文章基本信息

标题：Nondeterministic Quantum Communication Complexity: the Cyclic Equality Game and Iterated Matrix Multiplication
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作者：Harry Buhrman ; Matthias Christandl ; Jeroen Zuiddam 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2017
卷号：67
页码：24:1-24:18
DOI：10.4230/LIPIcs.ITCS.2017.24
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：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 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.
关键词：quantum communication complexity; broadcast channel; number-in-hand; matrix multiplication; support rank