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

homological stability for automorphism groups abstract given a family of groups admitting a braided monoidal structure satisfying mild assumptions we construct a family of spaces on which the groups act and whose connectivity yields via a classical argument of quillen homological stability for the family of groups we show that stability also holds with both polynomial and abelian twisted coefficients with no further assumptions this new construction of a family of spaces from a family of groups recovers known spaces in the classical examples of stable families of groups such as the symmetric groups general linear groups and mapping class groups by making systematic the proofs of classical stability results we show that they all hold with the same type of coefficient systems obtaining in particular without any further work new stability theorems with twisted coefficients for the symmetric groups braid groups automorphisms of free groups unitary groups mapping class groups of non orientable surfaces and mapping class groups of 3 manifolds our construction can also be applied to families of groups not considered before in the context of homological stability as a byproduct of our work we construct the braided analogue of the category fi of finite sets and injections relevant to the present context and define polynomiality for functors in the context of pre braided monoidal categories

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

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

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

a dichotomy theorem for constraint satisfaction problems on a 3 element set the constraint satisfaction problem csp provides a common framework for many combinatorial problems the general csp is known to be np complete however certain restrictions on a possible form of constraints may affect the complexity and lead to tractable problem classes there is therefore a fundamental research direction aiming to separate those subclasses of the csp that are tractable and those which remain np complete schaefer gave an exhaustive solution of this problem for the csp on a 2 element domain in this article we generalise this result to a classification of the complexity of the csp on a 3 element domain the main result states that every subproblem of the csp is either tractable or np complete and the criterion separating them is that conjectured in bulatov et al 2005 and bulatov and jeavons 2001b we also characterize those subproblems for which standard constraint propagation techniques provide a decision procedure finally we exhibit a polynomial time algorithm which for a given set of allowed constraints outputs if this set gives rise to a tractable problem class to obtain the main result and the algorithm we extensively use the algebraic technique for the csp developed in jeavons 1998b bulatov et al 2005 and bulatov and jeavons 2001b

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

the complexity of the counting constraint satisfaction problem the counting constraint satisfaction problem csp h over a finite relational structure h can be expressed as follows given a relational structure g over the same vocabulary determine the number of homomorphisms from g to h in this article we characterize relational structures h for which csp h can be solved in polynomial time and prove that for all other structures the problem is p complete

stability of the homology of the mapping class groups of orientable surfaces the mapping class group of f fgs r is f rgs wo a where a is the topological group of orientation preserving diffeomorphisms of f which are the identity on df and fix the s punctures when r 0 and 2g s 3 an important feature of f is that it acts properly discontinuously on the teichmiller space us the quotient is the moduli space f of isometry classes of complete hyperbolic metrics of finite area on f since us is homeomorphic to euclidean space and f is virtually torsion free we have

a complexity dichotomy for partition functions with mixed signs partition functions also known as homomorphism functions form a rich family of graph invariants that contain combinatorial invariants such as the number of k colorings or the number of independent sets of a graph and also the partition functions of certain spin glass models of statistical physics such as the ising model building on earlier work by dyer and greenhill random structures algorithms 17 2000 pp 260 289 and bulatov and grohe theoret comput sci 348 2005 pp 148 186 we completely classify the computational complexity of partition functions our main result is a dichotomy theorem stating that every partition function is either computable in polynomial time or p complete partition functions are described by symmetric matrices with real entries and we prove that it is decidable in polynomial time in terms of the matrix whether a given partition function is in polynomial time or p complete while in general it is very complicated to give an explicit algebraic or combinatorial description of the tractable cases for partition functions described by hadamard matrices these turn out to be central in our proofs we obtain a simple algebraic tractability criterion which says that the tractable cases are those representable by a quadratic polynomial over the field mathbb f 2

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

iterated monoidal categories abstract we develop a notion of an n fold monoidal category and show that it corresponds in a precise way to the notion of an n fold loop space specifically the group completion of the nerve of such a category is an n fold loop space and free n fold monoidal categories give rise to a finite simplicial operad of the same homotopy type as the classical little cubes operad used to parametrize the higher h space structure of an n fold loop space we also show directly that this operad has the same homotopy type as the n th smith filtration of the barratt eccles operad and the n th filtration of berger s complete graph operad moreover this operad contains an equivalent preoperad which gives rise to milgram s small model for 2 2 x when n 2 and is very closely related to milgram s model of n n x for n 2

robust iterative inversion for the one dimensional acoustic wave equation seismic inversion can be formulated by considering a linearized integral relation which is deduced from the wave equation this born inversion approach is equivalent to linear least squares inversion for a particular parameterization of the medium the least squares solution is a member of a family of generalized lp norm solutions which are deduced from a maximum likelihood formulation this formulation allows design of various statistical inversion solutions we present two iterative solutions to the one dimensional 1 d seismic inverse problem the iterative least squares ils and the iterative reweighted least squares irls methods the ils method involves solving a distorted background velocity problem after the initial least squares solution is obtained the irls method is used as a robust regression technique which is better suited for dealing with certain types of noise and is computationally faster than ils several numerical examples demonstrate that the irls method accurately estimates impeda

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 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

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

homological stability for moduli spaces of high dimensional manifolds i s galatius was partially supported by nsf grants dms 1105058 and dms 1405001 the european research council erc under the european union s horizon 2020 research and innovation pro gramme grant agreement no 682922 as well as the danish national research foundation through the centre for symmetry and deformation dnrf92 and erc 682992 o randal williams was partially supported by the herchel smith fund and epsrc grant ep m027783 1

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

generalized linear inversion and the first born theory for acoustic media a procedure is derived which incorporates a generalized linear inverse viewpoint within a multidimensional born inversion method the method we present is a more general born theory which can accommodate insufficient and inaccurate data this general method reduces to the ordinary born procedure when the data requirements of the latter technique are satisfied

stabilization for mapping class groups of 3 manifolds we prove that the homology of the mapping class group of any 3manifold stabilizes under connected sum and boundary connected sum with an arbitrary 3 manifold when both manifolds are compact and orientable the stabilization also holds for the quotient group by twists along spheres and disks and includes as particular cases homological stability for symmetric automorphisms of free groups automorphisms of certain free products and handlebody mapping class groups our methods also apply to manifolds of other dimensions in the case of stabilization by punctures the main result of this paper is a homological stability theorem for mapping class groups of 3 manifolds where the stabilization is by connected sum with an arbitrary 3 manifold more precisely we show that given any two compact connected oriented 3 manifolds n and p with n 6 the homology group hi 0diff n p p rel n z is independent of the number n of copies of p in the connected sum as long as n 2i 2 i e each homology group stabilizes with p we also prove an analogous result for boundary connected sum and a version for the quotient group of the mapping class group by twists along spheres and disks a group closely related to the automorphism group of the fundamental group of the manifold homological stability theorems were first found in the sixties for symmetric groups by nakaoka 36 and linear groups by quillen and now form the foundation of modern algebraic k theory see for example 28 part iv and 42 stability theorems for mapping class groups of surfaces were obtained in the eighties by harer and ivanov 14 25 and recently turned out to be a key ingredient to a solution of the mumford conjecture about the homology of the riemann moduli space 30 the other main examples of families of groups for which stability has been known are braid groups 1 and automorphism groups of free groups 17 18 the present paper extends significantly the class of groups for which homological stability is known to hold it suggests that it is a widespread phenomenon among families of groups containing enough symmetries in addition to the already mentioned stability theorems for mapping class groups of 3 manifolds corollaries of our main result include stability for handlebody subgroups of surface mapping class groups symmetric automorphism groups of free groups and automorphism groups of free products ng for many groups g using similar techniques we obtain stability results also for mapping class groups 0diff m n points rel m for m any m dimensional manifold with boundary m 2 even the case m 2 is new here as well as for the fundamental group 1conf m n of the configuration space of n unordered points in m our paper thus also unifies previous known results as we recover stability for braid groups as 1conf d n symmetric groups as 1conf d n and automorphism groups of free groups as aut nz

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

nearfield inverse scattering formalism for the three dimensional wave equation the inclusion of apriori velocity information we address the problem of identifying a local velocity c r in a three dimensional scalar wave equation from nearfield data the implicit model assumptions of previous nearfield inverse scattering procedures are examined these assumptions cannot be reasonably assumed in certain applications therefore a method for extending the range of validity of nearfield inverse scattering procedures is presented in addition we present a technique for incorporating a priori information or estimates into this formal wave equation inversion procedure an approximate perturbative solution for this formalism has the ability to accommodate selective and potentially troublesome multiples within the born model

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

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

reflection positivity rank connectivity and homomorphism of graphs it is shown that a graph parameter can be realized as the number of homomorphisms into a fixed weighted graph if and only if it satisfies two linear algebraic conditions reflection positivity and exponential rank connectivity in terms of statistical physics this can be viewed as a characterization of partition functions of vertex models

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

the complexity of weighted boolean csp this paper gives a dichotomy theorem for the complexity of computing the partition function of an instance of a weighted boolean constraint satisfaction problem the problem is parameterized by a finite set mathcal f of nonnegative functions that may be used to assign weights to the configurations feasible solutions of a problem instance classical constraint satisfaction problems correspond to the special case of 0 1 valued functions we show that computing the partition function i e the sum of the weights of all configurations is text sf fp text sf p complete unless either 1 every function in mathcal f is of product type or 2 every function in mathcal f is pure affine in the remaining cases computing the partition function is in p

poset fiber theorems suppose that f p q is a poset map whose fibers f 1 qless than or equal to q are sufficiently well connected our main result is a formula expressing the homotopy type of p in terms of q and the fibers several fiber theorems from the literature due to babson baclawski and quillen are obtained as consequences or special cases homology cohen macaulay and equivariant versions are given and some applications are discussed

on the edge of the stable range we prove a general homological stability theorem for certain families of groups equipped with product maps followed by two theorems of a new kind that give information about the last two homology groups outside the stable range these last two unstable groups are the edge in our title applying our results to automorphism groups of free groups yields a new proof of homological stability with an improved stable range a description of the last unstable group up to a single ambiguity and a lower bound on the rank of the penultimate unstable group we give similar applications to the general linear groups of the integers and of the field of order 2 this time recovering the known stability range the results can also be applied to general linear groups of arbitrary principal ideal domains symmetric groups and braid groups our methods require us to use field coefficients throughout

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