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

a lagrangean heuristic for the capacitated concave minimum cost network flow problem we propose a heuristic solution technique for the capacitated concave minimum cost network flow problem based on a lagrangean dualization of the problem despite its dual character the algorithm guarantees the generation of primal feasible solutions which are local optima and therefore candidates of being the global optimum the lagrangean dual is solved by a subgradient search procedure and provides a lower bound to the optimal value the lower bound is in general stronger than the one obtained by a linear approximation of the original problem it can be used as a judgement of the quality of the solution or in a branch and bound procedure computational results from randomly generated problems are presented

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

quasiclassical computation abstract the chemical kinetic description of time evolution where the phase is random but the states are discrete is discussed as a basis for a computational approach this proposed scheme uses numbers in the entire range of 0 to 1 to represent boolean propositions in the implementation by chemical kinetics these numbers are the mole fractions of different species vibrational relaxation in a mixture of hcl and dci is the physical system that is used to illustrate the approach energy exchange in such a mixture corresponds to two strongly coupled two level systems a search problem previously discussed in the quantum computing literature is solved as an example the solution requires the same number of function evaluations as in the quantal case the action of the oracle is described in detail

on the relationship between continuous and discrete time quantum walk quantum walk is one of the main tools for quantum algorithms defined by analogy to classical random walk a quantum walk is a time homogeneous quantum process on a graph both random and quantum walks can be defined either in continuous or discrete time but whereas a continuous time random walk can be obtained as the limit of a sequence of discrete time random walks the two types of quantum walk appear fundamentally different owing to the need for extra degrees of freedom in the discrete time case

probabilistic local search algorithms for concave cost transportation network problems in practice concave cost transportation problems are characterized as np hard therefore cost functions are usually simplified as linear in order to facilitate problem solving however linear cost functions may not reflect actual operations which generally results in decreased operational performance this research employs the techniques of simulated annealing and threshold accepting to develop several heuristics that would efficiently solve these concave cost transportation network problems a network generator has also been designed to generate many instances on an hp workstation to test the heuristics the preliminary results show that these heuristics are potentially useful

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

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

constructive quantum shannon decomposition from cartan involutions the work presented here extends upon the best known universal quantum circuit the quantum shannon decomposition proposed by shende et al 2006 ieee trans comput aided des integr circuits syst 25 1000 we obtain the basis of the circuit s design in a pair of cartan decompositions this insight gives a simple constructive factoring algorithm in terms of the cartan involutions corresponding to these decompositions

experimental analysis of simulated annealing based algorithms for the layout problem abstract in this paper results from an experimental analysis involving the 2 way and 3 way exchange algorithms a modified penalty algorithm the simulated annealing algorithm and a hybrid simulated annealing algorithm are presented the hybrid simulated annealing algorithm uses the modified penalty algorithm to generate an initial solution and then improves it using simulated annealing it is tested on single row layout problems with facilities of unequal area and multi row layout problems with facilities of equal area the algorithm produced solutions better than those previously reported in the literature for two of the larger single row problems and five multi row layout problems for all the other single row problems and eight of ten multi row layout test problems it produced the best known solutions the hybrid simulated annealing algorithm appears to produce better quality solutions than the simulated annealing and tabu navigation algorithms but requires more computation time than simulated annealing

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

constructing carmichael numbers through improved subset product algorithms we have constructed a carmichael number with 10 333 229 505 prime factors and have also constructed carmichael numbers with k prime factors for every k between 3 and 19 565 220 these computations are the product of implementations of two new algorithms for the subset product prob lem that exploit the non uniform distribution of primes p with the property that p 1 divides a highly composite

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

threshold accepting a general purpose optimization algorithm appearing superior to simulated annealing abstract a new general purpose algorithm for the solution of combinatorial optimization problems is presented the new threshold accepting method is even simpler structured than the wellknown simulated annealing approach the power of the new algorithm is demonstrated by computational results concerning the traveling salesman problem and the problem of the construction of error correcting codes moreover deterministic versions of the new heuristic turn out to perform nearly equally well consuming only a fraction of the computing time of the stochastic versions as an example the deterministic threshold accepting method yields very near to optimum tours for the famous 442 cities traveling salesman problem of grotschel within 1 to 2 s of cpu time

hidden translation and translating coset in quantum computing we give efficient quantum algorithms for the problems of hidden translation and hidden subgroup in a large class of nonabelian solvable groups including solvable groups of constant exponent and of constant length derived series our algorithms are recursive for the base case we solve efficiently hidden translation in mathbb z p n whenever p is a fixed prime for the induction step we introduce the problem translating coset generalizing both hidden translation and hidden subgroup and prove a powerful self reducibility result translating coset in a finite solvable group g is reducible to instances of translating coset in g n and n for appropriate normal subgroups n of g our self reducibility framework combined with kuperberg s subexponential quantum algorithm for solving hidden translation in any abelian group leads to subexponential quantum algorithms for hidden translation and hidden subgroup in any solvable group

witten reshetikhin turaev invariants of seifert manifolds for seifert homology spheres we derive a holomorphic function of k whose value at integer k is the sl2 witten reshetikhin turaev invariant zk at q exp 2 i k this function is expressed as a sum of terms which can be naturally corresponded to the contributions of flat connections in the stationary phase expansion of the witten chern simons path integral the trivial connection contribution is found to have an asymptotic expansion in powers of k 1 which for k an odd prime power converges k adically to the exact total value of the invariant zk at that root of unity evaluations at rational k are also discussed using similar techniques an expression for the coloured jones polynomial of a torus knot is obtained providing a trivial connection contribution which is an analytic function of the colour this demonstrates that the stationary phase expansion of the chern simons witten theory is exact for seifert manifolds and for torus knots in s3 the possibility of generalising such results is also discussed

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

scaling and better approximating quantum fourier transform by higher radices quantum fourier transform qft plays a principal role in the development of efficient quantum algorithms since the number of quantum bits that can be built is limited while many quantum technologies are inherently three or more valued we consider extending the reach of the realistic quantum systems by building a qft over ternary quantum digits compared to traditional binary qft the q valued transform improves approximation properties and increases the state space by a factor of q 2 n further we use nonbinary qft derivation to generalize and improve the approximation bounds for qft

a subexponential time quantum algorithm for the dihedral hidden subgroup problem we present a quantum algorithm for the dihedral hidden subgroup problem dhsp with time and query complexity 2 o sqrt log n in this problem an oracle computes a function f on the dihedral group d n which is invariant under a hidden reflection in d n by contrast the classical query complexity of dhsp is o sqrt n the algorithm also applies to the hidden shift problem for an arbitrary finitely generated abelian group the algorithm begins as usual with a quantum character transform which in the case of d n is essentially the abelian quantum fourier transform this yields the name of a group representation of d n which is not by itself useful and a state in the representation which is a valuable but indecipherable qubit the algorithm proceeds by repeatedly pairing two unfavorable qubits to make a new qubit in a more favorable representation of d n once the algorithm obtains certain target representations direct measurements reveal the hidden subgroup

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

on the virasoro constraints for torus knots we construct a virasoro algebra of differential operators for the matrix model for torus knots these operators generate various relations between wilson loops then we discuss the operators constructed and corresponding relations in the stability limit we also give a series of examples

a decomposition method using a pricing mechanism for min concave cost flow problems with a hierarchical structure in this paper we develop a decomposition method using a pricing mechanism which has been widely applied to linear and convex programs for a class of nonconvex optimization problems that are min concave cost flow problems under directed uncapacitated networks with a hierarchical structure

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

schur polynomials and biorthogonal random matrix ensembles the study of the average of schur polynomials over a stieltjes wigert ensemble has been carried out by dolivet and tierz j math phys 48 023507 2007 e print arxiv hep th 0609167 where it was shown that it is equal to quantum dimensions using the same approach we extend the result to the biorthogonal case we also study using the littlewood richardson rule some particular cases of the quantum dimension result finally we show that the notion of giambelli compatibility of schur averages introduced by borodin et al adv appl math 37 209 2006 e print arxiv math ph 0505021 also holds in the biorthogonal setting

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

the power of strong fourier sampling quantum algorithms for affine groups and hidden shifts many quantum algorithms including shor s celebrated factoring and discrete log algorithms proceed by reduction to a hidden subgroup problem in which an unknown subgroup h of a group g must be determined from a quantum state psi over g that is uniformly supported on a left coset of h these hidden subgroup problems are typically solved by fourier sampling the quantum fourier transform of psi is computed and measured when the underlying group is nonabelian two important variants of the fourier sampling paradigm have been identified the weak standard method where only representation names are measured and the strong standard method where full measurement i e the row and column of the representation in a suitably chosen basis as well as its name occurs it has remained open whether the strong standard method is indeed stronger that is whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak or any other known method in this article we settle this question in the affirmative we show that hidden subgroups h of the q hedral groups i e semidirect products mathbb z q ltimes mathbb z p where q mid p 1 and in particular the affine groups a p can be information theoretically reconstructed using the strong standard method moreover if h p rm polylog p these subgroups can be fully reconstructed with a polynomial amount of quantum and classical computation we compare our algorithms to two weaker methods that have been discussed in the literature the forgetful abelian method and measurement in a random basis and show that both of these are weaker than the strong standard method thus at least for some families of groups it is crucial to use the full power of representation theory and nonabelian fourier analysis namely to measure the high dimensional representations in an adapted basis that respects the group s subgroup structure we apply our algorithm for the hidden subgroup problem to new families of cryptographically motivated hidden shift problems generalizing the work of van dam hallgren and ip on shifts of multiplicative characters finally we close by proving a simple closure property for the class of groups over which the hidden subgroup problem can be solved efficiently

a random activity network generator exact and heuristic procedures are often developed to obtain optimal and near optimal solutions to decision problems modeled as activity networks testing the accuracy and efficiency of these procedures requires the use of activity networks with various sizes structures and parameters the size of the network is determined by its number of nodes and arcs where the structure is chosen from the set of all structures for the specified network size the network parameters depend on the nature of the decision problem often it is desirable for test problems to be generated at random from the space of all feasible networks this paper deals with the problem of generating the size and structure of the network at random from the space of all feasible networks it develops a theory which guarantees the randomness of the network structure the theory is the basis for two methods one can be used to generate dense networks where the other is used to generate nondense networks the methods which are practical a

on the empirical scaling of run time for finding optimal solutions to the travelling salesman problem abstract the travelling salesman problem tsp is one of the most prominent np hard combinatorial optimisation problems after over fifty years of intense study the tsp continues to be of broad theoretical and practical interest using a novel approach to empirical scaling analysis which in principle is applicable to solvers for many other problems we demonstrate that some of the most widely studied types of tsp instances tend to be much easier than expected from previous theoretical and empirical results in particular we show that the empirical median run time required for finding optimal solutions to so called random uniform euclidean rue instances one of the most widely studied classes of tsp instances scales substantially better than 2 n with the number n of cities to be visited the concorde solver for which we achieved this result is the best performing exact tsp solver we are aware of and has been applied to a broad range of real world problems furthermore we show that even when applied to a broad range of instances from the prominent tsplib benchmark collection for the tsp concorde exhibits run times that are surprisingly consistent with our empirical model of concorde s scaling behaviour on rue instances this result suggests that the behaviour observed for the simple random structure underlying rue is very similar to that obtained on the structured instances arising in various applications

the sign rank of ac 0 the sign rank of a matrix a a ij with pm1 entries is the least rank of a real matrix b b ij with a ij b ij 0 for all i j we obtain the first exponential lower bound on the sign rank of a function in mathsf ac 0 namely let f x y bigwedge i 1 dots m bigvee j 1 dots m 2 x ij wedge y ij we show that the matrix f x y x y has sign rank exp omega m this in particular implies that sigma 2 cc not subseteq mathsf upp cc which solves a longstanding open problem in communication complexity posed by babai frankl and simon proceedings of the 27th symposium on foundations of computer science focs 1986 pp 337 347 our result additionally implies a lower bound in learning theory specifically let phi 1 dots phi r 0 1 n to mathbb r be functions such that every dnf formula f 0 1 n to 1 1 of polynomial size has the representation f equiv mathrm sgn a 1 phi 1 dots a r phi r for some reals a 1 dots a r we prove that then r geqslant exp omega n 1 3 which essentially matches an upper bound of exp tilde o n 1 3 due to klivans and servedio j comput system sci 68 2004 pp 303 318 finally our work yields the first exponential lower bound on the size of threshold of majority circuits computing a function in mathsf ac 0 this substantially generalizes and strengthens the results of krause and pudlak theoret comput sci 174 1997 pp 137 156

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

quantum algorithms and the fourier transform the quantum algorithms of deutsch simon and shor are described in a way which highlights their dependence on the fourier transform the general construction of the fourier transform on an abelian group is outlined and this provides a unified way of understanding the efficacy of the algorithms finally we describe an efficient quantum factoring algorithm based on a general formalism of kitaev and contrast its structure to the ingredients of shor9 algorithm

chern simons matrix models and stieltjes wigert polynomials employing the random matrix formulation of chern simons theory on seifert manifolds we show how the stieltjes wigert orthogonal polynomials are useful in exact computations in chern simons matrix models we construct a biorthogonal extension of the stieltjes wigert polynomials not available in the literature necessary to study chern simons matrix models when the geometry is a lens space we also study the relationship between stieltjes wigert and rogers szego polynomials and the corresponding equivalence with a unitary matrix model finally we give a detailed proof of a result that relates quantum dimensions with averages of schur polynomials in the stieltjes wigert ensemble

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

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

send and split method for minimum concave cost network flows many problems from inventory production and capacity planning and from network design exhibit scale economies and can be formulated in terms of finding minimum additive concave cost nonnegative network flows we reduce the problem to an equivalent one in which the arc flow costs are nonnegative and give a dynamic programming method called the send and split method to solve it the main work of the method entails repeatedly solving set splitting and minimum cost chain problems in uncapacitated networks with n nodes a arcs and d 1 demand nodes i e nodes with nonzero exogenous demand the algorithm requires up to n2 13d s2d operations additions and comparisons where s n log2n 3a is the number of operations required to solve a minimum cost chain problem with nonnegative arc costs on the augmented graph formed by appending a node and an arc thereto from each node in the graph if also the network is k planar i e the graph is planar with all demand nodes lying on the boundary of k faces the method requires at most n2 kd3k sd2k operations the algorithm can be applied to capacitated networks because they can be reduced to equivalent uncapacitated ones these results unify significantly generalize e g to cyclic problems and sometimes improve upon e g for tandem facilities known polynomial time dynamic programming algorithms for wagner and whitin s wagner h m whitin t m 1958 dynamic version of the economic lot size model management sci 5 89 96 dynamic economic order quantity problem zangwill s zangwill w i 1969 a backlogging model and a multi echelon model of a dynamic economic lot size production system a network approach management sci 15 506 527 generalization to tandem facilities and veinott s veinott jr a f 1969 minimum concave cost solution of leontief substitution models of multi facility inventory systems oper res 17 262 291 increasing capacity warehousing problem the networks for the finite period versions of these problems are each 1 planar the method improves upon zangwill s zangwill w i 1968 minimum concave cost flows in certain networks management sci 14 429 450 related oand running time dynamic programming method for finding minimum additive concave cost non negative flows in circuitless single source networks we also implement the method to solve in polynomial time the d 1 demand node and k planar versions of the minimum cost forest and steiner problems in graphs the running time for the d 1 demand node steiner problem in graphs is comparable to that of dreyfus and wagner s dreyfus s e wagner r a 1971 the steiner problem in graphs networks1 195 207 method

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

solving systems of diagonal polynomial equations over finite fields we present an algorithm to solve a system of diagonal polynomial equations over finite fields when the number of variables is greater than some fixed polynomial of the number of equations whose degree depends only on the degree of the polynomial equations our algorithm works in time polynomial in the number of equations and the logarithm of the size of the field whenever the degree of the polynomial equations is constant as a consequence we design polynomial time quantum algorithms for two algebraic hidden structure problems for the hidden subgroup problem in certain semidirect product p groups of constant nilpotency class and for the multi dimensional univariate hidden polynomial graph problem when the degree of the polynomials is constant 1

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

on the hecke algebras and the colored homfly polynomial the colored homfly polynomial is the quantum invariant of oriented links in s 3 associated with irreducible representations of the quantum group u q sl n in this paper using an approach to calculate quantum invariants of links via the cabling projection rule we derive a formula for the colored homfly polynomial in terms of the characters of the hecke algebras and schur polynomials the technique leads to a fairly simple formula for the colored homfly polynomial of torus links this formula allows us to test the labastida marino vafa conjecture which reveals a deep relationship between chern simons gauge theory and string theory on torus links

quantum algorithms for testing properties of distributions suppose one has access to oracles generating samples from two unknown probability distributions p and q on some n element set how many samples does one need to test whether the two distributions are close or far from each other in the l1 norm this and related questions have been extensively studied during the last years in the field of property testing in the present paper we study quantum algorithms for testing properties of distributions it is shown that the l1 distance p q 1 can be estimated with a constant precision using only o n1 2 queries in the quantum settings whereas classical computers need n1 o 1 queries we also describe quantum algorithms for testing uniformity and orthogonality with query complexity o n1 3 the classical query complexity of these problems is known to be n1 2

quantum computation and lattice problems we present the first explicit connection between quantum computation and lattice problems namely our main result is a solution to the unique shortest vector problem svp under the assumption that there exists an algorithm that solves the hidden subgroup problem on the dihedral group by coset sampling additionally we present an approach to solving the hidden sub group problem on the dihedral group by using an average case subset sum routine

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

the symmetric group defies strong fourier sampling the dramatic exponential speedups of quantum algorithms over their best existing classical counterparts were ushered in by the technique of fourier sampling introduced by bernstein and vazirani and developed by simon and shor into an approach to the hidden subgroup problem this approach has proved successful for abelian groups leading to efficient algorithms for factoring extracting discrete logarithms and other number theoretic problems we show however that this method cannot resolve the hidden subgroup problem in the symmetric groups even in the weakest information theoretic sense in particular we show that the graph isomorphism problem cannot be solved by this approach our work implies that any quantum approach based upon the measurement of coset states must depart from the original framework by using entangled measurements on multiple coset states

complexity classes of equivalence problems revisited to determine if two lists of numbers are the same set we sort both lists and see if we get the same result the sorted list is a canonical form for the equivalence relation of set equality other canonical forms arise in graph isomorphism algorithms to determine if two graphs are cospectral have the same eigenvalues we compute their characteristic polynomials and see if they are equal the characteristic polynomial is a complete invariant for cospectrality finally an equivalence relation may be decidable in p without either a complete invariant or canonical form blass and gurevich 1984 asked whether these conditions on equivalence relations having an fp canonical form having an fp complete invariant and being in p are distinct they showed that this question requires non relativizing techniques to resolve we extend their results and give new connections to probabilistic and quantum computation