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 with communication tensor Tf, egin{enumerate} item in the Number-On-Forehead model, the cost is upper-bounded by the logarithm of nrank(Tf); item in the Number-In-Hand model, the cost is lower-bounded by the logarithm of nrank(Tf). end{enumerate} 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
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机译:在本文中,我们研究了多方通信中的量子不确定性。量子计算中存在三种(可能)不同类型的不确定性:i)强,ii)具有经典证明的弱性,以及iii)具有量子证明的弱性。在这里,我们专注于第一个。一个强的量子不确定性协议以正概率接受正确的输入,并以概率1拒绝错误的输入。在这项工作中,我们将强量子不确定性的多方通信复杂性与前额数和数字中的通信张量的等级相关联。手中模型。特别地,通过将de Wolf提出的定义扩展到{ it undeterministic tensor-rank}(等级),我们表明,对于具有通信张量Tf的任何布尔函数f,Number-On-中的 begin {enumerate} item前额模型的成本以nrank(Tf)的对数为上限; Number of Hand-Hand模型中的 item,其成本由nrank(Tf)的对数降低。 end {enumerate}这自然地概括了该领域中的先前结果,并且首次将(高阶)张量秩的概念与量子通信相关。此外,我们表明,强量子不确定性可以比经典的多方不确定性通信成指数地强。为此,我们将结果应用于矩阵乘法问题
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