Reducing the number of inputs in nonlocal games

TL;DR

This paper introduces a vector-valued empirical method to reduce the number of inputs in nonlocal games while maintaining the ratio of quantum to classical bias, demonstrated on the Khot-Vishnoi game.

Contribution

It presents a novel vector-valued empirical approach to decrease input complexity in nonlocal games without losing key bias ratios.

Findings

Reduced questions from exponential to polynomial in the Khot-Vishnoi game

Maintained a quantum over classical bias of (n/log^2 n)

Demonstrated effectiveness of the method on complex nonlocal games

Abstract

In this work we show how a vector-valued version of Schechtman's empirical method can be used to reduce the number of inputs in a nonlocal game $G$ while preserving the quotient $\beta^*(G)/\beta(G)$ of the quantum over the classical bias. We apply our method to the Khot-Vishnoi game, with exponentially many questions per player, to produce another game with polynomially many ($N\approx n^8$) questions so that the quantum over the classical bias is $\Omega (n/\log^2 n)$.