tensor norms and the classical communication complexity of nonlocal quantum measurement we initiate the study of quantifying nonlocality of a bipartite measurement by the minimum amount of classical communication required to simulate the measurement we derive general upper bounds in terms of some tensor norms of the measurement operator as applications we show that a if the amount of communication is a constant then quantum and classical communication protocols with an unlimited amount of shared entanglement or shared randomness compute the same class of functions and b it requires only a constant amount of communication to classically generate an approximation of the output distribution resulting from local measurements on an entangled quantum state as long as the number of measurement outcomes is a constant

quantum finite automata we first outline main steps and achievements along bertoni s research path in quantum finite automata theory from the very basic definitions of the models of quantum finite automata throughout the investigation of their computational and descriptional power next we choose to focus on bertoni s studies on quantum finite automata descriptional complexity in particular we expand on a statistical framework for the synthesis of succinct quantum finite automata discussing its adaptation to the case of multiperiodic events and languages we then improve such a framework to obtain even more succinct quantum finite automata for some multiperiodic languages finally we introduce some promise problems for multiperiodic inputs showing that even on this class of problems the descriptional power of quantum finite automata greatly outperforms that of equivalent classical finite automata we build small size monte carlo quantum finite automata for multiperiodic languages we analyze the modular architecture of such monte carlo quantum finite automata we solve promise problems on multiperiodic inputs by small quantum finite automata

fault tolerant quantum computation with constant error rate this paper shows that quantum computation can be made fault tolerant against errors and inaccuracies when eta the probability for an error in a qubit or a gate is smaller than a constant threshold eta c this result improves on shor s result proceedings of the 37th symposium on the foundations of computer science ieee los alamitos ca 1996 pp 56 65 which shows how to perform fault tolerant quantum computation when the error rate eta decays polylogarithmically with the size of the computation an assumption which is physically unreasonable the cost of making the quantum circuit fault tolerant in our construction is polylogarithmic in time and space our result holds for a very general local noise model which includes probabilistic errors decoherence amplitude damping depolarization and systematic inaccuracies in the gates moreover we allow exponentially decaying correlations between the errors both in space and in time fault tolerant computation can be performed with any universal set of gates the result also holds for quantum particles with p 2 states namely p qudits and is also generalized to one dimensional quantum computers with only nearest neighbor interactions no measurements or classical operations are required during the quantum computation we estimate the threshold of our construction to be eta c simeq 10 6 in the best case by this we show that local noise is in principle not an obstacle for scalable quantum computation the main ingredient of our proof is the computation on states encoded by a quantum error correcting code qecc to this end we introduce a special class of calderbank shor steane css codes called polynomial codes the quantum analogue of reed solomon codes their nice algebraic structure allows all of the encoded gates to be transversal we also provide another version of the proof which uses more general css codes but its encoded gates are slightly less elegant to achieve fault tolerance we encode the quantum circuit by another circuit by using one of these qeccs this step is repeated polyloglog many times each step slightly improving the effective error rate to achieve the desired reliability the resulting circuit exhibits a hierarchical structure and for the analysis of its robustness we borrow terminology from khalfin and tsirelson found phys 22 1992 pp 879 948 and gacs advances in computing research a research annual randomness and computation jai press greenwich ct 1989 the paper is to a large extent self contained in particular we provide simpler proofs for many of the known results we use such as the fact that it suffices to correct for bit flips and phase flips the correctness of css codes and the fact that two qubit gates are universal together with their extensions to higher dimensional particles we also provide full proofs of the universality of the sets of gates we use the proof of universality was missing in shor s paper this paper thus provides a self contained and complete proof of universal fault tolerant quantum computation in the presence of local noise

a crossed dipole reciprocity paradox the principle of seismic reciprocity states that the roles of sources and receivers in seismic experiments can be interchanged and the same signal as a function of time will be recorded in both cases in a multicomponent zero offset shear wave experiment the sources and receivers are at the same location and the reciprocal experiment is the same as the original reciprocity thus predicts that the recorded xy inline source to crossline receiver and yx crossline source to inline receiver sections should be identical this should be true regardless of anisotropy attenuation or heterogeneity in the earth

complexity of solving nonlinear equations in the deterministic randomized and quantum settings we consider the root finding of a real valued function f defined on the d dimensional unit cube we assume that f has r continuous partial derivatives with all partial derivatives of order r being holder functions with the exponent r we study the e complexity of this problem in three settings deterministic randomized and quantum it is known that with the root error criterion the deterministic e complexity is infinite i e the problem is unsolvable we show that the same holds in the randomized and quantum settings under the residual error criterion we show that the deterministic and randomized e complexity is of order e d r r in the quantum setting the e complexity is shown to be of order e d 2 r r this means that a quadratic speed up is achieved on a quantum computer

quantum de finetti theorems under local measurements with applications quantum de finetti theorems are a useful tool in the study of correlations in quantum multipartite states in this paper we prove two new quantum de finetti theorems both showing that under tests formed by local measurements in each of the subsystems one can get an exponential improvement in the error dependence on the dimension of the subsystems we also obtain similar results for non signaling probability distributions we give several applications of the results to quantum complexity theory polynomial optimization and quantum information theory the proofs of the new quantum de finetti theorems are based on information theory in particular on the chain rule of mutual information the results constitute improvements and generalizations of a recent de finetti theorem due to brandao christandl and yard

product formulas for exponentials of commutators we provide a recursive method for systematically constructing product formula approximations to exponentials of commutators giving approximations that are accurate to arbitrarily high order using these formulas we show how to approximate unitary exponentials of possibly nested commutators using exponentials of the elementary operators and we upper bound the number of elementary exponentials needed to implement the desired operation within a given error tolerance by presenting an algorithm for quantum search using evolution according to a commutator we show that the scaling of the number of exponentials in our product formulas with the evolution time is nearly optimal finally we discuss applications of our product formulas to quantum control and to implementing anticommutators providing new methods for simulating many body interaction hamiltonians

multipartite nonlocality as a resource and quantum correlations having indefinite causal order the characterization of quantum correlations in terms of information theoretic resource has been a fruitful approach to understand the power of quantum correlations as a resource while bipartite entanglement and bell inequality violation in this setting have been extensively studied relatively little is known about their multipartite counterpart in this paper we apply and adapt the recently proposed definitions of multipartite nonlocality bancal et al 2013 phys rev a 88 014102 to the three and four partite scenarios to gain new insight on the resource aspect of multipartite nonlocal quantum correlations specifically we show that reproducing certain tripartite quantum correlations requires mixtures of classical resources be it the ability to change the groupings or the time orderings of measurements thus when seen from the perspective of biseparable one way classical signaling resources certain tripartite quantum correlations do not admit a definite causal order in the four partite scenario we obtain a superset description of the set of biseparable correlations which can be produced by allowing two groups of bipartite non signaling resources quantum violation of the resulting bell like inequalities is investigated as a byproduct we obtain some new examples of device independent witnesses for genuine four partite entanglement and also device independent witnesses that allow one to infer the structure of the underlying multipartite entanglement this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

quantum entanglement and the communication complexity of the inner product function we consider the communication complexity of the binary inner product function in a variation of the two party scenario where the parties have an a priori supply of particles in an entangled quantum state we prove linear lower bounds for both exact protocols as well as for protocols that determine the answer with bounded error probability our proofs employ a novel kind of quantum reduction from a quantum information theory problem to the problem of computing the inner product the communication required for the former problem can then be bounded by an application of holevo s theorem we also give a specific example of a probabilistic scenario where entanglement reduces the communication complexity of the inner product function by one bit

on the distributional complexity of disjointness we prove that the distributional communication complexity of the predicate disjointness with respect to a very simple measure on inputs is n

a monte carlo study of polymer adsorption random copolymers and random surfaces we consider a model of random copolymer adsorption in which an n edge self avoiding walk in three dimensions interacts with a plane defining a half space to which the walk is confined each vertex of the walk is randomly labelled a with probability p or b with probability 1 p and only vertices labelled a are attracted to the surface plane the system is quenched i e the labelling is fixed and then the thermodynamic properties are computed we use monte carlo methods to investigate the behaviour of this system we observe self averaging of the energy as n increases and investigate the location of the adsorption transition for various values of p in addition we compare the behaviour of this system with that of a homopolymer adsorbing at a randomly heterogeneous surface consisting of two types of sites only one of which interacts with the monomers of the polymer

pulling polymers adsorbed on a striped surface we consider the adsorption of a linear homopolymer on a striped surface where the monomers have different energies of interaction with each of the two kinds of stripe in addition the polymer is subject to a force that tends to desorb the polymer and the force can be applied at different angles we consider both a partially directed walk model and a self avoiding walk model we solve the problem for the partially directed walk problem essentially completely and give phase diagrams in the force temperature plane for various values of the direction in which the force is applied the phase diagrams show re entrance phenomena and for some force directions degenerate ground states we also present a low temperature treatment that helps to interpret the results from the combinatorial solution for the self avoiding walk model we present monte carlo results and show that the phase diagrams for the partially directed walk and self avoiding walk are qualitatively similar and at low temperatures are in quantitative agreement

quantum and classical complexity classes separations collapses and closure properties we study the complexity of quantum complexity classes such as eqp bqp and nqp quantum analogs of p bpp and np respectively using classical complexity classes such as zpp wpp and c p the contributions of this paper are threefold first via oracle constructions we show that no relativizable proof technique can improve the best known classical upper bound for bqp bqp awpp journal of computer and system sciences 59 2 1999 240 to bqp wpp and the best known classical lower bound for eqp p eqp to zpp eqp second we prove that there are oracles a and b such that relative to a corp is immune to nqp and relative to b bqp is immune to pc p extending a result of de graaf and valiant technical report quant ph 0211179 quantum physics 2002 we construct a relativized world where eqp is immune to modpkp third motivated by the fact that counting classes e g lwpp awpp etc are the best known classical upper bounds on quantum complexity classes we study properties of these counting classes we prove that wpp is closed under polynomial time truth table reductions while we construct an oracle relative to which wpp is not closed under polynomial time turing reductions the latter result implies that proving the equality of the similar appearing classes lwpp and wpp would require nonrelativizable proof techniques we also prove that both awpp and app are closed under tup reductions we use closure properties of wpp and awpp to prove interesting consequences in terms of the complexity of the polynomial hierarchy of the following hypotheses nqp bqp and eqp nqp

quantum computational networks the theory of quantum computational networks is the quantum generalization of the theory of logic circuits used in classical computing machines quantum gates are the generalization of classical logic gates a single type of gate the universal quantum gate together with quantum unit wires is adequate for constructing networks with any possible quantum computational property

splitting information securely with entanglement although many quantum security tools have been proposed many of them are quantum secret key distribution protocols however we naturally wish to construct quantum cryptosystems different from the conventional cryptosystems in this paper first we show simple methods with prior entanglement in order to split a party s message among some parties securely moreover as an application we also show that by using our splitting methods we can construct not only quantum secret key distribution protocols but also quantum cryptosystems some of our quantum cryptosystems are different from the conventional cryptosystems because they are executed interactively

self avoiding walks interacting with a surface discusses two models of polymer adsorption in one model a self avoiding walk on a d dimensional lattice interacts with a d 1 dimensional hyperplane in the other model this walk must also lie in or on one side of this hyperplane both models exhibit non analytic behaviour corresponding to a phase transition but these phase transitions do not occur at the same point the authors give numerical estimates of the locations of these transitions

on the power of non local boxes a non local box is a virtual device that has the following property given that alice inputs a bit at her end of the device and that bob does likewise it produces two bits one at alice s end and one at bob s end such that the xor of the outputs is equal to the and of the inputs this box inspired from the chsh inequality was first proposed by popescu and rohrlich to examine the question given that a maximally entangled pair of qubits is non local why is it not maximally non local we believe that understanding the power of this box will yield insight into the non locality of quantum mechanics it was shown recently by cerf gisin massar and popescu that this imaginary device is able to simulate correlations from any measurement on a singlet state here we show that the non local box can in fact do much more through the simulation of the magic square pseudo telepathy game and the mermin ghz pseudo telepathy game we show that the non local box can simulate quantum correlations that no entangled pair of qubits can in a bipartite scenario and even in a multi party scenario finally we show that a single non local box cannot simulate all quantum correlations and propose a generalization for a multi party non local box in particular we show quantum correlations whose simulation requires an exponential amount of non local boxes in the number of maximally entangled qubit pairs

approximating fractional time quantum evolution an algorithm is presented for approximating the arbitrary powers of a black box unitary operation where t is a real number and is a black box implementing an unknown unitary the complexity of this algorithm is calculated in terms of the number of calls to the black box the errors in the approximation and a certain gap parameter for general and large t one should apply a total of t times followed by our procedure for approximating the fractional power an example is also given where for large integers t this method is more efficient than direct application of t copies of further applications and related algorithms are also discussed

strengths and weaknesses of quantum computing recently a great deal of attention has been focused on quantum computation following a sequence of results bernstein and vazirani in proc 25th annual acm symposium theory comput 1993 pp 11 20 siam j comput 26 1997 pp 1277 1339 simon in proc 35th annual ieee symposium foundations comput sci 1994 pp 116 123 siam j comput 26 1997 pp 1340 1349 shor in proc 35th annual ieee symposium foundations comput sci 1994 pp 124 134 suggesting that quantum computers are more powerful than classical probabilistic computers following shor s result that factoring and the extraction of discrete logarithms are both solvable in quantum polynomial time it is natural to ask whether all of np can be efficiently solved in quantum polynomial time in this paper we address this question by proving that relative to an oracle chosen uniformly at random with probability 1 the class np cannot be solved on a quantum turing machine qtm in time o 2 n 2 we also show that relative to a permutation oracle chosen uniformly at random with probability 1 the class np cap conp cannot be solved on a qtm in time o 2 n 3 the former bound is tight since recent work of grover in it proc 28 th annual acm symposium theory comput 1996 shows how to accept the class np relative to any oracle on a quantum computer in time o 2 n 2

a globalized newton method for the accurate solution of a dipole quantum control problem a theoretical and computational framework is presented to obtain accurate controls for fast quantum state transitions that are needed in a host of applications such as nanoelectronic devices and quantum computing this method is based on a reduced hessian krylov newton scheme applied to a norm preserving discrete model of a dipole quantum control problem the use of second order numerical methods for solving the control problem is justified proving the existence of optimal solutions and analyzing first and second order optimality conditions criteria for the discretization of the nonconvex optimization problem and for the formulation of the hessian are given to ensure accurate gradients and a symmetric hessian robustness of the newton approach is obtained using a globalization strategy with a robust linesearch procedure results of numerical experiments demonstrate that the newton approach presented in this paper is able to provide fast and accurate controls for high energy state transitions

a self avoiding walk model of random copolymer adsorption there is an error in the proof of lemma 3 4 which can be remedied by using 3 22 and 3 13 directly the proof of lemma 3 2 can also be simplified please see pdf for details

discrete quantum walks hit exponentially faster this paper addresses the question what processes take polynomial time on a quantum computer that require exponential time classically we show that the hitting time of the discrete time quantum walk on the n bit hypercube from one corner to its opposite is polynomial in n this gives the first exponential quantum classical gap in the hitting time of discrete quantum walks we provide the basic framework for quantum hitting time and give two alternative definitions to set the ground for its study on general graphs we outline a possible application to sequential packet routing

counting by quantum eigenvalue estimation for every computation there corresponds the physical task of manipulating a starting state into an output state with a desired property as the classical theory of physics has been replaced by quantum physics it is interesting to consider the capabilities of a computer that can exploit the distinctive quantum features of nature the extra capabilities seem enormous for example with only an expected o square rootn evaluations of a function f 0 1 n 1 0 1 we can find a solution to f x 1 provided one exists another example is the ability to 1nd efficiently the order of an element g in a group by using a quantum computer to estimate a random eigenvalue of the unitary operator that multiplies by g in the group by using this eigenvalue estimation algorithm to estimate an eigenvalue of the unitary operator used in quantum searching we can approximately count the number of solutions to f x 1 this paper describes this eigenvector approach to quantum counting and related algorithms

hidden symmetry subgroup problems we advocate a new approach for addressing hidden structure problems and finding efficient quantum algorithms we introduce and investigate the hidden symmetry subgroup problem hssp which is a generalization of the well studied hidden subgroup problem hsp given a group acting on a set and an oracle whose level sets define a partition of the set the task is to recover the subgroup of symmetries of this partition inside the group the hssp provides a unifying framework that besides the hsp encompasses a wide range of algebraic oracle problems including quadratic hidden polynomial problems while the hssp can have provably exponential quantum query complexity we obtain efficient quantum algorithms for various interesting cases to achieve this we present a general method for reducing the hssp to the hsp which works efficiently in several cases related to symmetries of polynomials the hssp therefore connects in a rather surprising way certain hidden polynomial problems with the hsp using this co

nondeterministic quantum query and communication complexities we study nondeterministic quantum algorithms for boolean functions f such algorithms have positive acceptance probability on input x iff f x 1 in the setting of query complexity we show that the nondeterministic quantum complexity of a boolean function is equal to its nondeterministic polynomial degree we also prove a quantum vs classical gap of 1 vs n for nondeterministic query complexity for a total function in the setting of communication complexity we show that the nondeterministic quantum complexity of a two party function is equal to the logarithm of the rank of a nondeterministic version of the communication matrix this implies that the quantum communication complexities of the equality and disjointness functions are n 1 if we do not allow any error probability we also exhibit a total function in which the nondeterministic quantum communication complexity is exponentially smaller than its classical counterpart

an optical model of computation we prove computability and complexity results for an original model of computation called the continuous space machine our model is inspired by the theory of fourier optics we prove our model can simulate analog recurrent neural networks thus establishing a lower bound on its computational power we also define a log2n unordered search algorithm with our model

lower bounds of a quantum search for an extreme point we show that durr and hoyer9s quantum algorithm of searching for an extreme point of an integer function cannot be sped up for functions that are chosen randomly any other algorithm acting in a substantially shorter time o 2 n n gives an incorrect answer for the functions phi with the single point of maximum chosen randomly with probability p err 1 the lower bound as 2 n b is established for the quantum search for solution of the equation f x 1 where f is a boolean function with b such solutions chosen at random with asymptotic probability 1 n

cryptographic hardness based on the decoding of reed solomon codes in this paper we investigate the decoding problem of reed solomon rs codes also known as the polynomial reconstruction problem pr from a cryptographic hardness perspective namely we deal with samplable pr instances over parameter choices for which decoding is not known to be feasibly solvable and where part of the solution polynomial is the hidden input we put forth a natural decisional intractability assumption that relates to this decoding problem distinguishing between a single randomly chosen error location and a single randomly chosen nonerror location for a given corrupted rs codeword with random noise we prove that under this assumption pr instances are entirely pseudorandom i e they are indistinguishable from random vectors over the underlying finite field moreover under the same assumption we show that it is hard to extract any partial information related to the hidden input encoded by the corrupted pr instance i e pr instances hide their message polynomial solution in the semantic security sense the above results lay a framework for the exploitation of pr as an intractability assumption for provable security of cryptographic primitives based on this framework we present provably secure cryptographic constructions for 1 a pseudorandom number generator 2 a semantically secure version of the oblivious polynomial evaluation ope protocol and 3 a stateful cipher with a set of interesting properties that include semantic security forward secrecy error correcting decryption and an array of random self reducibility properties with respect to the plaintext choice key choice and partial domain choice

quantum existence testing and its application for finding extreme values in unsorted databases many information processing and computing problems can be traced back to find the extreme value of a database or a function unfortunately classical solutions suffer from high computational complexity if the database is unsorted or equivalently the function has many local minimum maximum points proposed quantum computing based solutions involve the repeated application of grower s searching algorithm in this paper we introduce a new technique exploiting the parallel processing capabilities of quantum computing in a different way we derive a special case of quantum counting we call it quantum existence testing which allows adapting the classical logarithmic search algorithm so that it is suitable for structured databases to unstructured ones the paper analyzes the required number of database queries the corresponding computational complexity and the probability of error and their relationship

unbiased bits from sources of weak randomness and probabilistic communication complexity a new model for weak random physical sources is presented the new model strictly generalizes previous models e g the santha and vazirani model 27 the sources considered output strings according to probability distributions in which no single string is too probable the new model provides a fruitful viewpoint on problems studied previously such as extracting almost perfect bits from sources of weak randomness the question of possibility as well as the question of efficiency of such extraction schemes are addressed probabilistic communication complexity it is shown that most functions have linear communication complexity in a very strong probabilistic sense robustness of bpp with respect to sources of weak randomness generalizing a result of vazirani and vazirani 32 33

spectral gap amplification many problems can be solved by preparing a specific eigenstate of some hamiltonian h the generic cost of quantum algorithms for these problems is determined by the inverse spectral gap of h for that eigenstate and the cost of evolving with h for some fixed time the goal of spectral gap amplification is to construct a hamiltonian h with the same eigenstate as h but a bigger spectral gap requiring that constant time evolutions with h and h are implemented with nearly the same cost we show that a quadratic spectral gap amplification is possible when h satisfies a frustration free property and give h for these cases this results in quantum speedups for optimization problems it also yields improved constructions for adiabatic simulations of quantum circuits and for the preparation of projected entangled pair states which play an important role in quantum many body physics defining a suitable black box model we establish that the quadratic amplification is optimal for frustration f

quantum lower bounds by polynomials we examine the number of queries to input variables that a quantum algorithm requires to compute boolean functions on 0 1 n in the black box model we show that the exponential quantum speed up obtained for partial functions i e problems involving a promise on the input by deutsch and jozsa simon and shor cannot be obtained for any total function if a quantum algorithm computes some total boolean function f with small error probability using t black box queries then there is a classical deterministic algorithm that computes f exactly with o ts6 queries we also give asymptotically tight characterizations of t for all symmetric f in the exact zero error and bounded error settings finally we give new precise bounds for and or and parity our results are a quantum extension of the so called polynomial method which has been successfully applied in classical complexity theory and also a quantum extension of results by nisan about a polynomial relationship between randomized and deterministic decision tree complexity

determining acceptance possibility for a quantum computation is hard for the polynomial hierarchy abstract it is shown that determining whether a quantum computation has a non zero probability of accepting is at least as hard as the polynomial time hierarchy this hardness result also applies to determining in general whether a given quantum basis state appears with nonzero amplitude in a superposition or whether a given quantum bit has positive expectation value at the end of a quantum computation this result is achieved by showing that the complexity class nqp of adleman demarrais and huang a quantum analog of np is equal to the counting class co c equals p

threesomes degenerates and love triangles the 3sum problem is to decide given a set of n real numbers whether any three sum to zero it is widely conjectured that a trivial o n2 time algorithm is optimal on the real ram and optimal even in the nonuniform linear decision tree model over the years the consequences of this conjecture have been revealed this 3sum conjecture implies n2 lower bounds on numerous problems in computational geometry and a variant of the conjecture for integer inputs implies strong lower bounds on triangle enumeration dynamic graph algorithms and string matching data structures in this article we refute the conjecture that 3sum requires n2 in the real ram and refute more forcefully the conjecture that its complexity is n2 in the linear decision tree model in particular we prove that the decision tree complexity of 3sum is o n3 2s log n and give two subquadratic 3sum algorithms a deterministic one running in o n2 log n log log n 2 3 time and a randomized one running in o n2 log log n 2 log n time with high probability our results lead directly to improved bounds on the decision tree complexity of k variate linear degeneracy testing for all odd k 3 finally we give a subcubic algorithm for a generalization of the min product over real valued matrices and apply it to the problem of finding zero weight triangles in edge weighted graphs we give a depth o n5 2s log n decision tree for this problem as well as a deterministic algorithm running in time o n3 log log n 2 log n

unbounded violation of tripartite bell inequalities we prove that there are tripartite quantum states constructed from random unitaries that can lead to arbitrarily large violations of bell inequalities for dichotomic observables as a consequence these states can withstand an arbitrary amount of white noise before they admit a description within a local hidden variable model this is in sharp contrast with the bipartite case where all violations are bounded by grothendieck s constant we will discuss the possibility of determining the hilbert space dimension from the obtained violation and comment on implications for communication complexity theory moreover we show that the violation obtained from generalized greenberger horne zeilinger ghz states is always bounded so that in contrast to many other contexts ghz states do not lead to extremal quantum correlations in this case in order to derive all these physical consequences we will have to obtain new mathematical results in the theories of operator spaces and tensor norms in particular we will prove the existence of bounded but not completely bounded trilinear forms from commutative c algebras finally we will relate the existence of diagonal states leading to unbounded violations with a long standing open problem in the context of banach algebras

quantum walk algorithm for element distinctness we use quantum walks to construct a new quantum algorithm for element distinctness and its generalization for element distinctness the problem of finding two equal items among n given items we get an o n 2 3 query quantum algorithm this improves the previous o n 3 4 quantum algorithm of buhrman et al siam j comput 34 2005 pp 1324 1330 and matches the lower bound of aaronson and shi j acm 51 2004 pp 595 605 we also give an o n k k 1 query quantum algorithm for the generalization of element distinctness in which we have to find k equal items among n items

alford rotation ray theory and crossed dipole geometry two generalizations of alford rotation have been proposed for processing 2 2 component data containing nonorthogonal split shear waves singular value decomposition svd and eigenvector eigenvalue decomposition eed using a simple crossed dipole synthetic model we demonstrate that the physical model behind the eed method is invalid it incorrectly assumes that a vector source aligned with the particle motion of an anisotropic pure mode will excite only that one mode ray theory shows that a vector point force source embedded in a homogeneous anisotropic medium instead excites all those modes with particle motions that are not perpendicular to the direction of the applied force just as a vector point receiver detects all modes with polarizations that are not perpendicular to the receiver correctly generalized alford rotation synthesizes vector sources and receivers such that each component is perpendicular to all but one of the pure modes of the medium although this ray theory result does not allo

quantum communication and complexity in the setting of communication complexity two distributed parties want to compute a function depending on both their inputs using as little communication as possible the required communication can sometimes be significantly lowered if we allow the parties the use of quantum communication we survey the main results of the young area of quantum communication complexity its relation to teleportation and dense coding the main examples of fast quantum communication protocols lower bounds and some applications

quantum query complexity of some graph problems quantum algorithms for graph problems are considered both in the adjacency matrix model and in an adjacency list like array model we give almost tight lower and upper bounds for the bounded error quantum query complexity of connectivity strong connectivity minimum spanning tree and single source shortest paths for example we show that the query complexity of minimum spanning tree is in theta n 3 2 in the matrix model and in theta sqrt nm in the array model while the complexity of connectivity is also in theta n 3 2 in the matrix model but in theta n in the array model the upper bounds utilize search procedures for finding minima of functions under various conditions

one complexity theorist s view of quantum computing the complexity of quantum computation remains poorly understood while physicists attempt to find ways to create quantum computers we still do not have much evidence one way or the other as to how useful these machines will be the tools of computational complexity theory should come to bear on these important questions quantum computing often scares away many potential researchers in computer science because of the apparent background need in quantum mechanics and the alien looking notation used in papers on the topic this paper will give an overview of quantum computation from the point of view of a complexity theorist we will see that one can think of bqp as yet another complexity class and study its power without focusing on the physical aspects behind it

an exact solution of two friendly interacting directed walks near a sticky wall we find the exact solution of two interacting friendly directed walks modelling polymers on the square lattice these walks are confined to the quarter plane by a horizontal attractive surface to capture the effects of dna denaturation and adsorption we find the solution to the model s corresponding generating function by means of the obstinate kernel method specifically we apply this technique in two different instances to establish partial solutions for two simplified generating functions of the same underlying model that ignore either surface or shared contacts we then subsequently combine our two partial solutions to find the solution for the full generating function in terms of the two simpler variants this expression guides our analysis of the model where we find the system exhibits four phases and proceed to delineate the full phase diagram showing that all observed phase transitions are second order

quantum speedup of monte carlo methods monte carlo methods use random sampling to estimate numerical quantities which are hard to compute deterministically one important example is the use in statistical physics of rapidly mixing markov chains to approximately compute partition functions in this work we describe a quantum algorithm which can accelerate monte carlo methods in a very general setting the algorithm estimates the expected output value of an arbitrary randomized or quantum subroutine with bounded variance achieving a near quadratic speedup over the best possible classical algorithm combining the algorithm with the use of quantum walks gives a quantum speedup of the fastest known classical algorithms with rigorous performance bounds for computing partition functions which use multiple stage markov chain monte carlo techniques the quantum algorithm can also be used to estimate the total variation distance between probability distributions efficiently

magnetic lieb thirring inequalities we study the generalizations of the well known lieb thirring inequality for the magnetic schrodinger operator with nonconstant magnetic field our main result is the naturally expected magnetic lieb thirring estimate on the moments of the negative eigenvalues for a certain class of magnetic fields including even some unbounded ones we develop a localization technique in path space of the stochastic feynman kac representation of the heat kernel which effectively estimates the oscillatory effect due to the magnetic phase factor

a monte carlo study of adsorption of random copolymers on random surfaces we study the adsorption problem of a random copolymer on a random surface in which a self avoiding walk in three dimensions interacts with a plane defining a half space to which the walk is confined each vertex of the walk is randomly labelled a with probability pp or b with probability 1 pp and only vertices labelled a are attracted to the surface plane each lattice site on the plane is also labelled either a with probability ps or b with probability 1 ps and only lattice sites labelled a interact with the walk we study two variations of this model in the first case the a vertices of the walk interact only with the a sites on the surface in the second case the constraint of selective binding is removed that is any contact between the walk and the surface that involves an a labelling either from the surface or from the walk is counted as a visit to the surface the system is quenched in both cases i e the labellings of the walk and of the surface are fixed as thermodynamic properties are computed we present monte carlo simulation results which provide evidence for second order transitions in both cases we observe that in both cases the adsorption location depends on both ps and pp the dilution of the interactive sites on the surface and the walk we compare critical properties for the two cases and find that the adsorption location varies for different models while the crossover exponent is independent of the details of the models

quantum entanglement and communication complexity we consider a variation of the communication complexity scenario where the parties are supplied with an extra resource particles in an entangled quantum state we note that quantum nonlocality can be naturally expressed in the language of communication complexity these are communication complexity problems where the output is embodied in the correlations between the outputs of the individual parties without entanglement the parties must communicate to produce the required correlations whereas with entanglement no communication is necessary to produce the correlations in this sense nonlocality proofs can also be viewed as communication complexity problems where the presence of quantum entanglement reduces the amount of necessary communication we show how to transform examples of nonlocality into more traditional communication complexity problems where the output is explicitly determined by each individual party the resulting problems require communication with or without entanglement but the required communication is less when entanglement is available all these results are a noteworthy contrast to the well known fact that entanglement cannot be used to actually simulate or compress classical communication between remote parties

statistical mechanics of directed models of polymers in the square lattice directed square lattice models of polymers and vesicles have received considerable attention in the recent mathematical and physical sciences literature these are idealized geometric directed lattice models introduced to study phase behaviour in polymers and include dyck paths partially directed paths directed trees and directed vesicles models directed models are closely related to models studied in the combinatorics literature and are often exactly solvable they are also simplified versions of a number of statistical mechanics models including the self avoiding walk lattice animals and lattice vesicles the exchange of approaches and ideas between statistical mechanics and combinatorics have considerably advanced the description and understanding of directed lattice models and this will be explored in this review the combinatorial nature of directed lattice path models makes a study using generating function approaches most natural in contrast the statistical mechanics approach would introduce partition functions and free energies and then investigate these using the general framework of critical phenomena generating function and statistical mechanics approaches are closely related for example questions regarding the limiting free energy may be approached by considering the radius of convergence of a generating function and the scaling properties of thermodynamic quantities are related to the asymptotic properties of the generating function in this review the methods for obtaining generating functions and determining free energies in directed lattice path models of linear polymers is presented these methods include decomposition methods leading to functional recursions as well as the temperley method that is implemented by creating a combinatorial object one slice at a time a constant term formulation of the generating function will also be reviewed the thermodynamic features and critical behaviour in models of directed paths may be informative about the underlying properties that determine phase diagrams for wider classes of models including physical models of polymers of particular interest are adsorption and collapse transitions in models of polymers and copolymers the properties of thermodynamic quantities in those models are described by tricritical scaling this is reviewed for directed path models and the generating function approaches can be used to apply tricritical scaling to models of adsorbing inflating and collapsing directed lattice paths critical exponents for a variety of models can be obtained in this manner and with it a better understanding and a classification of the models

limitations of quantum coset states for graph isomorphism it has been known for some time that graph isomorphism reduces to the hidden subgroup problem hsp what is more most exponential speedups in quantum computation are obtained by solving instances of the hsp a common feature of the resulting algorithms is the use of quantum coset states which encode the hidden subgroup an open question has been how hard it is to use these states to solve graph isomorphism it was recently shown by moore et al 2005 that only an exponentially small amount of information is available from one or a pair of coset states a potential source of power to exploit are entangled quantum measurements that act jointly on many states at once we show that entangled quantum measurements on at least n log n coset states are necessary to get useful information for the case of graph isomorphism matching an information theoretic upper bound this may be viewed as a negative result because in general it seems hard to implement a given highly entangled measurement our main theorem is very general and also rules out using joint measurements on few coset states for some other groups such as gl n fpm and gn where g is finite and satisfies a suitable property

eisner s reciprocity paradox and its resolution the principle of reciprocity says that when a vertical source a vibrator and a vertical receiver a geophone are interchanged the same seismogram will be recorded in each case in a field study conducted by d fenati and f rocca in 1984 it was found that the reciprocal principle also applies surprisingly well even when the required conditions are technically violated such as when an isotropic source dynamite is used with a vertical receiver again a geophone

quantum and classical tradeoffs we initiate the study of quantifying the quantumness of a quantum circuit by the number of gates that do not preserve the computational basis as a means to understand the nature of quantum algorithmic speedups intuitively a reduction in the quantumness requires an increase in the amount of classical computation thus giving a quantum and classical tradeoff in this paper we present two results on this measure of quantumness the first gives almost matching upper and lower bounds on the question what is the minimum number of non basis preserving gates required to generate a good approximation to a given state this question is the quantum analogy of the following classical question how many fair coins are needed to generate a given probability distribution which was studied and resolved by knuth and yao in 1976 algorithms and complexity new directions and recent results academic press new york 1976 pp 357 428 our second result shows that any quantum algorithm that solves grover s problem of size n using k queries and l levels of non basis preserving gates must have kl n

quantum lower bounds for the collision and the element distinctness problems given a function f as an oracle the collision problem is to find two distinct indexes i and j such that f i f j under the promise that such indexes exist since the security of many fundamental cryptographic primitives depends on the hardness of finding collisions our lower bounds provide evidence for the existence of cryptographic primitives that are immune to quantum cryptanalysis we prove that any quantum algorithm for finding a collision in an r to one function must evaluate the function n r 1 3 times where n is the size of the domain and r n this matches an upper bound of brassard hoyer and tapp no lower bound better than constant was previously known our result also implies a quantum lower bound of n2 3 queries for the element distinctness problem which is to determine whether n integers are all distinct the best previous lower bound was sn queries

analogies and differences between quantum and stochastic automata abstract we analyze some features of the behaviour of quantum automata providing analogies and differences with the corresponding stochastic models in particular we prove there is a quantum automaton where the change of state depends on unitary transformations defined by matrices with nonnull amplitudes that accepts a non regular language with cut point zero and inverse error polynomially bounded stochastic automata with matrices having nonnull elements and with polynomial bounds on the inverse error recognize only regular languages the class of stochastic languages contains the class of quantum languages quantum languages are empty or contain an infinite number of words the class of quantum languages is not closed under complementation

quantum functional oracles the limitation on the size of quantum computers makes it important to reuse qubits for auxiliary registers even though they are entangled with others and are occupied by other computational processes we construct a quantum algorithm that performs the functional phase rotation which is the generalized form of the conventional conditional phase transforms using the functional evaluation oracle the constructed algorithm works without any a priori knowledge of the state of an auxiliary register at the beginning and it recovers the initial state of an auxiliary register at the end this provides ample scope to choose qubits for auxiliary registers at will

a fast algorithm for approximating the ground state energy on a quantum computer estimating the ground state energy of a multiparticle system with relative error using deterministic classical algorithms has cost that grows exponentially with the number of particles the problem depends on a number of state variablesd that is proportional to the number of particles and suers from the curse of dimensionality quantum computers can vanquish this curse in particular we study a ground state eigenvalue problem and exhibit a quantum algorithm that achieves relative error using a number of qubits c 0 d log 1 with total cost number of queries plus other quantum operations cd 3 where 0 is arbitrarily small and c and c 0 are inde

an oracle builder s toolkit we show how to use various notions of genericity as tools in oracle creation in particular 1 we give an abstract definition of genericity that encompasses a large collection of different generic notions 2 we consider a new complexity class awpp which contains bqp quantum polynomial time and infer several strong collapses relative to sp generics 3 we show that under additional assumptions these collapses also occur relative to cohen generics 4 we show that relative to sp generics ulin co ulin dtime n k for any k where ulin is unambiguous linear time despite the fact that up np co np p relative to these generics 5 we show that there is an oracle relative to which np 1 co np 1 np co np poly and 6 we use a specialized notion of genericity to create an oracle relative to which np bpp ma

rapid solution of problems by quantum computation a class of problems is described which can be solved more efficiently by quantum computation than by any classical or stochastic method the quantum computation solves the problem with certainty in exponentially less time than any classical deterministic computation

the pivot algorithm a highly efficient monte carlo method for the self avoiding walk the pivot algorithm is a dynamic monte carlo algorithm first invented by lal which generates self avoiding walks saws in a canonical fixed n ensemble with free endpoints heren is the number of steps in the walk we find that the pivot algorithm is extraordinarily efficient one effectively independent sample can be produced in a computer time of ordern this paper is a comprehensive study of the pivot algorithm including a heuristic and numerical analysis of the acceptance fraction and autocorrelation time an exact analysis of the pivot algorithm for ordinary random walk a discussion of data structures and computational complexity a rigorous proof of ergodicity and numerical results on self avoiding walks in two and three dimensions our estimates for critical exponents are 0 7496 0 0007 ind 2 and 0 592 0 003 ind 3 95 confidence limits based on saws of lengths 200 n 10000 and 200 n 3000 respectively

classical quantum and nonsignalling resources in bipartite games we study bipartite games that arise in the context of nonlocality with the help of graph theory our main results are alternate proofs that deciding whether a no communication classical winning strategy exists for certain games called forbidden edge and covering games is np complete while the problem of deciding if these games admit a nonsignalling winning strategy is in p we discuss relations between quantum winning strategies and orthogonality graphs we also show that every pseudotelepathy game yields both a proof of the bell kochen specker theorem and an instance of a two prover interactive proof system that is classically sound but that becomes unsound when provers use shared entanglement

the statistical mechanics of random copolymers random copolymers are polymers with two or more types of monomer where the monomer sequence is determined by some random process once determined the sequence is fixed so random copolymers are an example of a system with quenched randomness we review the statistical mechanics of random copolymers focusing on self avoiding walk models where there are two types of monomers a and b which are randomly distributed along the polymer chain theoretical approximate and numerical results are reviewed for models of the random copolymer adsorption localization and collapse phase transitions we concentrate on what is known about the existence of phase transitions the morita approximation and results about self averaging we also discuss in less detail the replica trick and numerical methods including monte carlo methods exact enumeration and transfer matrix methods important open problems are identified throughout and highlighted in the conclusions

family of bell like inequalities as device independent witnesses for entanglement depth we present a simple family of bell inequalities applicable to a scenario involving arbitrarily many parties each of which performs two binary outcome measurements we show that these inequalities are members of the complete set of full correlation bell inequalities discovered by werner wolf ifmmode dot z else z fi ukowski brukner for scenarios involving a small number of parties we further verify that these inequalities are facet defining for the convex set of bell local correlations moreover we show that the amount of quantum violation of these inequalities naturally manifests the extent to which the underlying system is genuinely many body entangled in other words our bell inequalities when supplemented with the appropriate quantum bounds naturally serve as device independent witnesses for entanglement depth allowing one to certify genuine k partite entanglement in an arbitrary n ensuremath ge k partite scenario without relying on any assumption about the measurements being performed or the dimension of the underlying physical system a brief comparison is made between our witnesses and those based on some other bell inequalities as well as quantum fisher information a family of witnesses for genuine k partite nonlocality applicable to an arbitrary n ensuremath ge k partite scenario based on our bell inequalities is also presented

quantum theory the church turing principle and the universal quantum computer it is argued that underlying the church turing hypothesis there is an implicit physical assertion here this assertion is presented explicitly as a physical principle every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means classical physics and the universal turing machine because the former is continuous and the latter discrete do not obey the principle at least in the strong form above a class of model computing machines that is the quantum generalization of the class of turing machines is described and it is shown that quantum theory and the universal quantum computer are compatible with the principle computing machines resembling the universal quantum computer could in principle be built and would have many remarkable properties not reproducible by any turing machine these do not include the computation of non recursive functions but they do include quantum parallelism a method by which certain probabilistic tasks can be performed faster by a universal quantum computer than by any classical restriction of it the intuitive explanation of these properties places an intolerable strain on all interpretations of quantum theory other than everett s some of the numerous connections between the quantum theory of computation and the rest of physics are explored quantum complexity theory allows a physically more reasonable definition of the complexity or knowledge in a physical system than does classical complexity theory

bounds on the power of proofs and advice in general physical theories quantum theory presents us with the tools for computational and communication advantages over classical theory one approach to uncovering the source of these advantages is to determine how computation and communication power vary as quantum theory is replaced by other operationally defined theories from a broad framework of such theories such investigations may reveal some of the key physical features required for powerful computation and communication in this paper we investigate how simple physical principles bound the power of two different computational paradigms which combine computation and communication in a non trivial fashion computation with advice and interactive proof systems we show that the existence of non trivial dynamics in a theory implies a bound on the power of computation with advice moreover we provide an explicit example of a theory with no non trivial dynamics in which the power of computation with advice is unbounded finally we show that the power of simple interactive proof systems in theories where local measurements suffice for tomography is non trivially bounded this result provides a proof that formula see text is contained in formula see text which does not make use of any uniquely quantum structure such as the fact that observables correspond to self adjoint operators and thus may be of independent interest

average case quantum query complexity we compare classical and quantum query complexities of total boolean functions it is known that for worst case complexity the gap between quantum and classical can be at most polynomial we show that for average case complexity under the uniform distribution quantum algorithms can be exponentially faster than classical algorithms under non uniform distributions the gap can even be super exponential we also prove some general bounds for average case complexity and show that the average case quantum complexity of majority under the uniform distribution is nearly quadratically better than the classical complexity a preliminary version of this paper appeared in the proceedings of the 17th annual symposium on theoretical aspects of computer science stacs 2000 lecture notes in computer science vol 1770 berlin springer

computing algebraic formulas using a constant number of registers it is shown that over an arbitrary ring the functions computed by polynomial size algebraic formulas are also computed by polynomial length algebraic straight line programs that use only three registers this was previously known for boolean formulas d a barrington j comput system sci 38 1989 pp 150 164 which are equivalent to algebraic formulas over the ring gf 2 for formulas over arbitrary rings the result is an improvement over previous methods that require the number of registers to be logarithmic in the size of the formulas in order to obtain polynomial length straight line programs moreover the straight line programs that arise in these constructions have the property that they consist of statements whose actions on the registers are linear and bijective a consequence of this is that the problem of determining the iterated product of n3 times 3 matrices is complete under p projections for algebraic nc 1 also when the ring is gf 2 the programs that arise in the c