Abstract
We study nondeterministic multiparty quantum communication with a quantum generalization of broadcasts. We show that, with numberinhand 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 logrank 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 Strassenâ€™s laser method to show that asymptotically there exist nontrivial protocols for every oddplayer cyclic equality problem. We exhibit an efficient protocol for the 5player problem for small inputs, and we show how Young flattenings yield nontrivial complexity lower bounds.
BibTeX  Entry
@InProceedings{buhrman_et_al:LIPIcs:2017:8181,
author = {Harry Buhrman and Matthias Christandl and Jeroen Zuiddam},
title = {{Nondeterministic Quantum Communication Complexity: the Cyclic Equality Game and Iterated Matrix Multiplication}},
booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
pages = {24:124:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770293},
ISSN = {18688969},
year = {2017},
volume = {67},
editor = {Christos H. Papadimitriou},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/8181},
URN = {urn:nbn:de:0030drops81812},
doi = {10.4230/LIPIcs.ITCS.2017.24},
annote = {Keywords: quantum communication complexity, broadcast channel, numberinhand, matrix multiplication, support rank}
}
Keywords: 

quantum communication complexity, broadcast channel, numberinhand, matrix multiplication, support rank 
Collection: 

8th Innovations in Theoretical Computer Science Conference (ITCS 2017) 
Issue Date: 

2017 
Date of publication: 

28.11.2017 