Bounds on oblivious multiparty quantum communication complexity

The main conceptual contribution of this paper is investigating quantum multiparty communication complexity in the setting where communication is \emph{oblivious}. This requirement, which to our knowledge is satisfied by all quantum multiparty protocols in the literature, means that the communication pattern, and in particular the amount of communication exchanged between each pair of players at each round is fixed \emph{independently of the input} before the execution of the protocol. We show, for a wide class of functions, how to prove strong lower bounds on their oblivious quantum $k$-party communication complexity using lower bounds on their \emph{two-party} communication complexity. We apply this technique to prove tight lower bounds for all symmetric functions with \textsf{AND} gadget, and in particular obtain an optimal $\Omega(k\sqrt{n})$ lower bound on the oblivious quantum $k$-party communication complexity of the $n$-bit Set-Disjointness function. We also show the tightness of these lower bounds by giving (nearly) matching upper bounds.Comment: 13 pages, an accepted paper of LATIN 202

Quantum communication complexity of linear regression

Dequantized algorithms show that quantum computers do not have exponential speedups for many linear algebra problems in terms of time and query complexity. In this work, we show that quantum computers can have exponential speedups in terms of communication complexity for some fundamental linear algebra problems. We mainly focus on solving linear regression and Hamiltonian simulation. In the quantum case, the task is to prepare the quantum state of the result. To allow for a fair comparison, in the classical case the task is to sample from the result. We investigate these two problems in two-party and multiparty models, propose near-optimal quantum protocols and prove quantum/classical lower bounds. In this process, we propose an efficient quantum protocol for quantum singular value transformation, which is a powerful technique for designing quantum algorithms. As a result, for many linear algebra problems where quantum computers lose exponential speedups in terms of time and query complexity, it is possible to have exponential speedups in terms of communication complexity.Comment: 28 page

A Lower Bound for Sampling Disjoint Sets

Suppose Alice and Bob each start with private randomness and no other input, and they wish to engage in a protocol in which Alice ends up with a set x subseteq[n] and Bob ends up with a set y subseteq[n], such that (x,y) is uniformly distributed over all pairs of disjoint sets. We prove that for some constant beta0 of the uniform distribution over all pairs of disjoint sets of size sqrt{n}

Nondeterministic quantum communication complexity: the cyclic equality game and iterated matrix multiplication

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

Separating NOF communication complexity classes RP and NP

We provide a non-explicit separation of the number-on-forehead communication complexity classes RP and NP when the number of players is up to \delta log(n) for any \delta<1. Recent lower bounds on Set-Disjointness [LS08,CA08] provide an explicit separation between these classes when the number of players is only up to o(loglog(n))

A Hypercontractive Inequality for Matrix-Valued Functions with Applications to Quantum Computing and LDCs

The Bonami-Beckner hypercontractive inequality is a powerful tool in Fourier analysis of real-valued functions on the Boolean cube. In this paper we present a version of this inequality for matrix-valued functions on the Boolean cube. Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also present a number of applications. First, we analyze maps that encode $n$ classical bits into $m$ qubits, in such a way that each set of $k$ bits can be recovered with some probability by an appropriate measurement on the quantum encoding; we show that if $m<0.7 n$, then the success probability is exponentially small in $k$. This result may be viewed as a direct product version of Nayak's quantum random access code bound. It in turn implies strong direct product theorems for the one-way quantum communication complexity of Disjointness and other problems. Second, we prove that error-correcting codes that are locally decodable with 2 queries require length exponential in the length of the encoded string. This gives what is arguably the first ``non-quantum'' proof of a result originally derived by Kerenidis and de Wolf using quantum information theory, and answers a question by Trevisan.Comment: This is the full version of a paper that will appear in the proceedings of the IEEE FOCS 08 conferenc