Three-player entangled XOR games are NP-hard to approximate

TL;DR

This paper proves that approximating the entangled value of three-player XOR games within any constant factor is NP-hard, extending classical hardness results to quantum and entangled settings.

Contribution

It establishes the first constant-factor NP-hardness of approximation for entangled games under P ≠ NP, using novel analysis of low-degree tests against entangled players.

Findings

NP-hardness of approximating entangled XOR games within any constant factor

Extension of classical hardness results to quantum entangled settings

Simplified analysis of multilinearity tests for entangled players

Abstract

We show that for any eps>0 the problem of finding a factor (2-eps) approximation to the entangled value of a three-player XOR game is NP-hard. Equivalently, the problem of approximating the largest possible quantum violation of a tripartite Bell correlation inequality to within any multiplicative constant is NP-hard. These results are the first constant-factor hardness of approximation results for entangled games or quantum violations of Bell inequalities shown under the sole assumption that P \neq NP. They can be thought of as an extension of Hastad's optimal hardness of approximation results for MAX-E3-LIN2 (JACM'01) to the entangled-player setting. The key technical component of our work is a soundness analysis of a point-vs-plane low-degree test against entangled players. This extends and simplifies the analysis of the multilinearity test by Ito and Vidick (FOCS'12). Our results demonstrate the possibility for efficient reductions between entangled-player games and our techniques may lead to further hardness of approximation results.