Algorithms, bounds, and strategies for entangled XOR games

Abstract

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 ω∗ 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 ω∗ = 1, and use these criteria to derive the following results: 1. An algorithm for symmetric games that decides in polynomial time whether ω∗ = 1 or ω∗ < 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 ω∗ < 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 Ckunsat depending only on the number k of players, such that a random k-XOR game over an alphabet of size n has ω∗ < 1 with high probability when the number of clauses is above Ckunsat n. 4. A lower bound of Ω(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.