depend continuously on the contrast by proving that the interior transmission operator has upper triangular compact resolvent, and that the spectrum of these operators share many of the properties of operators with compact resolvent. In particular, the spectrum is discrete and the generalized eigenspaces are finite dimensional. Our main hypothesis is a coercivity condition on the contrast that must hold only in a neighborhood of the boundary." }

[]

[ "Setting u = v − w, instead of the interior transmission problem we consider the equivalent problem for u ∈ H k 0 (Ω) and w:", "Here u, in a sense, satisfies many \"boundary conditions\", but w satisfies none.", "We first consider the \"Born approximation\" by simply striking out the term Qu or λV u, depending on whether ν = 0 or ν = 1.", "Once the properties of the resolvent operators of the simpler case has been dealt with, the missing term can be brought back in with a perturbation argument.", "This is particularly simple in the Schrödinger case but requires more work in the Helmholtz situation." ]

[ "Theorems" ]

{ "title": "Discreteness of Transmission Eigenvalues for Higher-Order Main Terms and Perturbations", "abstract": "In this paper we extend Sylvester's approach via upper triangular compact operators to establish the discreteness of transmission eigenvalues for higher-order main terms and higher-order perturbations. The coefficients of the perturbations must be sufficiently smooth and the coefficients of the higher-order terms of the perturbation must vanish in a neighbourhood of the boundary of the underlying domain. The zeroeth order term must satisfy a suitable coercivity condition in a neighbourhood of the boundary." }

{ "title": "Discreteness of Transmission Eigenvalues via Upper Triangular Compact Operators", "abstract": "Transmission eigenvalues are points in the spectrum of the interior transmission operator, a coupled 2x2 system of elliptic partial differential equations, where one unknown function must satisfy two boundary conditions and the other must satisfy none. We show that the interior transmission eigenvalues are discrete and depend continuously on the contrast by proving that the interior transmission operator has upper triangular compact resolvent, and that the spectrum of these operators share many of the properties of operators with compact resolvent. In particular, the spectrum is discrete and the generalized eigenspaces are finite dimensional. Our main hypothesis is a coercivity condition on the contrast that must hold only in a neighborhood of the boundary." }

[ "The goal of this paper is to obtain an analogue of the Weyl law for the signed counting function of positive ITEs and establish an important connection between positive ITEs and the scattering matrix.", "Due to the lack of symmetry (and ellipticity), the discreteness of the spectrum of the ITE problem, the existence of real eigenvalues, and their asymptotics can not be obtained by soft arguments.", "Moreover, the existence of non-real ITEs was shown in #OTHEREFR , and an example of an elliptic ITE problem where the set of ITEs is not discrete can be found in #OTHEREFR Examples 1, #OTHEREFR .", "There is extensive literature (see the review #OTHEREFR ) on the properties of ITEs and corresponding eigenfunctions.", "The following results are most closely related to our study." ]

[ "The latter condition (which means that the inhomogeneity has a sharp boundary) will be assumed to hold in our study.", "It was shown in #OTHEREFR , #OTHEREFR , #OTHEREFR , #OTHEREFR , #OTHEREFR 1 that the standard Weyl estimate holds for the complex ITEs located in an arbitrary cone containing the real positive semi-axis:", "where ω d is the volume of the unit ball in R d .", "Earlier, in a series of articles #OTHEREFR , #OTHEREFR , #OTHEREFR , we have shown that if", "then the set of positive ITEs (which are the most important for applications) is infinite, and moreover," ]

[ "boundary" ]

{ "title": "Sharp Weyl Law for Signed Counting Function of Positive Interior Transmission Eigenvalues", "abstract": "We consider the interior transmission eigenvalue (ITE) problem, which arises when scattering by inhomogeneous media is studied. The ITE problem is not selfadjoint. We show that positive ITEs are observable together with plus or minus signs that are defined by the direction of motion of the corresponding eigenvalues of the scattering matrix (when the latter approach z = 1). We obtain a Weyl type formula for the counting function of positive ITEs, which are taken together with ascribed signs." }

{ "title": "Discreteness of Transmission Eigenvalues via Upper Triangular Compact Operators", "abstract": "Transmission eigenvalues are points in the spectrum of the interior transmission operator, a coupled 2x2 system of elliptic partial differential equations, where one unknown function must satisfy two boundary conditions and the other must satisfy none. We show that the interior transmission eigenvalues are discrete and depend continuously on the contrast by proving that the interior transmission operator has upper triangular compact resolvent, and that the spectrum of these operators share many of the properties of operators with compact resolvent. In particular, the spectrum is discrete and the generalized eigenspaces are finite dimensional. Our main hypothesis is a coercivity condition on the contrast that must hold only in a neighborhood of the boundary." }

[ "After (42) is proved for all λ except a fixed exceptional set { λ s }, we can change the value of t to another value t = t s for which λ = λ s is not an eigenvalue of the impedance problem (22) with t = t s .", "Then (42) will be justified for λ = λ s .", "In fact, we can find a value of t = t that can be used simultaneously for all points λ s , but we do not need to do it.", "The proof of Theorems 1.1 and 1.3 is complete.", "Proof of Theorem 1.4." ]

[ "Indeed, if λ > 0 is not a pole of the operator R −1 (λ), then R −1 (λ) maps an arbitrary function f ∈ H 3/2 (∂O) into", "where (u, v) is the solution of the problem ∆u + λu = 0, u ∈ H 2 (O),", "One can express solution (u, v) of (51)-(53) through the resolvent of the ITE problem by looking for (u, v) as a sum of two terms, where the first term (u 1 , v 1 ) satisfies only the boundary conditions, and the second term is the solution of the problem (51)- (53) with homogeneous boundary conditions and the right-hand side in the equations defined by the first term.", "Hence the operator f → (u, v) has a pole of at most first order at λ = λ 0 if the resolvent of the ITE problem has a pole of the first order at λ 0 .", "Therefore, (50) implies that the eigenvalues of the operator R(λ) may have zeroes only of the first order at λ = λ 0 ." ]

[ "eigenvalues" ]

{ "title": "Sharp Weyl Law for Signed Counting Function of Positive Interior Transmission Eigenvalues", "abstract": "We consider the interior transmission eigenvalue (ITE) problem, which arises when scattering by inhomogeneous media is studied. The ITE problem is not selfadjoint. We show that positive ITEs are observable together with plus or minus signs that are defined by the direction of motion of the corresponding eigenvalues of the scattering matrix (when the latter approach z = 1). We obtain a Weyl type formula for the counting function of positive ITEs, which are taken together with ascribed signs." }

{ "title": "Discreteness of Transmission Eigenvalues via Upper Triangular Compact Operators", "abstract": "Transmission eigenvalues are points in the spectrum of the interior transmission operator, a coupled 2x2 system of elliptic partial differential equations, where one unknown function must satisfy two boundary conditions and the other must satisfy none. We show that the interior transmission eigenvalues are discrete and depend continuously on the contrast by proving that the interior transmission operator has upper triangular compact resolvent, and that the spectrum of these operators share many of the properties of operators with compact resolvent. In particular, the spectrum is discrete and the generalized eigenspaces are finite dimensional. Our main hypothesis is a coercivity condition on the contrast that must hold only in a neighborhood of the boundary." }

[ "However, depending on the type of perturbation, the use of a system may be quite inconvenient, or may not work at all (e.g.", "in #OTHEREFR the formula does not work for complex eigenvalues).", "The ability to use a nonlinear formulation leaves us a number of options. The simplest may be to invert A, and use", "If we have a simple nonlinear eigenvalue λ 0 and some perturbation indexed by h, then Corollary 4.1 (assuming all hypotheses are met) yields", "21) where one expects the square of the norms to be asymptotically smaller than the correction term." ]

[ "is analytic with respect to λ away from the negative real axis.", "Perturbations in η which correspond to material defects, or numerical discretizations of such an operator, can be handled by the theory presented here for the case of simple eigenvalues.", "Since the eigenvalue problem is not polynomial, converting to a linear system would require the use of an infinite system and would potentially be far more complicated." ]

[ "generalized eigenvalue problem" ]

{ "title": "Nonlinear eigenvalue approximation for compact operators", "abstract": "In [13] a general spectral approximation theory was developed for compact operators on a Banach space which does not require that the operators be self-adjoint and also provides a first order correction term. Here we extend some of the results of that paper to nonlinear eigenvalue problems. We present examples of its application that arise in electromagnetics." }

{ "title": "Discreteness of Transmission Eigenvalues via Upper Triangular Compact Operators", "abstract": "Transmission eigenvalues are points in the spectrum of the interior transmission operator, a coupled 2x2 system of elliptic partial differential equations, where one unknown function must satisfy two boundary conditions and the other must satisfy none. We show that the interior transmission eigenvalues are discrete and depend continuously on the contrast by proving that the interior transmission operator has upper triangular compact resolvent, and that the spectrum of these operators share many of the properties of operators with compact resolvent. In particular, the spectrum is discrete and the generalized eigenspaces are finite dimensional. Our main hypothesis is a coercivity condition on the contrast that must hold only in a neighborhood of the boundary." }

[ "Researchers at Twitter developed a similar measure called Who To Follow (WTF) that uses a bipartite graph where the left hand side is constructed form a personalized \"circle-of-trust\" and the right hand side is created by the those Twitter users that the query vertex follows #OTHEREFR .", "WTF then performs SALSA-like iterations to calculate whom on the left-hand-side should be recommended for the user, i.e., the query node, to follow.", "SimRank Family Apart from personalized extensions of PageRank and SALSA, SimRank is also used for similarity search on networks based on the notion that \"objects are similar if they are referenced by similar objects\" #OTHEREFR .", "P-Rank generalizes SimRank by computing structural similarity based on in-neighbors only, on out-neighbors only, and then by making a linear combination of the two similarities #OTHEREFR .", "An important property of SimRank, P-Rank, and related measures #OTHEREFR are that they are symmetric, that is, the similarity between two nodes u and v is the same as the similarity of v and u." ]

[ "Each of the network similarity families have their own particular strengths and weaknesses.", "The PageRank family, in particular, has been successful in many applications, e.g., Web search, and has been personalized (PPR) to find nodes topologically similar to some query node #OTHEREFR .", "The SALSA family has shown some success in personalization, although the calculation of hubs and authorities scores is not naturally applicable to the computation of personalized similarity scores, and therefore requires a re-appropriation of its original intent in personalized cases.", "Other methods include SCAN #OTHEREFR , ObjectRank #OTHEREFR and PopRank #OTHEREFR , among many others #OTHEREFR .", "However, these similarity measures disregard the different roles and types of the objects and links." ]

[ "SimRank" ]

{ "title": "Forward Backward Similarity Search in Knowledge Networks", "abstract": "Similarity search is a fundamental problem in social and knowledge networks like GitHub, DBLP, Wikipedia, etc. Existing network similarity measures are limited because they only consider similarity from the perspective of the query node. However, due to the complicated topology of real-world networks, ignoring the preferences of target nodes often results in odd or unintuitive performance. In this work, we propose a dual perspective similarity metric called Forward Backward Similarity (FBS) that efficiently computes topological similarity from the perspective of both the query node and the perspective of candidate nodes. The effectiveness of our method is evaluated by traditional quantitative ranking metrics and large-scale human judgement on four large real world networks. The proposed method matches human preference and outperforms other similarity search algorithms on community overlap and link prediction. Finally, we demonstrate top-5 rankings for five famous researchers on an academic collaboration network to illustrate how our approach captures semantics more intuitively than other approaches. Computing the similarity of two or more objects in an information network is the main focus of a large amount of scientific research and technological development. Friendship recommendation in social networks is one example, but web search, community detection, general link prediction, list augmentation, and dozens of other application areas are all singularly dependent upon some notion of similarly in the underlying networks. Similarity is multi-faceted; various traits can be used to determine similarity depending on the specific problem domain. Entire fields of research are dedicated to the development of algorithms that effectively and efficiently retrieve objects similar to some query-object, e.g., information retrieval, computer vision, and databases (broadly speaking). Researchers and practitioners understand that network topology plays a critical role in the identification of object similarity [26, 27, 35 ]. An appreciation of the topological features has led to the development of models of network growth, clustering, prediction, and classification. Given a query vertex u, what we need is a network similarity metric that finds a target vertex v to be similar if they satisfy the following criteria: 1. u is highly connected to v, and 2. v is highly connected to u A typical approach used to compute personalized search is to measure the similarity between some query node and a set of candidate target nodes (maybe all other nodes). After the similarities of the candidate nodes have been found, the user is typically presented with a top-K list of candidate nodes ordered by their similarity scores. For example, in citation networks Case et al. had previously defined six citation behaviors [5], which we simplify into two categories: a) intra-domain citations and b) cross-domain citations. Intra-domain references often include related prior work that is directly related to the referencing paper, and are the type of references that a reader would expect to see included in the experimental comparison section of the referencing paper. On the other hand, cross-domain citations often represent paradigms, platforms, and data sets that come from a separate, loosely-related area. For example, the closely related references of this paper include references to personal PageRank [11]" }

{ "title": "Axiomatic ranking of network role similarity", "abstract": "A key task in analyzing social networks and other complex networks is role analysis: describing and categorizing nodes by how they interact with other nodes. Two nodes have the same role if they interact with equivalent sets of neighbors. The most fundamental role equivalence is automorphic equivalence. Unfortunately, the fastest algorithm known for graph automorphism is nonpolynomial. Moreover, since exact equivalence is rare, a more meaningful task is measuring the role similarity between any two nodes. This task is closely related to the link-based similarity problem that SimRank addresses. However, SimRank and other existing simliarity measures are not sufficient because they do not guarantee to recognize automorphically or structurally equivalent nodes. This paper makes two contributions. First, we present and justify several axiomatic properties necessary for a role similarity measure or metric. Second, we present RoleSim, a role similarity metric which satisfies these axioms and which can be computed with a simple iterative algorithm. We rigorously prove that RoleSim satisfies all the axiomatic properties and demonstrate its superior interpretative power on both synthetic and real datasets." }

[ "In Section 3, we describe notations in this paper and some preliminary knowledge.", "In Section 4, we present the proposed method and the fast computation strategies are introduced in Section 5.", "Extensive experiments are done to validate the proposed method in Section 6. Finally, Section 7 concludes this paper.", "Ranking is an important data mining task in network analysis. Many ranking methods have been proposed.", "For example, PageRank #OTHEREFR evaluates the importance of objects through a random walk process; HITS #OTHEREFR ranks objects using the authority and hub scores; SimRank #OTHEREFR evaluates the similarity of two objects by their neighbors' similarities." ]

[ "These approaches only consider the same type of objects in homogeneous networks, so they cannot be applied in heterogeneous networks.", "Some researches have begun to pay attention to the co-ranking on multiple types of objects. For example, Zhou et al.", "#OTHEREFR co-rank authors and their publications by coupling two random walk processes, and the co-HITS #OTHEREFR incorporates the bipartite graph with the content information and the constraints of relevance.", "Although these methods can rank different types of objects existing in HIN, they are restricted to bipartite graphs.", "Recently, MultiRank #OTHEREFR determines the importance of both objects and relations simultaneously for multi-relational data, and HAR #OTHEREFR is proposed to determine hub and authority scores of objects and relevance scores of relations in multi-relational data for query search." ]

[ "network structure", "role similarity" ]

{ "title": "HRank: A Path based Ranking Framework in Heterogeneous Information Network", "abstract": "Recently, there is a surge of interests on heterogeneous information network analysis. As a newly emerging network model, heterogeneous information networks have many unique features (e.g., complex structure and rich semantics) and a number of interesting data mining tasks have been exploited in this kind of networks, such as similarity measure, clustering, and classification. Although evaluating the importance of objects has been well studied in homogeneous networks, it is not yet exploited in heterogeneous networks. In this paper, we study the ranking problem in heterogeneous networks and propose the HRank framework to evaluate the importance of multiple types of objects and meta paths. Since the importance of objects depends upon the meta paths in heterogeneous networks, HRank develops a path based random walk process. Moreover, a constrained meta path is proposed to subtly capture the rich semantics in heterogeneous networks. Furthermore, HRank can simultaneously determine the importance of objects and meta paths through applying the tensor analysis. Extensive experiments on three real datasets show that HRank can effectively evaluate the importance of objects and paths together. Moreover, the constrained meta path shows its potential on mining subtle semantics by obtaining more accurate ranking results." }

{ "title": "Axiomatic ranking of network role similarity", "abstract": "A key task in analyzing social networks and other complex networks is role analysis: describing and categorizing nodes by how they interact with other nodes. Two nodes have the same role if they interact with equivalent sets of neighbors. The most fundamental role equivalence is automorphic equivalence. Unfortunately, the fastest algorithm known for graph automorphism is nonpolynomial. Moreover, since exact equivalence is rare, a more meaningful task is measuring the role similarity between any two nodes. This task is closely related to the link-based similarity problem that SimRank addresses. However, SimRank and other existing simliarity measures are not sufficient because they do not guarantee to recognize automorphically or structurally equivalent nodes. This paper makes two contributions. First, we present and justify several axiomatic properties necessary for a role similarity measure or metric. Second, we present RoleSim, a role similarity metric which satisfies these axioms and which can be computed with a simple iterative algorithm. We rigorously prove that RoleSim satisfies all the axiomatic properties and demonstrate its superior interpretative power on both synthetic and real datasets." }

[ "To use this random model assumes that we can only produce outputs within that restricted space, when in actuality Ω is the set of all partitions of N nodes.", "Furthermore, during evaluation, we hold our ground truth fixed, rather than marginalizing over possible ground truths.", "Were we to instead consider a distribution over T s, we would add noise from other possible generative processes which yield the same graph from different underlying partitions.", "In our average, we might be including scores on ground truths that better align with our notions of, say, core-periphery partitioning.", "For this reason, we take a one-sided expectation-over C, holding T fixed." ]

[ "This distribution is what we use for our AMI expectation, giving a measure denoted as AMI 1 all , which is recommended by #OTHEREFR . It takes the form", "The differences between M all and M perm are illustrated in Figure 2 under |V | = 3.", "We will now show that substituting M all for M perm , hence using AMI 1 all , allows for an exact No Free Lunch theorem." ]

[ "N nodes", "partitions" ]

{ "title": "An Exact No Free Lunch Theorem for Community Detection", "abstract": "A precondition for a No Free Lunch theorem is evaluation with a loss function which does not assume a priori superiority of some outputs over others. A previous result for community detection by Peel et al. (2017) relies on a mismatch between the loss function and the problem domain. The loss function computes an expectation over only a subset of the universe of possible outputs; thus, it is only asymptotically appropriate with respect to the problem size. By using the correct random model for the problem domain, we provide a stronger, exact No Free Lunch theorem for community detection. The claim generalizes to other set-partitioning tasks including core-periphery separation, k-clustering, and graph partitioning. Finally, we review the literature of proposed evaluation functions and identify functions which (perhaps with slight modifications) are compatible with an exact No Free Lunch theorem." }

{ "title": "The Impact of Random Models on Clustering Similarity", "abstract": "Clustering is a central approach for unsupervised learning. After clustering is applied, the most fundamental analysis is to quantitatively compare clusterings. Such comparisons are crucial for the evaluation of clustering methods as well as other tasks such as consensus clustering. It is often argued that, in order to establish a baseline, clustering similarity should be assessed in the context of a random ensemble of clusterings. The prevailing assumption for the random clustering ensemble is the permutation model in which the number and sizes of clusters are fixed. However, this assumption does not necessarily hold in practice; for example, multiple runs of K-means clustering returns clusterings with a fixed number of clusters, while the cluster size distribution varies greatly. Here, we derive corrected variants of two clustering similarity measures (the Rand index and Mutual Information) in the context of two random clustering ensembles in which the number and sizes of clusters vary. In addition, we study the impact of one-sided comparisons in the scenario with a reference clustering. The consequences of different random models are illustrated using synthetic examples, handwriting recognition, and gene expression data. We demonstrate that the choice of random model can have a drastic impact on the ranking of similar clustering pairs, and the evaluation of a clustering method with respect to a random baseline; thus, the choice of random clustering model should be carefully justified." }

[ "It is noteworthy that the improvements brought by the different encoders on this task roughly correspond to the performance on the pattern similarity task. This fact implies two potential impacts.", "First, the distributed representations of relational patterns are useful and easily transferable to other tasks such as knowledge base population.", "Second, the pattern similarity dataset provides a gauge to predict successes of distributed representations in another task.", "We could further improve the performance of SVM + GAC by incorporating external resources in the similar manner as the previous studies did." ]

[ "Moreover, we could obtain 84.2 F1 score, using the ranking based loss function (dos #OTHEREFR and fine-tuning of the distributed representations initially trained by GAC.", "Currently, this is the second best score among the performance values reported in the previous studies on this task (the second group of Table 3).", "If we could use the negative sampling technique proposed by #OTHEREFR", "(2015) , we might improve the performance further 14 ." ]

[ "WordNet" ]

{ "title": "Composing Distributed Representations of Relational Patterns", "abstract": "Learning distributed representations for relation instances is a central technique in downstream NLP applications. In order to address semantic modeling of relational patterns, this paper constructs a new dataset that provides multiple similarity ratings for every pair of relational patterns on the existing dataset (Zeichner et al., 2012) . In addition, we conduct a comparative study of different encoders including additive composition, RNN, LSTM, and GRU for composing distributed representations of relational patterns. We also present Gated Additive Composition, which is an enhancement of additive composition with the gating mechanism. Experiments show that the new dataset does not only enable detailed analyses of the different encoders, but also provides a gauge to predict successes of distributed representations of relational patterns in the relation classification task." }

{ "title": "Task-Oriented Learning of Word Embeddings for Semantic Relation Classification", "abstract": "We present a novel learning method for word embeddings designed for relation classification. Our word embeddings are trained by predicting words between noun pairs using lexical relation-specific features on a large unlabeled corpus. This allows us to explicitly incorporate relationspecific information into the word embeddings. The learned word embeddings are then used to construct feature vectors for a relation classification model. On a well-established semantic relation classification task, our method significantly outperforms a baseline based on a previously introduced word embedding method, and compares favorably to previous state-ofthe-art models without syntactic information or manually constructed external resources. Furthermore, when incorporating external resources, our method outperforms the previous state of the art." }