Tensor Rank and Strong Quantum Nondeterminism in Multiparty Communication

TL;DR

This paper investigates the relationship between strong quantum nondeterministic multiparty communication complexity and tensor rank, establishing bounds and complexity class separations in quantum communication models.

Contribution

It introduces the nondeterministic tensor-rank concept and relates it to communication complexity, providing bounds and class separations in multiparty quantum communication.

Findings

Communication cost bounds are expressed in terms of nondeterministic tensor rank.

In the Number-On-Forehead model, cost is upper-bounded by log of nrank.

For small number of players, NQP is not contained in BQP in this setting.

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 {\it 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 that $NQP\nsubseteq BQP$ for Number-On-Forehead communication.