License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2015.512
URN: urn:nbn:de:0030-drops-50727
URL: https://drops.dagstuhl.de/opus/volltexte/2015/5072/
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### Parallel Repetition for Entangled k-player Games via Fast Quantum Search

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### Abstract

We present two parallel repetition theorems for the entangled value of multi-player, one-round free games (games where the inputs come from a product distribution). Our first theorem shows that for a k-player free game G with entangled value val^*(G) = 1 - epsilon, the n-fold repetition of G has entangled value val^*(G^(\otimes n)) at most (1 - epsilon^(3/2))^(Omega(n/sk^4)), where s is the answer length of any player. In contrast, the best known parallel repetition theorem for the classical value of two-player free games is val(G^(\otimes n)) <= (1 - epsilon^2)^(Omega(n/s)), due to Barak, et al. (RANDOM 2009). This suggests the possibility of a separation between the behavior of entangled and classical free games under parallel repetition. Our second theorem handles the broader class of free games G where the players can output (possibly entangled) quantum states. For such games, the repeated entangled value is upper bounded by (1 - epsilon^2)^(Omega(n/sk^2)). We also show that the dependence of the exponent on k is necessary: we exhibit a k-player free game G and n >= 1 such that val^*(G^(\otimes n)) >= val^*(G)^(n/k). Our analysis exploits the novel connection between communication protocols and quantum parallel repetition, first explored by Chailloux and Scarpa (ICALP 2014). We demonstrate that better communication protocols yield better parallel repetition theorems: in particular, our first theorem crucially uses a quantum search protocol by Aaronson and Ambainis, which gives a quadratic Grover speed-up for distributed search problems. Finally, our results apply to a broader class of games than were previously considered before; in particular, we obtain the first parallel repetition theorem for entangled games involving more than two players, and for games involving quantum outputs.

### BibTeX - Entry

@InProceedings{chung_et_al:LIPIcs:2015:5072,
author =	{Kai-Min Chung and Xiaodi Wu and Henry Yuen},
title =	{{Parallel Repetition for Entangled k-player Games via Fast Quantum Search}},
booktitle =	{30th Conference on Computational Complexity (CCC 2015)},
pages =	{512--536},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-81-1},
ISSN =	{1868-8969},
year =	{2015},
volume =	{33},
editor =	{David Zuckerman},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address =	{Dagstuhl, Germany},
URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5072},
URN =		{urn:nbn:de:0030-drops-50727},
doi =		{10.4230/LIPIcs.CCC.2015.512},
annote =	{Keywords: Parallel repetition, quantum entanglement, communication complexity}
}


 Keywords: Parallel repetition, quantum entanglement, communication complexity Collection: 30th Conference on Computational Complexity (CCC 2015) Issue Date: 2015 Date of publication: 06.06.2015

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