temporally unstructured quantum computation we examine theoretic architectures and an abstract model for a restricted class of quantum computation called here temporally unstructured instantaneous quantum computation because it allows for

matchgate and space bounded quantum computations are equivalent matchgates are an especially multiflorous class of two qubit nearest neighbour quantum gates defined by a set of algebraic constraints they occur for example in the theory of perfect matchings of graphs non interacting fermions and one dimensional spin chains we show that the computational power of circuits of matchgates is equivalent to that of space bounded quantum computation with unitary gates with space restricted to being logarithmic in the width of the matchgate circuit in particular for the conventional setting of polynomial sized logarithmic space generated families of matchgate circuits known to be classically simulatable we characterize their power as coinciding with polynomial time and logarithmic space bounded universal unitary quantum computation

nordhaus gaddum relations for proximity and remoteness in graphs the transmission of a vertex in a connected graph is the sum of all distances from that vertex to the others it is said to be normalized if divided by n 1 where n denotes the order of the graph the proximity of a graph is the minimum normalized transmission while the remoteness is the maximum normalized transmission in this paper we give nordhaus gaddum type inequalities for proximity and remoteness in graphs the extremal graphs are also characterized for each case

computational complexity of holant problems we propose and explore a novel alternative framework to study the complexity of counting problems called holant problems compared to counting constraint satisfaction problems csp it is a refinement with a more explicit role for the constraint functions both graph homomorphism and csp can be viewed as special cases of holant problems we prove complexity dichotomy theorems in this framework our dichotomy theorems apply to local constraint functions which are symmetric functions on boolean input variables and evaluate to arbitrary real or complex values we discover surprising tractable subclasses of counting problems which could not easily be specified in the csp framework when all unary functions are assumed to be free mathrm holant problems the tractable ones consist of functions that are degenerate or of arity at most two or holographic transformations of fibonacci gates when only two special unary functions the constant zero and constant one functions are assumed to be free mathrm holant c problems we further identify three special families of tractable cases then we prove that all other cases are p hard the main technical tool we use and develop is holographic reductions another technical tool used in combination with holographic reductions is polynomial interpolations

a collapse theorem for holographic algorithms with matchgates on domain size at most 4 holographic algorithms with matchgates are a novel approach to design polynomial time computation they use kasteleyn s algorithm for perfect matchings and more importantly a holographic reduction the two fundamental parameters of a holographic reduction are the domain size k of the underlying problem and the basis size a holographic reduction transforms the computation to matchgates by a linear transformation that maps to a tensor product space of a linear space of dimension 2 we prove a sharp basis collapse theorem which shows that for domain size 3 and 4 all non trivial holographic reductions can be expressed with a basis of size 1 or 2 respectively the main proof techniques are matchgate identities and a group property of matchgate signatures

variable neighborhood search principles and applications abstract systematic change of neighborhood within a possibly randomized local search algorithm yields a simple and effective metaheuristic for combinatorial and global optimization called variable neighborhood search vns we present a basic scheme for this purpose which can easily be implemented using any local search algorithm as a subroutine its effectiveness is illustrated by solving several classical combinatorial or global optimization problems moreover several extensions are proposed for solving large problem instances using vns within the successive approximation method yields a two level vns called variable neighborhood decomposition search vnds modifying the basic scheme to explore easily valleys far from the incumbent solution yields an efficient skewed vns svns heuristic finally we show how to stabilize column generation algorithms with help of vns and discuss various ways to use vns in graph theory i e to suggest disprove or give hints on how to prove conjectures an area where metaheuristics do not appear to have been applied before

clifford gates in the holant framework abstract we show that the clifford gates and stabilizer circuits in the quantum computing literature which admit efficient classical simulation are equivalent to affine signatures under a unitary condition the latter is a known class of tractable functions under the holant framework

exponential decay of correlations implies area law we prove that a finite correlation length i e exponential decay of correlations implies an area law for the entanglement entropy of quantum states defined on a line the entropy bound is exponential in the correlation length of the state thus reproducing as a particular case hastings s proof of an area law for groundstates of 1d gapped hamiltonians

some optimal inapproximability results we prove optimal up to an arbitrary e 0 inapproximability results for max e k sat for k 3 maximizing the number of satisfied linear equations in an over determined system of linear equations modulo a prime p and set splitting as a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously in particular for max e2 sat max cut max di cut and vertex cover

approximate counting for complex weighted boolean constraint satisfaction problems constraint satisfaction problems or csps have been extensively studied in for instance artificial intelligence database theory graph theory and statistical physics from a practical viewpoint it is beneficial to approximately solve those csps when one tries to approximate the total number of truth assignments that satisfy all boolean valued constraints for unweighted boolean csps there is a known trichotomy theorem by which all such counting problems are neatly classified into exactly three categories under polynomial time randomized approximation preserving reductions in contrast we obtain a dichotomy theorem of approximate counting for complex weighted boolean csps provided that all complex valued unary constraints are freely available to use it is the expressive power of free unary constraints that enables us to prove such a stronger complete classification theorem this discovery makes a step forward in the quest for the approximation complexity classification of all counting csps to deal with complex weights we employ proof techniques of factorization and arity reduction along the line of solving holant problems moreover we introduce a novel notion of t constructibility that naturally induces approximation preserving reducibility our result also gives an approximation analogue of the dichotomy theorem on the complexity of exact counting for complex weighted boolean csps

partition functions on k regular graphs with 0 1 vertex assignments and real edge functions we prove a complexity dichotomy theorem for a class of partition functions over k regular graphs for any fixed k these problems can be viewed as graph homomorphisms from an arbitrary k regular input graph g to the weighted two vertex graph on 0 1 defined by a real valued symmetric function h we completely classify the computational complexity of this problem we show that there are exactly the following alternatives for any given h depending on h over k regular graphs either 1 the problem is p hard even for planar graphs 2 the problem is p hard for general non planar graphs but solvable in polynomial time for planar graphs or 3 the problem is solvable in polynomial time for general graphs the dependence on h is an explicit criterion furthermore we show that in case 2 the problem is solvable in polynomial time over k regular planar graphs by exactly the theory of holographic algorithms using matchgates

holographic algorithms without matchgates abstract the theory of holographic algorithms which are polynomial time algorithms for certain combinatorial counting problems surprised the complexity community by showing certain problems very similar to p complete problems were in fact in the class p in particular the theory produces algebraic tests for a problem to be in p in this article we describe the geometric basis of these algorithms by i replacing the construction of graph fragments in the procedure by the direct construction of a skew symmetric matrix and ii replacing the computation of weighted perfect matchings of an auxiliary graph by computing the pfaffian of the directly constructed skew symmetric matrix this procedure indicates a more geometric approach to complexity classes it also leads to more general constructions where one replaces the grassmann plucker identities which test for admissibility by other algebraic tests natural problems treatable by these methods have been previously considered in a different context and we present one such example

on blockwise symmetric matchgate signatures and higher domain csp abstract for any n 3 and q 3 we prove that the equality function n on n variables over a domain of size q cannot be realized by matchgates under holographic transformations this is a consequence of our theorem on the structure of blockwise symmetric matchgate signatures this has the implication that the standard holographic algorithms based on matchgates a methodology known to be universal for csp over the boolean domain cannot produce p time algorithms for planar csp over any higher domain q 3

spin systems on k regular graphs with complex edge functions let k 1 be an integer and let h h 0 0 h 0 1 h 1 0 h 1 1 be a complex valued symmetric function on domain 0 1 i e where h 0 1 h 1 0 we introduce a new technique called a syzygy and prove a dichotomy theorem for the following class of problems specified by k and h given an arbitrary k regular graph g v e where the function h is attached to each edge compute z g email protected s v 0 1 u v eh s u s v z is known as the partition function of the spin system also known as counting graph homomorphisms on domain size two and is a special case of holant problems the dichotomy theorem gives a complete classification of the computational complexity of this problem depending on k and h the dependence on k and h is explicit

quantum computing postselection and probabilistic polynomial time i study the class of problems efficiently solvable by a quantum computer given the ability to postselect on the outcomes of measurements i prove that this class coincides with a classical complexity class called pp or probabilistic polynomial time using this result i show that several simple changes to the axioms of quantum mechanics would let us solve pp complete problems efficiently the result also implies as an easy corollary a celebrated theorem of beigel reingold and spielman that pp is closed under intersection as well as a generalization of that theorem due to fortnow and reingold this illustrates that quantum computing can yield new and simpler proofs of major results about classical computation

clifford groups of quantum gates bn pairs and smooth cubic surfaces the recent proposal planat and kibler 2008 arxiv 0807 3650 quant ph of representing clifford quantum gates in terms of unitary reflections is revisited in this communication the geometry of a clifford group g is expressed as a bn pair i e a pair of subgroups b and n that generate g is such that intersection h b n is normal in g the group w n h is a coxeter group and two extra axioms are satisfied by the double cosets acting on b the bn pair used in this decomposition relies on the swap and match gates already introduced for classically simulating quantum circuits jozsa and miyake 2008 arxiv 0804 4050 quant ph the two and three qubit cases are related to the configuration with 27 lines on a smooth cubic surface

computing the tutte polynomial of lattice path matroids using determinantal circuits we give a quantum inspired o n 4 algorithm computing the tutte polynomial of a lattice path matroid where n is the size of the ground set of the matroid furthermore this can be improved to o n 2 arithmetic operations if we evaluate the tutte polynomial on a given input fixing the values of the variables the best existing algorithm found in 2004 was o n 5 and the problem has only been known to be polynomial time since 2003 conceptually our algorithm embeds the computation in a determinant using a recently demonstrated equivalence of categories useful for counting problems such as those that appear in simulating quantum systems

unknown quantum states the quantum de finetti representation we present an elementary proof of the quantum de finetti representation theorem a quantum analog of de finetti s classical theorem on exchangeable probability assignments this contrasts with the original proof of hudson and moody z wahrschein verw geb 33 343 1976 which relies on advanced mathematics and does not share the same potential for generalization the classical de finetti theorem provides an operational definition of the concept of an unknown probability in bayesian probability theory where probabilities are taken to be degrees of belief instead of objective states of nature the quantum de finetti theorem in a closely analogous fashion deals with exchangeable density operator assignments and provides an operational definition of the concept of an unknown quantum state in quantum state tomography this result is especially important for information based interpretations of quantum mechanics where quantum states like probabilities are taken to be states of knowledge rather than

simulating quantum computation by contracting tensor networks the treewidth of a graph is a useful combinatorial measure of how close the graph is to a tree we prove that a quantum circuit with t gates whose underlying graph has a treewidth d can be simulated deterministically in t o 1 exp o d time which in particular is polynomial in t if d o log t among many implications we show efficient simulations for log depth circuits whose gates apply to nearby qubits only a natural constraint satisfied by most physical implementations we also show that one way quantum computation of raussendorf and briegel phys rev lett 86 2001 pp 5188 5191 a universal quantum computation scheme with promising physical implementations can be efficiently simulated by a randomized algorithm if its quantum resource is derived from a small treewidth graph with a constant maximum degree the requirement on the maximum degree was removed in i l markov and y shi preprint quant ph 0511069

impossibility of classically simulating one clean qubit model with multiplicative error the one clean qubit model or the deterministic quantum computation with one quantum bit model is a restricted model of quantum computing where all but a single input qubits are maximally mixed it is known that the probability distribution of measurement results on three output qubits of the one clean qubit model cannot be classically efficiently sampled within a constant multiplicative error unless the polynomial time hierarchy collapses to the third level t morimae k fujii and j f fitzsimons phys rev lett 112 130502 2014 prltao0031 900710 1103 physrevlett 112 130502 it was open whether we can keep the no go result while reducing the number of output qubits from three to one here we solve the open problem affirmatively we also show that the third level collapse of the polynomial time hierarchy can be strengthened to the second level one the strengthening of the collapse level from the third to the second also holds for other subuniversal models such as the instantaneous quantum polynomial model m bremner r jozsa and d j shepherd proc r soc a 467 459 2011 prlaaz1364 502110 1098 rspa 2010 0301 and the boson sampling model s aaronson and a arkhipov stoc 2011 p 333 we additionally study the classical simulatability of the one clean qubit model with further restrictions on the circuit depth or the gate types

on traces of tensor representations of diagrams abstract let t be an abstract set of types and let o t z a t diagram is a locally ordered directed graph g equipped with a function v g t such that each vertex v of g has indegree v and outdegree o v a directed graph is locally ordered if at each vertex v linear orders of the edges entering v and of the edges leaving v are specified let v be a finite dimensional f linear space where f is an algebraically closed field of characteristic 0 a function r on t assigning to each t t a tensor r t v t v o t is called a tensor representation of t the trace or partition function of r is the f valued function p r on the collection of t diagrams obtained by decorating each vertex v of a t diagram g with the tensor r v and contracting tensors along each edge of g while respecting the order of the edges entering v and leaving v in this way we obtain a tensor network we characterize which functions on t diagrams are traces and show that each trace comes from a unique strongly nondegenerate tensor representation the theorem applies to virtual knot diagrams chord diagrams and group representations

capturing exponential variance using polynomial resources applying tensor networks to nonequilibrium stochastic processes estimating the expected value of an observable appearing in a nonequilibrium stochastic process usually involves sampling if the observable s variance is high many samples are required in contrast we show that performing the same task without sampling using tensor network compression efficiently captures high variances in systems of various geometries and dimensions we provide examples for which matching the accuracy of our efficient method would require a sample size scaling exponentially with system size in particular the high variance observable e ensuremath ensuremath beta w motivated by jarzynski s equality with w the work done quenching from equilibrium at inverse temperature ensuremath beta is exactly and efficiently captured by tensor networks

matchgates and classical simulation of quantum circuits let g a b denote the two qubit gate that acts as the one qubit su 2 gates a and b in the even and odd parity subspaces respectively of two qubits using a clifford algebra formalism we show th

quantum formulas a lower bound and simulation we show that nechiporuk s method i wegener the complexity of boolean functions teubner wiley new york 1987 for proving lower bounds for boolean formulas can be extended to the quantum case this leads to an omega n 2 log 2 n lower bound for quantum formulas computing an explicit function the only known previous explicit lower bound for quantum formulas a yao proceedings of 34th ieee symposium on foundations of computer science ieee computer society press los alamitos ca 1993 pp 352 361 states that the majority function does not have a linear size quantum formula we also show that quantum formulas can be simulated by boolean circuits of almost the same size

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

quantum circuits that can be simulated classically in polynomial time a model of quantum computation based on unitary matrix operations was introduced by feynman and deutsch it has been asked whether the power of this model exceeds that of classical turing machines we show here that a significant class of these quantum computations can be simulated classically in polynomial time in particular we show that two bit operations characterized by 4 4 matrices in which the sixteen entries obey a set of five polynomial relations can be composed according to certain rules to yield a class of circuits that can be simulated classically in polynomial time this contrasts with the known universality of two bit operations and demonstrates that efficient quantum computation of restricted classes is reconcilable with the polynomial time turing hypothesis therefore it is possible that the techniques introduced bring the quantum computational model within the realm of algebraic complexity theory in a manner consistent with one view of quantum physics the wave function is simulated deterministically and randomization arises only in the course of making measurements the results generalize the quantum model in that they do not require the matrices to be unitary in a different direction these techniques also yield deterministic polynomial time algorithms for the decision and parity problems for certain classes of read twice boolean formulae all our results are based on the use of gates that are defined in terms of their graph matching properties

analyticity and asymptotics for the stark wannier states it is proved that the stark wannier states as functions of the electric field are analytic in a disc tangential to the real axis at the origin with asymptotic expansion to the second order which coincides with the wannier approximation up to the first order

generalized counting constraint satisfaction problems with determinantal circuits abstract generalized counting constraint satisfaction problems include holant problems with planarity restrictions polynomial time algorithms for such problems include matchgates and matchcircuits which are based on pfaffians in particular they use gates which are expressible in terms of a vector of sub pfaffians of a skew symmetric matrix we introduce a new type of circuit based instead on determinants with seemingly different expressive power in these determinantal circuits a gate is represented by the vector of all minors of an arbitrary matrix determinantal circuits permit a different class of gates applications of these circuits include proofs of theorems from algebraic graph theory including the chung langlands formula for the number of rooted spanning forests of a graph and computing tutte polynomials of certain matroids they also give a strategy for simulating quantum circuits with closed timelike curves monoidal category theory provides a useful language for discussing such counting problems turning combinatorial restrictions into categorical properties we introduce the counting problem in monoidal categories and count preserving functors as a way to study fp subclasses of problems in settings which are generally p hard using this machinery we show that surprisingly determinantal circuits can be simulated by pfaffian circuits at quadratic cost

complexity classification of the six vertex model abstract we prove a complexity dichotomy theorem for the six vertex model for every setting of the parameters of the model we prove that computing the partition function is either solvable in polynomial time or p hard the dichotomy criterion is explicit

threshold computation and cryptographic security threshold machines are turing machines whose acceptance is determined by what portion of the machine s computation paths are accepting paths probabilistic machines are turing machines whose acceptance is determined by the probability weight of the machine s accepting computation paths in 1975 simon proved that for unbounded error polynomial time machines these two notions yield the same class pp perhaps because simon s result seemed to collapse the threshold and probabilistic modes of computation the relationship between threshold and probabilistic computing for the case of bounded error has remained unexplored in this paper we compare the bounded error probabilistic class bpp with the analogous threshold class bpppath and more generally we study the structural properties of bpppath we prove that rm bpp path contains both np bpp and p rm np log and that rm bpp path is contained in p rm sigma 2 p log rm bpp np and pp we conclude that unless the polynomial hierarchy collapses bounded error threshold computation is strictly more powerful than bounded error probabilistic computation we also consider the natural notion of secure access to a database an adversary who watches the queries should gain no information about the input other than perhaps its length we show for both bpp and bpppath that if there is any database for which this formalization of security differs from the security given by oblivious database access then p neq pspace it follows that if any set lacking small circuits can be securely accepted then p neq pspace

constrained codes as networks of relations we address the well known problem of determining the capacity of constrained coding systems while the one dimensional case is well understood to the extent that there are techniques for rigorously deriving the exact capacity in contrast computing the exact capacity of a two dimensional constrained coding system is still an elusive research challenge the only known exception in the two dimensional case is an exact however not rigorous solution to the run length limited rll system on the hexagonal lattice furthermore only exponential time algorithms are known for the related problem of counting the exact number of constrained two dimensional information arrays we present the first known rigorous technique that yields an exact capacity of a two dimensional constrained coding system in addition we devise an efficient polynomial time algorithm for counting the exact number of constrained arrays of any given size our approach is a composition of a number of ideas and techniques describing the capacity problem as a solution to a counting problem in networks of relations graph theoretic tools originally developed in the field of statistical mechanics techniques for efficiently simulating quantum circuits as well as ideas from the theory related to the spectral distribution of toeplitz matrices using our technique we derive a closed form solution to the capacity related to the path cover constraint in a two dimensional triangular array the resulting calculated capacity is path cover is a generalization of the well known one dimensional rll constraint for which the capacity is known to be

holographic algorithms by fibonacci gates abstract we introduce fibonacci gates as a polynomial time computable primitive and develop a theory of holographic algorithms based on these gates the fibonacci gates play the role of matchgates in valiant s theory valiant 2008 19 they give rise to polynomial time computable counting problems on general graphs while matchgates mainly work over planar graphs only we develop a signature theory and characterize all realizable signatures for fibonacci gates for bases of arbitrary dimensions we prove a basis collapse theorem we apply this theory to give new polynomial time algorithms for certain counting problems we also use this framework to prove that some slight variations of these counting problems are p hard holographic algorithms with fibonacci gates prove to be useful as a general tool for classification results of counting problems dichotomy theorems cai et al 2009 7

classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy we consider quantum computations comprising only commuting gates known as iqp computations and provide compelling evidence that the task of sampling their output probability distributions is unli

tensor network contractions for sat the computational cost of counting the number of solutions satisfying a boolean formula which is a problem instance of sat has proven subtle to quantify even when finding individual satisfying solutions is computationally easy e g 2 sat which is in mathsf p determining the number of solutions can be mathsf p hard recently computational methods simulating quantum systems experienced advancements due to the development of tensor network algorithms and associated quantum physics inspired techniques by these methods we give an algorithm using an axiomatic tensor contraction language for n variable sat instances with complexity o g cd o 1 2 c where c is the number of copy tensors g is the number of gates and d is the maximal degree of any copy tensor thus n variable counting problems can be solved efficiently when their tensor network expression has at most o log n copy tensors and polynomial fan out this framework also admits an intuitive proof of a variant of the tovey conjecture the r 1 sat instance of the dubois tovey theorem this study increases the theory expressiveness and application of tensor based algorithmic tools and provides an alternative insight on these problems which have a long history in statistical physics and computer science

on the extremal properties of the average eccentricity the eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity ecc g of a graph g is the mean value of eccentricities of all vertices of g the average eccentricity is deeply connected with a topological descriptor called the eccentric connectivity index defined as a sum of products of vertex degrees and eccentricities in this paper we analyze extremal properties of the average eccentricity introducing two graph transformations that increase or decrease ecc g furthermore we resolve four conjectures obtained by the system autographix about the average eccentricity and other graph parameters the clique number and the independence number refute one autographix conjecture about the average eccentricity and the minimum vertex degree and correct one autographix conjecture about the domination number

a dichotomy theorem for the approximate counting of complex weighted bounded degree boolean csps we determine the computational complexity of approximately counting the total weight of variable assignments for every complex weighted boolean constraint satisfaction problem or csp with any number of additional unary i e arity 1 constraints particularly when degrees of input instances are bounded from above by a fixed constant all degree 1 counting csps are obviously solvable in polynomial time when the instance s degree is more than two we present a dichotomy theorem that classifies all counting csps admitting free unary constraints into exactly two categories this classification theorem extends to complex weighted problems an earlier result on the approximation complexity of unweighted counting boolean csps of bounded degree the framework of the proof of our theorem is based on a theory of signature developed from valiant s holographic algorithms that can efficiently solve seemingly intractable counting csps despite the use of arbitrary complex weight our proof of the classification theorem is rather elementary and intuitive due to an extensive use of a novel notion of limited t constructibility for the remaining degree 2 problems in contrast they are as hard to approximate as holant problems which are a generalization of counting csps

holographic algorithms the power of dimensionality resolved valiant s theory of holographic algorithms is a novel methodology to achieve exponential speed ups in computation a fundamental parameter in holographic algorithms is the dimension of the linear basis vectors we completely resolve the problem of the power of higher dimensional bases we prove that 2 dimensional bases are universal for holographic algorithms

a complete dichotomy rises from the capture of vanishing signatures we prove a complexity dichotomy theorem for holant problems over an arbitrary set of complex valued symmetric constraint functions mathcal f on boolean variables this extends and unifies all previous dichotomies for holant problems on symmetric constraint functions taking values without a finite modulus we define and characterize all symmetric vanishing signatures they turn out to be essential to the complete classification of holant problems the dichotomy theorem has an explicit tractability criterion expressible in terms of holographic transformations a holant problem defined by a set of constraint functions mathcal f is solvable in polynomial time if it satisfies this tractability criterion and is p hard otherwise the tractability criterion can be intuitively stated as follows a set mathcal f is tractable if 1 every function in mathcal f has arity at most two or 2 mathcal f is transformable to an affine type or 3 mathcal f is transformable to a product type o

explicit lower bounds on strong quantum simulation we consider the problem of classical strong amplitude wise simulation of n qubit quantum circuits and identify a subclass of simulators we call monotone this subclass encompasses almost all prominent simulation techniques we prove an unconditional i e without relying on any complexity theoretic assumptions and explicit n 2 2 n 3 1 lower bound on the running time of simulators within this subclass assuming the strong exponential time hypothesis seth we further remark that a universal simulator computing any amplitude to precision 2 n 2 must take at least 2 n o n time we then compare strong simulators to existing sat solvers and identify the time complexity below which a strong simulator would improve on state of the art general sat solving finally we investigate clifford t quantum circuits with t t gates using the sparsification lemma we identify a time complexity lower bound of 2 2 2451 times 10 8 t below which a strong simulator would improve on state of the art 3 sat solving this also yields a conditional exponential lower bound on the growth of the stabilizer rank of magic states

beyond csp a dichotomy for counting weighted eulerian orientations with ars abstract we define and explore a notion of unique prime factorization for constraint functions and use this as a new tool to prove a complexity classification for counting weighted eulerian orientation problems with arrow reversal symmetry ars we prove that all such problems are either polynomial time computable or p hard we show that the class of weighted eulerian orientation problems subsumes all weighted counting constraint satisfaction problems csp on boolean variables more significantly we establish a novel connection between csp and counting weighted eulerian orientation problems that is global in nature this connection is based on a structural determination of all half weighted affine linear subspaces over z 2 which is proved using mobius inversion

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

pp is closed under truth table reductions beigel reingold and spielman j comput system sci 50 191 202 1995 showed that pp is closed under intersection and a variety of special cases of polynomial time truth table closure we extend their techniques to show that pp is closed under general polynomial time truth table reductions we also show that pp is closed under constant round truth table reductions

horn of singularities for the stark wannier ladders the authors prove that the small field asymptotic behaviour of the stark wannier ladders near the real direction is generically highly singular this result is in agreement with the conjecture of a chaotic behaviour of the lifetime of the states because of infinitely many crossings

existence of stark ladder resonances we show that resonances in the translation analytic sense of herbst and howland exist for the one dimensional stark hamiltonian d2 dx2 q x ex withq x a trigonometric polynomial providede is sufficiently large

a computational proof of complexity of some restricted counting problems we explore a computational approach to proving the intractability of certain counting problems these problems can be described in various ways and they include concrete problems such as counting the number of vertex covers or independent sets for 3 regular graphs the high level principle of our approach is algebraic which provides sufficient conditions for interpolation to succeed another algebraic component is holographic reductions we then analyze in detail polynomial maps on r 2 induced by some combinatorial constructions these maps define sufficiently complicated dynamics of r 2 that we can only analyze them computationally in this paper we use both numerical computation as intuitive guidance and symbolic computation as proof theoretic verification to derive that a certain collection of combinatorial constructions in myriad combinations fulfills the algebraic requirements of proving p hardness the final result is a dichotomy theorem for a class of counting problems this includes a class of generic holant problems with an arbitrary real valued edge signature over 2 3 regular undirected graphs in particular it includes all partition functions with 0 1 vertex assignments and an arbitrary real valued edge function over all 3 regular undirected graphs

expressiveness of matchgates matchgates have been used to simulate classically in polynomial time assemblies of quantum gates in this paper it is shown that every matchgate for 2 input 2 output functions has to obey a certain set of five polynomial identities it is also shown that no matchgate can realize a nontrivial control gate for any number of inputs on the other hand it is proved that classical boolean formulae can be expressed as matchcircuits of polynomial size

making classical ground state spin computing fault tolerant we examine a model of classical deterministic computing in which the ground state of the classical system is a spatial history of the computation this model is relevant to quantum dot cellular automata as well as to recent universal adiabatic quantum computing constructions in its most primitive form systems constructed in this model cannot compute in an error free manner when working at nonzero temperature however by exploiting a mapping between the partition function for this model and probabilistic classical circuits we are able to show that it is possible to make this model effectively error free we achieve this by using techniques in fault tolerant classical computing and the result is that the system can compute effectively error free if the temperature is below a critical temperature we further link this model to computational complexity and show that a certain problem concerning finite temperature classical spin systems is complete for the complexity class merlin arthur this provides an interesting connection between the physical behavior of certain many body spin systems and computational complexity

quantum circuits and low degree polynomials over in this work we explore a correspondence between quantum circuits and low degree polynomials over the finite field any quantum circuit made up of hadamard z controlled z and controlled controlled z gates gives rise to a degree 3 polynomial over such that calculating quantum circuit amplitudes is equivalent to counting zeroes of the corresponding polynomial we exploit this connection which is especially clean and simple for this particular gate set in two directions first we give proofs of classical hardness results based on quantum circuit concepts second we find efficient classical simulation algorithms for certain classes of quantum circuits based on efficient algorithms for classes of polynomials

quantum theory of probability and decisions the probabilistic predictions of quantum theory are conventionally obtained from a special probabilistic axiom but that is unnecessary because all the practical consequences of such predictions fo

holographic algorithm with matchgates is universal for planar csp over boolean domain we prove a complexity classification theorem that classifies all counting constraint satisfaction problems csp over boolean variables into exactly three classes 1 polynomial time solvable 2

quantum matchgate computations and linear threshold gates the theory of matchgates is of interest in various areas in physics and computer science matchgates occur for example in the study of fermions and spin chains in the theory of holographic algorithms and in several recent works in quantum computation in this paper we completely characterize the class of boolean functions computable by unitary two qubit matchgate circuits with some probability of success we show that this class precisely coincides with that of the linear threshold gates the latter is a fundamental family that appears in several fields such as the study of neural networks using the above characterization we further show that the power of matchgate circuits is surprisingly trivial in those cases where the computation is to succeed with high probability in particular the only functions that are matchgate computable with success probability greater than 3 4 are functions depending on only a single bit of the input