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Algorithms, Bounds, and Strategies for Entangled XOR Games

Authors Adam Bene Watts, Aram W. Harrow, Gurtej Kanwar, Anand Natarajan

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  • 18 pages

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Author Details

Adam Bene Watts
  • MIT Center for Theoretical Physics, 77 Massachusetts Ave, 6-304, Cambridge, MA, USA
Aram W. Harrow
  • MIT Center for Theoretical Physics, 77 Massachusetts Ave, 6-304, Cambridge, MA, USA
Gurtej Kanwar
  • MIT Center for Theoretical Physics, 77 Massachusetts Ave, 6-304, Cambridge, MA, USA
Anand Natarajan
  • California Institute of Technology, 1200 E. California Blvd, Pasadena, CA, USA

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Adam Bene Watts, Aram W. Harrow, Gurtej Kanwar, and Anand Natarajan. Algorithms, Bounds, and Strategies for Entangled XOR Games. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 10:1-10:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


Entangled games are a quantum analog of constraint satisfaction problems and have had important applications to quantum complexity theory, quantum cryptography, and the foundations of quantum mechanics. Given a game, the basic computational problem is to compute its entangled value: the supremum success probability attainable by a quantum strategy. We study the complexity of computing the (commuting-operator) entangled value omega^* of entangled XOR games with any number of players. Based on a duality theory for systems of operator equations, we introduce necessary and sufficient criteria for an XOR game to have omega^* = 1, and use these criteria to derive the following results: 1) An algorithm for symmetric games that decides in polynomial time whether omega^* = 1 or omega^* < 1, a task that was not previously known to be decidable, together with a simple tensor-product strategy that achieves value 1 in the former case. The only previous candidate algorithm for this problem was the Navascués-Pironio-Acín (also known as noncommutative Sum of Squares or ncSoS) hierarchy, but no convergence bounds were known. 2) A family of games with three players and with omega^* < 1, where it takes doubly exponential time for the ncSoS algorithm to witness this. By contrast, our algorithm runs in polynomial time. 3) Existence of an unsatisfiable phase for random (non-symmetric) XOR games. We show that there exists a constant C_k^{unsat} depending only on the number k of players, such that a random k-XOR game over an alphabet of size n has omega^* < 1 with high probability when the number of clauses is above C_k^{unsat} n. 4) A lower bound of Omega(n log(n)/log log(n)) on the number of levels in the ncSoS hierarchy required to detect unsatisfiability for most random 3-XOR games. This is in contrast with the classical case where the (3n)^{th} level of the sum-of-squares hierarchy is equivalent to brute-force enumeration of all possible solutions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • Nonlocal games
  • XOR Games
  • Pseudotelepathy games
  • Multipartite entanglement


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