Tensor rank and strong quantum nondeterminism in multiparty communication

  • Authors:

  • Marcos Villagra;Masaki Nakanishi;Shigeru Yamashita;Yasuhiko Nakashima

  • Affiliations:

  • Nara Institute of Science and Technology, Nara, Japan;Yamagata University, Yamagata, Japan;Ritsumeikan University, Shiga, Japan;Nara Institute of Science and Technology, Nara, Japan

  • Venue:
  • TAMC'12 Proceedings of the 9th Annual international conference on Theory and Applications of Models of Computation
  • Year:
  • 2012

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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 , 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 ). This naturally generalizes previous results in the field and relates for the first time the concept of (high-order) tensor rank to quantum communication. Furthermore, we show that strong quantum nondeterminism can be exponentially stronger than classical multiparty nondeterministic communication. We do so by applying our results to the matrix multiplication problem.