Tensor Rank and Strong Quantum Nondeterminism in Multiparty Communication

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

doi:10.1587/transinf.e96.d.1

Cited by 3 publications

(3 citation statements)

References 28 publications

(37 reference statements)

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“…However, asymptotically in the input size, the increase is only k/2 and this relationship is tight. This solves a nondeterministic multiplayer quantum log-rank conjecture in the message-passing model of Villagra et al [VNYN13]. This topic is covered in Section 3.…”

Section: Our Resultsmentioning

confidence: 94%

“…Multiparty nondeterministic quantum communication with message passing has been studied by Villagra et al [VNYN13]. They show that the logarithm of the support rank of the communication tensor is a lower bound for the message-passing complexity and conjecture that this lower bound is polynomially related to the message-passing complexity.…”

Section: Our Resultsmentioning

confidence: 99%

“…Villagra et al [VNYN13] show that NQ 0 (f ) is at least the logarithm of the support rank of f . They furthermore conjecture that NQ 0 (f ) is upper bounded by a polynomial in the logarithm of the support rank.…”

Section: Nondeterministic Log-rank Conjecture For Message-passing Pro...mentioning

confidence: 99%

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Nondeterministic quantum communication complexity: the cyclic equality game and iterated matrix multiplication

Buhrman,

Christandl,

Zuiddam

2016

Preprint

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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.

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Section: Our Resultsmentioning

confidence: 94%

Section: Our Resultsmentioning

confidence: 99%