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

Abstract

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.

Asymptotic tensor rank of graph tensors: beyond matrix multiplication

“…Our proof of Theorem 2.1.6 follows the structure of the proof for the cycle graph case in [BCZ16] which builds upon the exposition in [Blä13].…”

Multi-partite entanglement

“…The conversion rate from the GHZ state to this tensor is directly related to the computational complexity of matrix multiplication [25], a well-known open problem in classical computer science. Much recent work on multi-partite entanglement transformations has been motivated by this connection [21,162,163]. These intrinsic challenges motivate after all to consider more pragmatic approaches to grasp multi-partite entanglement.…”

“…Bob subsequently carries out a POVM measurement on his system and obtains a classical outcome Y . Such a measurement can always be realized by first applying an isometry B → Y C, where C is an auxiliary subsystem, and subsequently discarding C. Using (21), we find that I(X : Y ) ≤ I(X : Y C). But I(X : Y C) = I(X : B) ≤ N , since N = log dim B is the number of qubits in B.…”

Entanglement Distillation from Greenberger–Horne–Zeilinger Shares

“…In particular, ref. [9] shows that our results imply new protocols and bounds in nondeterministic multiparty quantum communication complexity. Now let H be an arbitrary hypergraph and suppose that instead of GHZ states, the parties wish to distill EPR pairs shared between a specified pair AB.…”