Quasi-Polynomial Time Algorithms for Free Quantum Games in Bounded Dimension
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(đť’Ş(T^{12}(logÂ˛(AT)+log(Q)log(AT))/ÎµÂ˛)) to compute additive Îµ-approximations on the value of two-player free games with TĂ— 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.
non-local game
semidefinite programming
quantum correlation
approximation algorithm
Lasserre hierarchy
de Finetti theorem
Theory of computation
82:1-82:20
Track A: Algorithms, Complexity and Games
https://arxiv.org/abs/2005.08883
Hyejung H.
Jee
Hyejung H. Jee
Department of Computing, Imperial College London, UK
Carlo
Sparaciari
Carlo Sparaciari
Department of Computing, Imperial College London, UK
Department of Physics and Astronomy, University College London, UK
Omar
Fawzi
Omar Fawzi
Univ Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP, F-69342, Lyon Cedex 07, France
Mario
Berta
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
10.4230/LIPIcs.ICALP.2021.82
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Hyejung H. Jee, Carlo Sparaciari, Omar Fawzi, and Mario Berta
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