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arXiv:1202.6444 · IEICE Transactions on Information and Systems

Tensor Rank and Strong Quantum Nondeterminism in Multiparty Communication

Marcos Villagra, Masaki Nakanishi, Shigeru Yamashita, Yasuhiko Nakashima

Published Dec 31, 2012• 4 citations
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Abstract

In this paper we study quantum nondeterminism in multiparty communication. There are three (possibly) different types of nondeterminism in quantum computation: i) strong, ii) weak with classical proofs, and iii) weak with quantum proofs. Here we focus on the first one. A strong quantum nondeterministic protocol accepts a correct input with positive probability and rejects an incorrect input with probability 1. In this work we relate strong quantum nondeterministic multiparty communication complexity to the rank of the communication tensor in the Number-On-Forehead and Number-In-Hand models. In particular, by extending the definition proposed by de Wolf to nondeterministic tensor-rank (nrank), we show that for any boolean function f when there is no prior shared entanglement between the players, 1) in the Number-On-Forehead model the cost is upper-bounded by the logarithm of nrank(f); 2) in the Number-In-Hand model the cost is lower-bounded by the logarithm of nrank(f). Furthermore, we show that when the number of players is o(log log n), we have $NQP\nsubseteq BQP$ for Number-On-Forehead communication.

DOI
10.1587/transinf.e96.d.1

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In this paper we study quantum nondeterminism in multiparty communication. There are three (possibly) different types of nondeterminism in quantum computation: i) strong, ii) weak with classical proofs, and iii) weak with quantum proofs. Here we focus on the first one. A strong quantum nondeterministic protocol accepts a correct input with positive probability and rejects an incorrect input with probability 1. In this work we relate strong quantum nondeterministic multiparty communication complexity to the rank of the communication tensor in the Number-On-Forehead and Number-In-Hand models. In particular, by extending the definition proposed by de Wolf to nondeterministic tensor-rank (nrank), we show that for any boolean function f when there is no prior shared entanglement between the players, 1) in the Number-On-Forehead model the cost is upper-bounded by the logarithm of nrank(f); 2) in the Number-In-Hand model the cost is lower-bounded by the logarithm of nrank(f). Furthermore, we show that when the number of players is o(log log n), we have NQP⊈BQPNQP\nsubseteq BQPNQP⊈BQP for Number-On-Forehead communication.
There are three different types of nondeterminism in quantum communication: i) $\nqp$-communication, ii) $\qma$-communication, and iii) $\qcma$-communication. In this \redout{paper} we show that multiparty $\nqp$-communication can be exponentially stronger than $\qcma$-communication. This also implies an exponential separation with respect to classical multiparty nondeterministic communication complexity. We argue that there exists a total function that is hard for $\qcma$-communication and easy for $\nqp$-communication. The proof of it involves an application of the pattern tensor method and a new lower bound for polynomial threshold degree. Another important consequence of this result is that nondeterministic rank can be exponentially lower than the discrepancy bound.
We study nondeterministic multiparty quantum communication with a quantum generalization of broadcasts. We show that, with number-in-hand classical inputs, the communication complexity of a Boolean function in this communication model equals the logarithm of the support rank of the corresponding tensor, whereas the approximation complexity in this model equals the logarithm of the border support rank. This characterisation allows us to prove a log-rank conjecture posed by Villagra et al. for nondeterministic multiparty quantum communication with message-passing. The support rank characterization of the communication model connects quantum communication complexity intimately to the theory of asymptotic entanglement transformation and algebraic complexity theory. In this context, we introduce the graphwise equality problem. For a cycle graph, the complexity of this communication problem is closely related to the complexity of the computational problem of multiplying matrices, or more precisely, it equals the logarithm of the asymptotic support rank of the iterated matrix multiplication tensor. We employ Strassen's laser method to show that asymptotically there exist nontrivial protocols for every odd-player cyclic equality problem. We exhibit an efficient protocol for the 5-player problem for small inputs, and we show how Young flattenings yield nontrivial complexity lower bounds.
We study nondeterministic multiparty quantum communication with a quantum generalization of broadcasts. We show that, with number-in-hand classical inputs, the communication complexity of a Boolean function in this communication model equals the logarithm of the support rank of the corresponding tensor, whereas the approximation complexity in this model equals the logarithm of the border support rank. This characterisation allows us to prove a log-rank conjecture posed by Villagra et al. for nondeterministic multiparty quantum communication with message passing. The support rank characterization of the communication model connects quantum communication complexity intimately to the theory of asymptotic entanglement transformation and algebraic complexity theory. In this context, we introduce the graphwise equality problem. For a cycle graph, the complexity of this communication problem is closely related to the complexity of the computational problem of multiplying matrices, or more precisely, it equals the logarithm of the support rank of the iterated matrix multiplication tensor. We employ Strassen's laser method to show that asymptotically there exist nontrivial protocols for every odd-player cyclic equality problem. We exhibit an efficient protocol for the 5-player problem for small inputs, and we show how Young flattenings yield nontrivial complexity lower bounds.
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.
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.
We study nondeterministic quantum algorithms for Boolean functions f. Such algorithms have positive acceptance probability on input x iff f(x)=1. In the setting of query complexity, we show that the nondeterministic quantum complexity of a Boolean function is equal to its "nondeterministic polynomial" degree. We also prove a quantum-vs.-classical gap of 1 vs. n for nondeterministic query complexity for a total function. In the setting of communication complexity, we show that the nondeterministic quantum complexity of a two-party function is equal to the logarithm of the rank of a nondeterministic version of the communication matrix. This implies that the quantum communication complexities of the equality and disjointness functions are n+1 if we do not allow any error probability. We also exhibit a total function in which the nondeterministic quantum communication complexity is exponentially smaller than its classical counterpart.