Tensor Rank and Strong Quantum Nondeterminism in Multiparty Communication

Marcos VILLAGRA  Masaki NAKANISHI  Shigeru YAMASHITA  Yasuhiko NAKASHIMA  

Publication
IEICE TRANSACTIONS on Information and Systems   Vol.E96-D   No.1   pp.1-8
Publication Date: 2013/01/01
Online ISSN: 1745-1361
DOI: 10.1587/transinf.E96.D.1
Print ISSN: 0916-8532
Type of Manuscript: PAPER
Category: Fundamentals of Information Systems
Keyword: 
multiparty communication complexity,  quantum computation,  quantum nondeterminism,  tensor rank,  

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Summary: 
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 BQP for Number-On-Forehead communication.