Provable Advantage for Quantum Strategies in Random Symmetric XOR Games

TL;DR

This paper demonstrates that in almost all symmetric XOR games with n players, quantum strategies significantly outperform classical ones, with the quantum advantage growing as the number of players increases, based on asymptotic analysis.

Contribution

The paper establishes the asymptotic behavior of the entangled value in symmetric XOR games and quantifies the classical-quantum gap for large n, revealing a provable quantum advantage.

Findings

Quantum value scales as Θ(√ln n)/n^{1/4}

Classical-quantum gap scales as Θ(√ln n)

Quantum strategies outperform classical strategies in large symmetric XOR games

Abstract

Non-local games are widely studied as a model to investigate the properties of quantum mechanics as opposed to classical mechanics. In this paper, we consider a subset of non-local games: symmetric XOR games of $n$ players with 0-1 valued questions. For this class of games, each player receives an input bit and responds with an output bit without communicating to the other players. The winning condition only depends on XOR of output bits and is constant w.r.t. permutation of players. We prove that for almost any $n$-player symmetric XOR game the entangled value of the game is $\Theta (\frac{\sqrt{\ln{n}}}{n^{1/4}})$ adapting an old result by Salem and Zygmund on the asymptotics of random trigonometric polynomials. Consequently, we show that the classical-quantum gap is $\Theta (\sqrt{\ln{n}})$ for almost any symmetric XOR game.