arXiv:cs/9910010 · IEEE Conference on Computational Complexity (CCC)

Communication complexity lower bounds by polynomials

Harry Buhrman, Ronald de Wolf

Published Nov 13, 2002• 127 citations

Abstract

The quantum version of communication complexity allows the two communicating parties to exchange qubits and/or to make use of prior entanglement (shared EPR-pairs). Some lower bound techniques are available for qubit communication complexity, but except for the inner product function, no bounds are known for the model with unlimited prior entanglement. We show that the "log rank" lower bound extends to the strongest variant of quantum communication complexity (qubit communication+unlimited prior entanglement). By relating the rank of the communication matrix to properties of polynomials, we are able to derive some strong bounds for exact protocols. In particular, we prove both the "log rank conjecture" and the polynomial equivalence of quantum and classical communication complexity for various classes of functions. We also derive some weaker bounds for bounded-error quantum protocols.

DOI: 10.1109/ccc.2001.933879

Related papers

Renyi-entropic bounds on quantum communication

Wim van Dam, Patrick M. Hayden

In this article we establish new bounds on the quantum communication complexity of distributed problems. Specifically, we consider the amount of communication that is required to transform a bipartite state into another, typically more entangled, state. We obtain lower bounds in this setting by studying the Renyi entropy of the marginal density matrices of the distributed system. The communication bounds on quantum state transformations also imply lower bounds for the model of communication complexity where the task consists of the the distributed evaluation of a function f(x,y). Our approach encapsulates several known lower bound methods that use the log-rank or the von Neumann entropy of the density matrices involved. The technique is also effective for proving lower bounds on problems involving a promise or for which the "hard" distributions of inputs are correlated. As examples, we show how to prove a nearly tight bound on the bounded-error quantum communication complexity of the inner product function in the presence of unlimited amounts of EPR-type entanglement and a similarly strong bound on the complexity of the shifted quadratic character problem.

arXiv:quant-ph/0204093Apr 17, 200227 citations

An approximation algorithm for approximation rank

Troy Lee, Adi Shraibman

One of the strongest techniques available for showing lower bounds on quantum communication complexity is the logarithm of the approximation rank of the communication matrix--the minimum rank of a matrix which is entrywise close to the communication matrix. This technique has two main drawbacks: it is difficult to compute, and it is not known to lower bound quantum communication complexity with entanglement. Linial and Shraibman recently introduced a norm, called gamma_2^{alpha}, to quantum communication complexity, showing that it can be used to lower bound communication with entanglement. Here the parameter alpha is a measure of approximation which is related to the allowable error probability of the protocol. This bound can be written as a semidefinite program and gives bounds at least as large as many techniques in the literature, although it is smaller than the corresponding alpha-approximation rank, rk_alpha. We show that in fact log gamma_2^{alpha}(A)$ and log rk_{alpha}(A)$ agree up to small factors. As corollaries we obtain a constant factor polynomial time approximation algorithm to the logarithm of approximate rank, and that the logarithm of approximation rank is a lower bound for quantum communication complexity with entanglement.

arXiv:0809.2093Sep 11, 20081 citation

Approximation norms and duality for communication complexity lower bounds

Troy Lee, Adi Shraibman

Abstract: We will discuss a general norm based framework for showing lower bounds on communication complexity. An advantage of this approach is that one can use duality theory to obtain a lower bound quantity phrased as a maximization problem, which can be more convenient to work with in showing lower bounds. We discuss two applications of this approach. 1. The approximation rank of a matrix A is the minimum rank of a matrix close to A in ell_infty norm. The logarithm of approximation rank lower bounds quantum communication complexity and is one of the most powerful techniques available, albeit difficult to compute in practice. We show that an approximation norm known as gamma_2 is polynomially related to approximation rank. This results in a polynomial time algorithm to approximate approximation rank, and also shows that the logarithm of approximation rank lower bounds quantum communication complexity even with entanglement which was previously not known. 2. By means of an approximation norm which lower bounds multiparty number-on-the-forehead complexity, we show non-trivial lower bounds on the complexity of the disjointness function for up to c log log n players, c <1.

Dagstuhl Seminar ProceedingsJan 1, 20080 citations

Lower Bounds on Quantum Multiparty Communication Complexity

Troy Lee, Gideon Schechtman, Adi Shraibman

A major open question in communication complexity is if randomized and quantum communication are polynomially related for all total functions. So far, no gap larger than a power of two is known, despite significant efforts. We examine this question in the number-on-the-forehead model of multiparty communication complexity. We show that essentially all lower bounds known on randomized complexity in this model also hold for quantum communication. This includes bounds of size Omega(n/2k) for the k-party complexity of explicit functions, bounds for the generalized inner product function, and recent work on the multiparty complexity of disjointness. To the best of our knowledge, these are the first lower bounds of any kind on quantum communication in the general number-on-the-forehead model. We show this result in the following way. In the two-party case, there is a lower bound on quantum communication complexity in terms of a norm gamma2, which is known to subsume nearly all other techniques in the literature. For randomized complexity there is another natural bound in terms of a different norm mu which is also one of the strongest techniques available. A deep theorem in functional analysis, Grothendieck's inequality, implies that gamma2and mu are equivalent up to a constant factor. This connection is one of the major obstacles to showing a larger gap between randomized and quantum communication complexity in the two-party case. The lower bound technique in terms of the norm mu was recently extended to the multiparty number-on-the-forehead model. Here we show how the gamma2norm can be also extended to lower bound quantum multiparty complexity. Surprisingly, even in this general setting the two lower bounds, on quantum and classical communication, are still very closely related. This implies that separating quantum and classical communication in this setting will require the development of new techniques. The relation between these extensions of mu and gamma2is proved by a multi-dimensional version of Grothendieck's inequality.

IEEE Conference on Computational Complexity (CCC)Jul 1, 200921 citations

Quantum Communication Complexity and Nonlocality of Bipartite Quantum Operations.

Yufan Zhu

This dissertation is motivated by the following fundamental questions: (a) are there any exponential gaps between quantum and classical communication complexities? (b) what is the role of entanglement in assisting quantum communications? (c) how to characterize the nonlocality of quantum operations? We study four specific problems below. 1. The communication complexity of the Hamming Distance problem. The Hamming Distance problem is for two parties to determine whether or not the Hamming distance between two n-bit strings is more than a given threshold. We prove tighter quantum lower bounds in the general two-party, interactive communication model. We also construct an efficient classical protocol in the more restricted Simultaneous Message Passing model, improving previous results. 2. The Log-Equivalence Conjecture. A major open problem in communication complexity is whether or not quantum protocols can be exponentially more efficient than classical ones for computing a total Boolean function in the two-party, interactive model. The answer is believed to be No. Razborov proved this conjecture for the most general class of functions so far. We prove this conjecture for a broader class of functions that we called block-composed functions. Our proof appears to be the first demonstration of the dual approach of the polynomial method in proving new results. 3. Classical simulations of bipartite quantum measurement. We define a new ix concept that measures the nonlocality of bipartite quantum operations. From this measure, we derive an upper bound that shows the limitation of entanglement in reducing communication costs. 4. The maximum tensor norm of bipartite superoperators. We define a maximum tensor norm to quantify the nonlocality of bipartite superoperators. We show that a bipartite physically realizable superoperator is bi-local if and only if its maximum tensor norm is 1. Furthermore, the estimation of the maximum tensor norm can also be used to prove quantum lower bounds on communication complexities.

Jan 1, 20080 citations

On the Role of Shared Entanglement

Gavinsky, Dmytro

Despite the apparent similarity between shared randomness and shared entanglement in the context of Communication Complexity, our understanding of the latter is not as good as of the former. In particular, there is no known "entanglement analogue" for the famous theorem by Newman, saying that the number of shared random bits required for solving any communication problem can be at most logarithmic in the input length (i.e., using more than O(log n) shared random bits would not reduce the complexity of an optimal solution). In this paper we prove that the same is not true for entanglement. We establish a wide range of tight (up to a polylogarithmic factor) entanglement vs. communication tradeoffs for relational problems. The low end is: for any t>2, reducing shared entanglement from log^t(n) to o(log^{t-2}(n)) qubits can increase the communication required for solving a problem almost exponentially, from O(log^t(n)) to Ω(\sqrt n). The high end is: for any \eps>0, reducing shared entanglement from n^{1-\eps}log(n) to o(n^{1-\eps}/log(n)) can increase the required communication from O(n^{1-\eps}log(n)) to Ω(n^{1-\eps/2}/log(n)). The upper bounds are demonstrated via protocols which are exact and work in the \e{simultaneous message passing model}, while the lower bounds hold for \e{bounded-error protocols}, even in the more powerful \e{model of 1-way communication}. Our protocols use shared EPR pairs while the lower bounds apply to any sort of prior entanglement. We base the lower bounds on a strong direct product theorem for communication complexity of a certain class of relational problems. We believe that the theorem might have applications outside the scope of this work.

arXiv:quant-ph/0604052Apr 8, 20061 citation