on the uselessness of quantum queries 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

height probabilities in solid on solid models i for pt i see ibid vol 19 p l923 1986 the rsos 3 model is a sequence of two dimensional solid on solid models with integer heights restricted between 1 and r 1 and nearest neighbours differing by 0 1 or 1 it has an exact solution manifold with four regimes of distinct physical behaviour along which the height probabilities in the bulk of the system can be calculated using some results of the previous letter in this series the authors calculate the height probabilities for two of the regimes appropriate order parameters are defined and their critical exponents calculated

span programs are equivalent to quantum query algorithms span programs form a linear algebraic model of computation that is studied to prove classical lower bounds quantum query complexity is a coherent generalization for quantum algorithms of decision tree complexity it is characterized by a semidefinite program known as the general adversary bound we connect these classical and quantum models by proving that for any boolean function the optimal witness size of a span program equals the general adversary bound therefore span program witness size and quantum query complexity are equivalent measures in particular quantum algorithms can be designed based on span programs

quantum query complexity of entropy estimation estimation of shannon and renyi entropies of unknown discrete distributions is a fundamental problem in statistical property testing in this paper we give the first quantum algorithms for estimating alpha renyi entropies shannon entropy being 1 renyi entropy in particular we demonstrate a quadratic quantum speedup for shannon entropy estimation and a generic quantum speedup for alpha renyi entropy estimation for all alpha geq 0 values including tight bounds for the shannon entropy the hartley entropy alpha 0 and the collision entropy alpha 2 we also provide quantum upper bounds for estimating min entropy alpha infty as well as the kullback leibler divergence we complement our results with quantum lower bounds on alpha renyi entropy estimation for all alpha geq 0 values our approach is inspired by the pioneering work of bravyi harrow and hassidim bhh however with many new technical ingredients 1 we improve the error dependence of the bhh framework by a fine tuned error analysis together with montanaro s approach to estimating the expected output of quantum subroutines for alpha 0 1 2 we develop a procedure similar to cooling schedules in simulated annealing for general alpha geq 0 and 3 in the cases of integer alpha geq 2 and alpha infty we reduce the entropy estimation problem to the alpha distinctness and lceil log n rceil distinctness problems respectively

exact solubility of the self dual and the string melting points in the restricted solid on solid model the restricted solid on solid model which describes the roughening of two dimensional interfaces is found to be exactly soluble in two instances at a point where the model is self dual under the duality transformation of the five state clock model and at the string melting transition where the model belongs to the same universality class as an ising model on a transverse vibrating lattice both points are located in the subspace of the 19 vertex model solved by zamolodchikov and fateev 1980

an improved lower bound on the sensitivity complexity of graph properties turan 1984 11 initiated the study of the sensitivity complexity of graph properties he conjectured that for any non trivial graph properties on n vertices the sensitivity complexity is at least n 1 he proved an email protected lower bound for sensitivity in his paper turan 1984 11 wegener 1985 12 proved this conjecture for all monotone graph properties in this paper we improve turan s lower bound to 617n 0 35n we hope that this will shed some light on the proof of turan s conjecture

dual lower bounds for approximate degree and markov bernstein inequalities the e approximate degree of a boolean function f 1 1 n 1 1 is the minimum degree of a real polynomial that approximates f to within error e in the norm we prove several lower bounds on this important complexity measure by explicitly constructing solutions to the dual of an appropriate linear program our first result resolves the e approximate degree of the two level and or tree for any constant e 0 we show that this quantity is n closing a line of incrementally larger lower bounds the same lower bound was recently obtained independently by sherstov theory comput 2013 using related techniques our second result gives an explicit dual polynomial that witnesses a tight lower bound for the approximate degree of any symmetric boolean function addressing a question of spalek 2008 our final contribution is to reprove several markov type inequalities from approximation theory by constructing explicit dual solutions to natural linear programs these inequalities underly the proofs of many of the best known approximate degree lower bounds and have important uses throughout theoretical computer science

maximal sensitivity of boolean nested canalizing functions abstract boolean nested canalizing functions ncf have important applications in molecular regulatory networks engineering and computer science in the literature there are two sensitivities to measure the complexity of a boolean function one is average sensitivity the other one is maximal sensitivity we follow the tradition and omit the word maximal in other words in this paper sensitivity is always maximal sensitivity using the past work of the authors and their coauthors on a characterization of ncf we obtain the formula of the sensitivity of any ncf we find that the sensitivity of any ncf is between n 2 2 and n both lower and upper bounds are tight we prove that the block sensitivity hence the l block sensitivity is the same as the sensitivity for ncf it is well known that monotone boolean functions mbf also have this property we characterize all functions which are both monotone and nested canalizing mncf the closed formula of the cardinality of the set of mncfs is also provided

sensitivity block sensitivity and l block sensitivity of boolean functions sensitivity is one of the simplest and block sensitivity one of the most useful invariants of a boolean function nisan siam j comput 20 6 1991 999 and nisan and szegedy comput complexity 4 4 1994 301 have shown that block sensitivity is polynomially related to a number of measures of boolean function complexity the main open question is whether or not a polynomial relationship exists between sensitivity and block sensitivity we define the intermediate notion of l block sensitivity and show that for any fixed l this new quantity is polynomially related to sensitivity we then achieve an improved though still exponential upper bound on block sensitivity in terms of sensitivity as a corollary we also prove that sensitivity and block sensitivity are polynomially related when the block sensitivity is n

exact calculation of the local height probabilities in the body centred sos model the local height probabilities pa and the mean square height average h2 are calculated exactly for the smooth phase of the body centred solid on solid model near criticality pa approximately k t 1 4 and h2 approximately k t 1 2 where t is the deviation from criticality parameter and k k are constants specified in the text

q trinomial identities we obtain connection coefficients between q binomial and q trinomial coefficients using these one can transform q binomial identities into q trinomial identities and back again to demonstrate the usefulness of this procedure we rederive some known trinomial identities related to partition theory and prove many of the conjectures of berkovich mccoy and pearce which have recently arisen in their study of the 2 1 and 1 5 perturbations of minimal conformal field theory

crew prams and decision trees this paper gives a full characterization of the time needed to compute a boolean function on a crew pram with an unlimited number of processors the characterization is given in terms of a new compl

correlation bounds and sat algorithms for small linear size circuits we revisit the gate elimination method generalize it to prove correlation bounds of boolean circuits with parity and also derive deterministic satisfiability counting algorithms for small linear size circuits let b2 be the full binary basis and let u2 b2 we prove that for circuits over u2 of size 3nn for any constant 0 5 the correlation with parity is at most 2n 1 and there is a sat algorithm which counts the number of satisfying assignments running in time 2nn 1 for circuit size 3nn for 0 the correlation with parity is at most 2 n and there is a sat algorithm running in time 2n n similar correlation bounds and algorithms are also proved for circuits over b2 of size almost 2 5n

any and or formula of size n can be evaluated in time n1 2 o 1 on a quantum computer consider the problem of evaluating an and or formula on an n bit black box input we present a bounded error quantum algorithm that solves this problem in time n 1 2 o 1 in particular approximately balanced formulas can be evaluated in o n queries which is optimal the idea of the algorithm is to apply phase estimation to a discrete time quantum walk on a weighted tree whose spectrum encodes the value of the formula

further exact solutions of the eight vertex sos model and generalizations of the rogers ramanujan identities the restricted eight vertex solid on solid sos model is an exactly solvable class of two dimensional lattice models to each sitei of the lattice there is associated an integer heightli restricted to the range 1 li r 1 the boltzmann weights of the model are expressed in terms of elliptic functions of period 2k and involve a parameter in an earlier paper we considered the case k r here we generalize those considerations to the case sk r s an integer relatively prime tor we are again led to generalizations of the rogers ramanujan identities

quantum query as a state decomposition abstract the quantum query model is a framework that allows us to express most known quantum algorithms algorithms represented by this model consist of a set of unitary operators acting over a finite hilbert space and a final measurement step consisting of a set of projectors in this work we prove that the application of these unitary operators before the measurement step is equivalent to decomposing a unit vector into a sum of vectors and then inverting some of their relative phases we also prove that the vectors of that sum must fulfill a list of properties and we call such vectors a block set if we define the measurement step for the block set formulation similarly to the quantum query model then we prove that both formulations give the same gram matrix of output states although the block set formulation allows a much more explicit form therefore the block set reformulation of the quantum query model gives us an alternative interpretation of how quantum algorithms work finally we apply our approach to the analysis and complexity of quantum exact algorithms

complexity measures and decision tree complexity a survey we discuss several complexity measures for boolean functions certificate complexity sensitivity block sensitivity and the degree of a representing or approximating polynomial we survey the relations and biggest gaps known between these measures and show how they give bounds for the decision tree complexity of boolean functions on deterministic randomized and quantum computers

algorithms for boolean function query properties we investigate efficient algorithms for computing boolean function properties relevant to query complexity such properties include for example deterministic randomized and quantum query complexities block sensitivity certificate complexity and degree as a real polynomial the algorithms compute the properties given an n variable function s truth table of size n 2n as input our main results are the following o n log2 3 log n algorithms for many common properties an o n log2 5 log n algorithm for block sensitivity an o n algorithm for testing quasi symmetry a notion of a tree decomposition of a boolean function proof that the decomposition is unique and an o nlog2 3 log n algorithm for finding it a subexponential time approximation algorithm for space bounded quantum query complexity to develop this algorithm we give a new way to search systematically through unitary matrices using finite precision arithmetic the algorithms discussed have been implemented in a linkable library

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

backward and forward bisimulation minimization of tree automata we improve on an existing p a abdulla j hogberg l kaati bisimulation minimization of tree automata international journal of foundations of computer science 18 4 2007 699 713 bisimulation minimization algorithm for finite state tree automata by introducing backward and forward bisimulation and developing minimization algorithms for them minimization via forward bisimulation is also effective on deterministic tree automata faster than the previous algorithm and yields the minimal equivalent deterministic tree automaton minimization via backward bisimulation generalizes the previous algorithm and can yield smaller automata but is just as fast we demonstrate implementations of these algorithms on a typical task in natural language processing

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

improved average case lower bounds for de morgan formula size matching worst case lower bound we give an explicit function h 0 1 n to 0 1 such that every de morgan formula of size n 3 o 1 r 2 agrees with h on at most a fraction of frac 1 2 2 omega r of the inputs our technical contributions include a theorem that shows that the expected shrinkage result of h aastad siam j comput 27 1998 pp 48 64 actually holds with very high probability where the restrictions are chosen from a certain distribution that takes into account the structure of the formula using ideas of impagliazzo meka and zuckerman proceedings of focs 2012 pp 111 119

graphical approach to the nonintersecting string model star triangle equation inversion relation and exact solution abstract a general q state nonintersecting string model for q 2 is studied and solved using a graphical approach this model of which special cases were first introduced by strogonov and schultz has recently drawn further attention because of its relation to potts models in its simplest version a uniform separable nonintersecting string model on a quadratic lattice it corresponds to a q2 state potts model with potts spins on every other lattice face we first formulate the related star triangle equation which we solve in terms of a line variable parametrization such a parametrization of a solution of the star triangle equation leads to a simple and direct graphical derivation of an inversion relation for the partition function our graphical analysis also shows that the inversion relation holds if certain boundary effects can be neglected as we shall give a special finite lattice from which the inversion relation can be read off immediately this relation is next solved under appropriate analyticity assumptions again using a simple and direct approach and the result is applied to obtain exact solutions for specific string models

on the decisional complexity of problems over the reals we consider the role of randomness for the decisional complexity in algebraic decision or computation trees i e the number of comparisons ignoring all other computation recently ting and yao showed that the problem of finding the maximum of n elements has decisional complexity o log2n 1994 inform process lett 49 39 43 in contrast rabin showed in 1972 an n bound for the deterministic case 1972 j comput system sci 6 639 650 we point out that their technique is applicable to several problems for which corresponding n lower bounds hold we show that in general the randomized decisional complexity is logarithmic in the size of the decision tree we then turn to the question of the number of random bits needed to obtain the ting and yao result we provide a deterministic o k log n algorithm for finding the elements which are larger than a given element given a bound k on the number of these elements we use this algorithm to obtain an o log2n random bits and o log2n queries algorithm for finding the maximum

characterizing classes of functions computable by quantum parallelism computation by quantum parallelism involves associating a quantum state v f to each function f z m z n v f is formed from superpositions of states labelled by the values of f the standard choice being an equally weighted superposition of all the values of f a joint property g f of the values f 0 f m 1 is called computable by quantum parallelism qpc if there is an observable g which will reveal the value g f with non zero probability when applied to v f and will never show a false value it is shown that the problem of deciding which g s are qpc can be formulated entirely in terms of the linear relations which exist among the v f s in the case of f z m z 2 g z 2 m z 2 we explicitly describe all properties which are qpc and show that this includes only m 2 m 2 of the 2 2 m such properties by using a suitable nonstandard definition for v f this number can be increased to 2 2 m 2 m 1 for m 1

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

on the power of ambainis lower bounds the polynomial method and the ambainis lower bound or alb for short method are two main quantum lower bound techniques while recently ambainis showed that the polynomial method is not tight the present paper aims at studying the power and limitation of alb s we first use known alb s to derive n1 5 lower bounds for bipartiteness bipartiteness matching and graph matching in which the lower bound for bipartiteness improves the previous n one we then show that all the three known ambainis lower bounds have a limitation n min c0 f c1 f where c0 f and c1 f are the 0 and 1 certificate complexities respectively this implies that for many problems such as triangle k clique bipartiteness and bipartite graph matching which draw wide interest and whose quantum query complexities are still open the best known lower bounds cannot be further improved by using ambainis techniques another consequence is that all the ambainis lower bounds are not tight for total functions this upper bound for alb s can be further improved to min c0 f c1 f n ci f where ci f is the size of max intersection of a 0 and a 1 certificate set again this implies that alb s cannot improve the best known lower bound for some specific problems such as and or tree whose precise quantum query complexity is still open finally we generalize the three known alb s and give a new alb style lower bound method which may be easier to use for some problems

infinite conformal symmetry of critical fluctuations in two dimensions we study the massless quantum field theories describing the critical points in two dimensional statistical systems these theories are invariant with respect to the infinite dimensional group of conformal analytic transformations it is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of the virasoro algebra exactly solvable theories associated with degenerate representations are analized in these theories the anomalous dimensions are known exactly and the correlation functions satisfy the system of linear differential equations

quantum algorithms for finding constant sized sub hypergraphs we develop a general framework to construct quantum algorithms that detect if a 3 uniform hypergraph given as input contains a sub hypergraph isomorphic to a prespecified constant sized hypergraph this framework is based on the concept of nested quantum walks recently proposed by jeffery kothari and magniez 2013 12 and extends the methodology designed by lee magniez and santha 2013 18 for similar problems over graphs as applications we obtain a quantum algorithm for finding a 4 clique in a 3 uniform hypergraph on n vertices with query complexity o n1 883 and a quantum algorithm for determining if a ternary operator over a set of size n is associative with query complexity o n2 113

quantum query complexity of minor closed graph properties we study the quantum query complexity of minor closed graph properties which include such problems as determining whether an n vertex graph is planar is a forest or does not contain a path of a given length we show that most minor closed properties those that cannot be characterized by a finite set of forbidden subgraphs have quantum query complexity theta n 3 2 to establish this we prove an adversary lower bound using a detailed analysis of the structure of minor closed properties with respect to forbidden topological minors and forbidden subgraphs on the other hand we show that minor closed properties and more generally sparse graph properties that can be characterized by finitely many forbidden subgraphs can be solved strictly faster in o n 3 2 queries our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph finding problems

separations in query complexity based on pointer functions 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 2k bits defined by a complete binary tree of nand gates of depth k which achieves r0 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 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 n1 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 g on n variables that has zero error randomized query complexity n log n and bounded error randomized query complexity r g o n this is the first super linear separation between these two complexity measures the exact quantum query complexity of the same function is qe g o n these functions show that the relations d f o r1 f 2 and r0 f o r f 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 r0 a 3 2 power separation between qe 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 goos 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

the quantum query complexity of the abelian hidden subgroup problem simon in his focs 94 paper was the first to show an exponential gap between classical and quantum computation the problem he dealt with is now part of a well studied class of problems the hidden subgroup problems we study simon s problem from the point of view of quantum query complexity and give here a first non trivial lower bound on the query complexity of a hidden subgroup problem namely simon s problem more generally we give a lower bound which is optimal up to a constant factor for any abelian group

a tighter relation between sensitivity complexity and certificate complexity abstract the sensitivity conjecture proposed by nisan and szegedy in 1994 which asserts that for any boolean function its sensitivity complexity is polynomially related to the block sensitivity complexity is one of the most important and challenging problems in the study of decision tree complexity despite a lot of efforts the best known upper bound of block sensitivity as well as the certificate complexity is still exponential in terms of sensitivity in this paper we give a better upper bound for certificate complexity and block sensitivity b s f c f 8 9 o 1 s f 2 s f 1 where b s f c f and s f are the block sensitivity certificate complexity and sensitivity respectively the proof is based on a deep investigation on the structure of the sensitivity graph we also provide a tighter relationship between the 0 certificate complexity c 0 f and 0 sensitivity s 0 f for functions with small 1 sensitivity s 1 f

block sensitivity of minterm transitive functions boolean functions with a high degree of symmetry are interesting from a complexity theory perspective extensive research has shown that these functions if nonconstant must have high complexity according to various measures in a recent work of this type sun 2007 9 gave lower bounds on the block sensitivity of nonconstant boolean functions invariant under a transitive permutation group sun showed that all such functions satisfy bs f email protected n 1 3 he also showed that there exists such a function for which bs f o n 3 7lnn his example belongs to a subclass of transitively invariant functions called minterm transitive functions defined by chakraborty 2005 3 we extend these results in two ways first we show that nonconstant minterm transitive functions satisfy bs f email protected n 3 7 thus sun s example has nearly minimal block sensitivity for this subclass second we improve sun s example we exhibit a minterm transitive function for which bs f o n 3 7ln 1 7n

the spin 1 heisenberg chain with uniaxial anisotropy and its classical counterpart abstract the spin 1 heisenberg chain with uniaxial anisotropy is related to a two dimensional roughening model with anisotropic next nearest neighbor interactions using this relation some peculiar features of the phase diagram of the quantum chain can be explained naturally and predictions for the critical exponents are also obtained which agree with previous finite size scaling studies of the quantum model a numerical transfer matrix calculation of the free energy of the roughening model on infinite strips of small width n 6 yields a critical behavior consistent with the one found for the t 0 properties of the quantum chain

computing boolean functions from multiple faulty copies of input bits suppose we want to compute a boolean function f but instead of receiving the input we only get l e faulty copies of each input bit a typical solution in this case is to take the majority value of the faulty bits for each individual input bit and apply f on the majority values we call this the trivial construction we show that if f 0 1 n 0 1 and e are known the best function construction f is often not the trivial one in particular in many cases the best f cannot be written as a composition of f with some functions and in addition it is better to use a randomized f than a deterministic one we also prove that the trivial construction is optimal in some rough sense if we denote by l f the number of 1 10 biased copies we need from each input to reliably compute f using the best randomized recovery function f and we denote by ltriv f the analogous number for the trivial construction then ltriv f l f moreover both quantities are in log s f where s f is the sensitivity of f a quantity related to l f is dstat erand f min i 1n li where li is the number of 1 10 biased copies of xi such that the above number of readings is sufficient to recover f with high probability this quantity was first introduced by reischuk and schmeltz 14 in order to provide lower bounds for the noisy circuit size of f in this article we give a complete characterization of dstat erand f through a combinatorial lemma that can be interesting on its own right

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

pseudorandom generators hard for k dnf resolution and polynomial calculus resolution a pseudorandom generator gn f0 1g n f0 1g m is hard for a propositional proof system p if roughly speaking p cannot eciently

an nlogn algorithm for hyper minimizing a minimized deterministic automaton we improve a recent result a badr hyper minimization in o n 2 internat j found comput sci 20 4 2009 735 746 for hyper minimized finite automata namely we present an o nlogn algorithm that computes for a given deterministic finite automaton dfa an almost equivalent dfa that is as small as possible such an automaton is called hyper minimal here two finite automata are almost equivalent if and only if the symmetric difference of their languages is finite in other words two almost equivalent automata disagree on acceptance on finitely many inputs in this way we solve an open problem stated in a badr v geffert i shipman hyper minimizing minimized deterministic finite state automata rairo theor inf appl 43 1 2009 69 94 and by badr moreover we show that minimization linearly reduces to hyper minimization which shows that the time bound o nlogn is optimal for hyper minimization independently similar results were obtained in p gawrychowski a jez hyper minimisation made efficient in proc 34th int symp mathematical foundations of computer science in lncs vol 5734 springer 2009 pp 356 368

the pattern matrix method we develop a novel technique for communication lower bounds the pattern matrix method specifically fix an arbitrary function f 0 1 n to 0 1 and let a f be the matrix whose columns are each an application of f to some subset of the variables x 1 x 2 ldots x 4n we prove that a f has bounded error communication complexity omega d where d is the approximate degree of f this result remains valid in the quantum model regardless of prior entanglement in particular it gives a new and simple proof of razborov s breakthrough quantum lower bounds for disjointness and other symmetric predicates we further characterize the discrepancy approximate rank and approximate trace norm of a f in terms of well studied analytic properties of f broadly generalizing several recent results on small bias communication and agnostic learning the method of this paper has also enabled important progress in multiparty communication complexity

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

revisiting deutsch jozsa algorithm abstract the deutsch jozsa algorithm is essentially faster than any possible deterministic classical algorithm for solving a promise problem that is in fact a symmetric partial boolean function named as the deutsch jozsa problem the deutsch jozsa problem can be equivalently described as a partial function d j n 0 0 1 n 0 1 defined as d j n 0 x 1 for x n 2 d j n 0 x 0 for x 0 n and it is undefined for the remaining cases where n is even and x is the hamming weight of x the deutsch jozsa algorithm needs only one query to compute d j n 0 but the classical deterministic algorithm requires n 2 1 queries to compute it in the worse case we present all symmetric partial boolean functions with degree 1 and 2 we prove the exact quantum query complexity of all symmetric partial boolean functions with degree 1 and 2 we prove deutsch jozsa algorithm can compute any symmetric partial boolean function f with exact quantum 1 query complexity

efficient quantum algorithms for simulating sparse hamiltonians we present an efficient quantum algorithm for simulating the evolution of a quantum state for a sparse hamiltonian h over a given time t in terms of a procedure for computing the matrix entries of h in particular when h acts on n qubits has at most a constant number of nonzero entries in each row column and h is bounded by a constant we may select any positive integer k such that the simulation requires o log n t1 1 2k accesses to matrix entries of h we also show that the temporal scaling cannot be significantly improved beyond this because sublinear time scaling is not possible

hyper optimization for deterministic tree automata hyper minimization is a lossy minimization technique that allows a finite number of errors it was already demonstrated that hyper minimization can be performed efficiently for deterministic string automata and bottom up deterministic tree automata dtas the asymptotically fastest dta hyper minimization algorithms run in time o m log n where m is the size of the dta and n is the number of its states in this contribution the committed errors are investigated first the structure of all hyper minimal dtas for a given tree language is characterized which also yields a formula for the number of such hyper minimal dtas second an algorithm is developed that computes the number of errors that a given hyper minimal dta commits when compared to a given reference dta third it is shown that optimal hyper minimization i e computing a hyper minimal dta that commits the least number of errors of all hyper minimal dtas can be achieved in time o m n finally a discussion of various other error measures besides only their number is provided

on the parity complexity measures of boolean functions the parity decision tree model extends the decision tree model by allowing the computation of a parity function in one step we prove that the deterministic parity decision tree complexity of any boolean function is polynomially related to the non deterministic complexity of the function or its complement we also show that they are polynomially related to an analogue of the block sensitivity we further study parity decision trees in their relations with an intermediate variant of the decision trees as well as with communication complexity

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