We study a complexity model of quantum circuits analogous to the standard (acyclic) Boolean circuit model. It is shown that any function computable in polynomial time by a quantum Turing machine has a polynomial-size quantum circuit. This result also enables us to construct a universal quantum computer which can simulate, with a polynomial factor slowdown, a broader class of quantum machines than that considered by Bernstein and Vazirani [BV93], thus answering an open question raised in [BV93]. We also develop a theory of quantum communication complexity, and use it as a tool to prove that the majority function does not have a linear-size quantum formula. Keywords. Boolean circuit complexity, communication complexity, quantum communication complexity, quantum computation AMS subject classifications. 68Q05, 68Q15 1 This research was supported in part by the National Science Foundation under grant CCR-9301430. 1 Introduction One of the most intriguing questions in computation theroy ...

We show that any language recognized by an NC ’ circuit (fan-in 2, depth O(log n)) can be recognized by a width-5 polynomial-size branching program. As any bounded-width polynomial-size branching program can be simulated by an NC ’ circuit, we have that the class of languages recognized by such programs is exactly nonuniform NC’. Further, following

This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.) or open problems they would like publicized, should

We survey lower bounds established for the complexity of computing explicitly given Boolean functions by switching-and-rectifier networks, branching programs and switching networks. We first consider the unrestricted case and then proceed to various restricted models. Among these are monotone networks, bounded-width devices , oblivious devices and read-k times only devices. 1 Introduction The main goal of the Boolean complexity theory is to prove lower bounds on the complexity of computing "explicitly given" Boolean functions in interesting computational models. By "explicitly given" researchers usually mean "belonging to the class NP ". This is a very plausible interpretation since on the one hand this class contains the overwhelming majority of interesting Boolean functions and on the other hand it is small enough to prevent us from the necessity to take into account counting arguments. To illustrate the second point, let me remind the reader that already the class \Delta p 2 ,...

This is a survey of space-bounded probabilistic computation, summarizing the present state of knowledge about the relationships between the various complexity classes associated with such computation. The survey especially emphasizes recent progress in the construction of pseudorandom generators that fool probabilistic space-bounded computations, and the application of such generators to obtain deterministic simulations.

The "Number on the Forehead" model of multi-party communication complexity was first suggested by Chandra, Furst and Lipton. The best known lower bound, for an explicit function (in this model), is a lower bound of \Omega\Gamma n=2 k ), where n is the size of the input of each player, and k is the number of players (first proved by Babai, Nisan and Szegedy). This lower bound has many applications in complexity theory. Proving a better lower bound, for an explicit function, is a major open problem. Based on the result of BNS, Chung gave a sufficient criterion for a function to have large multi-partycommunication -complexity (up to \Omega\Gamma n=2 k )). In this paper, we use some of the ideas of BNS, and Chung, together with some new ideas, resulting in a new (easier and more modular) proof for the results of BNS and Chung. This gives a simpler way to prove lower bounds for the multi-party-communication-complexity of a function. 1 Multi-Party Communication Complexity Multi-party co...

In the multiparty communication game introduced by Chandra, Furst, and Lipton [CFL] (1983), k players wish to evaluate collaboratively a function f(x0 , ..., xk\Gamma1 ) for which player i sees all inputs except x i : The players have unlimited computational power. The objective is to minimize the amount of communication. We consider a restricted version of the multiparty communication game which we call the simultaneous messages model. The difference is that in this model, each of the k players simultaneously sends a message to a referee, who sees none of the input. The referee then announces the function value. We demonstrate an exponential gap between the Simultaneous Messages and the Communication models for up to (log n) 1\Gammaffl players, for any ffl ? 0: The separation is obtained by comparing the respective complexities of the generalized addressing function, GAFG;k , in each model. In addition, we give a nontrivial protocol for GAFG;k for G = Z t 2 ; which is very eff...

We use tools and techniques from information theory to study communication complexity problems in the one-way and simultaneous communication models. Our results include: (1) A tight characterization of multi-party one-way communication complexity for product distributions in terms of VC-dimension and shatter coefficients; (2) An equivalence of multi-party one-way and simultaneous communication models for product distributions; (3) A suite of lower bounds for specific functions in the simultaneous communication model, most notably an optimal lower bound for the multi-party set disjointness problem of Alon et al. [AMS99] and for the generalized addressing function problem of Babai et al. [BGKL96] for arbitrary groups. Methodologically, our main contribution is rendering communication complexity problems in the framework of information theory. This allows us access to the powerful calculus of information theory and the use of fundamental principles such as Fano's inequality and the Maximum Likelihood Estimate Principle.

In this paper we survey the major developments in understanding the complexity of the graph connectivity problem in several computational models, and highlight some challenging open problems. 1

. We investigate two methods for proving lower bounds on the size of small depth circuits, namely the approaches based on multiparty communication games and algebraic characterizations extending the concepts of the tensor rank and rigidity of matrices. Our methods are combinatorial, but we think that the main contribution concerns the algebraic concepts used in this area (tensor ranks and rigidity). Our main results are following. (i) An o(n) bit protocol for a communication game for computing shifts, which also gives an upper bound of o(n 2 ) on the contact rank of the tensor of multiplication of polynomials; this disproves some earlier conjectures. A related probabilistic construction gives o(n) upper bound for computing all permutations and O(n log log n) upper bound on the communication complexity of pointer jumping with permutations. (ii) A lower bound on certain restricted circuits of depth 2 which are related to the problem of proving a superlinear lower bound on the size of ...