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Quasi-Polynomial Time Algorithms for Free Quantum Games in Bounded Dimension

Authors Hyejung H. Jee, Carlo Sparaciari, Omar Fawzi, Mario Berta

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Author Details

Hyejung H. Jee
  • Department of Computing, Imperial College London, UK
Carlo Sparaciari
  • Department of Computing, Imperial College London, UK
  • Department of Physics and Astronomy, University College London, UK
Omar Fawzi
  • Univ Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP, F-69342, Lyon Cedex 07, France
Mario Berta
  • Department of Computing, Imperial College London, UK
  • IQIM, California Institute of Technology, Pasadena, CA, USA
  • AWS Center for Quantum Computing, Pasadena, CA, USA

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Hyejung H. Jee, Carlo Sparaciari, Omar Fawzi, and Mario Berta. Quasi-Polynomial Time Algorithms for Free Quantum Games in Bounded Dimension. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 82:1-82:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ICALP.2021.82

Abstract

In a recent landmark result [Ji et al., arXiv:2001.04383 (2020)], it was shown that approximating the value of a two-player game is undecidable when the players are allowed to share quantum states of unbounded dimension. In this paper, we study the computational complexity of two-player games when the dimension of the quantum systems is bounded by T. More specifically, we give a semidefinite program of size exp(𝒪(T^{12}(log²(AT)+log(Q)log(AT))/ε²)) to compute additive ε-approximations on the value of two-player free games with T× T-dimensional quantum entanglement, where A and Q denote the number of answers and questions of the game, respectively. For fixed dimension T, this scales polynomially in Q and quasi-polynomially in A, thereby improving on previously known approximation algorithms for which worst-case run-time guarantees are at best exponential in Q and A. For the proof, we make a connection to the quantum separability problem and employ improved multipartite quantum de Finetti theorems with linear constraints that we derive via quantum entropy inequalities.

Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • non-local game
  • semidefinite programming
  • quantum correlation
  • approximation algorithm
  • Lasserre hierarchy
  • de Finetti theorem

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