Parallel Repetition for $3$-Player XOR Games

TL;DR

This paper proves that for certain 3-player XOR games without Abelian embeddings, the probability of winning multiple repetitions decreases exponentially, extending previous results with new combinatorial and Fourier analysis techniques.

Contribution

It establishes exponential decay of game value for a broad class of 3-XOR games lacking Abelian embeddings, generalizing prior specific cases.

Findings

Exponential decay of game value for non-embedding 3-XOR games.

Extension of previous GHZ game results to a wider class.

Application of additive combinatorics and Fourier analysis methods.

Abstract

In a $3$-$\mathsf{XOR}$ game $\mathcal{G}$, the verifier samples a challenge $(x,y,z)\sim \mu$ where $\mu$ is a probability distribution over $\Sigma\times\Gamma\times\Phi$, and a map $t\colon \Sigma\times\Gamma\times\Phi\to\mathcal{A}$ for a finite Abelian group $\mathcal{A}$ defining a constraint. The verifier sends the questions $x$, $y$ and $z$ to the players Alice, Bob and Charlie respectively, receives answers $f(x)$, $g(y)$ and $h(z)$ that are elements in $\mathcal{A}$ and accepts if $f(x)+g(y)+h(z) = t(x,y,z)$. The value, $\mathsf{val}(\mathcal{G})$, of the game is defined to be the maximum probability the verifier accepts over all players' strategies. We show that if $\mathcal{G}$ is a $3$-$\mathsf{XOR}$ game with value strictly less than $1$, whose underlying distribution over questions $\mu$ does not admit Abelian embeddings into $(\mathbb{Z},+)$, then the value of the $n$-fold repetition of $\mathcal{G}$ is exponentially decaying. That is, there exists $c=c(\mathcal{G})>0$ such that $\mathsf{val}(\mathcal{G}^{\otimes n})\leq 2^{-cn}$. This extends a previous result of [Braverman-Khot-Minzer, FOCS 2023] showing exponential decay for the GHZ game. Our proof combines tools from additive combinatorics and tools from discrete Fourier analysis.