Parallel Repetition for k-Player XOR Games

Abstract

In a $k$-XOR game $Ψ$ over $\mathbb{F}_p$, a verifier samples challenges $(x_1,\ldots,x_k)\simμ$ where $μ$ is a distribution over $Σ_1\times\ldots\times Σ_k$, sends $x_i$ to player $i$ and receives an answer $f_i(x_i)\in\mathbb{F}_p$, and accepts if and only if $f_1(x_1)+\ldots+f_k(x_k)=t(x_1,\ldots,x_k)$, where $t\colon Σ_1\times\ldots\times Σ_k\to\mathbb{F}_p$ is a fixed map. We show that if $Ψ$ has value strictly less than $1$, then the value of the $n$-fold repeated game $Ψ^{\otimes n}$ is at most $2^{-cn}$ where $c = c(Ψ)>0$ is a fixed constant. This extends a previous result of [Bhangale-Braverman-Khot-Liu-Minzer] showing exponential decay for some $3$-XOR games. Our proof uses a relaxation of the value of games studied in the quantum setting, called the commuting value of a game. We show a transformation that maps a $k$-player game $Ψ$ with value strictly less than $1$ to a $2k$-player game $Ψ'$ with commuting value less than $1$, such that $\mathsf{val}(Ψ^{\otimes n})\leq \mathsf{val}((Ψ')^{\otimes n})$. We prove that $\mathsf{val}(Ψ')<1$ has a direct proof using nothing more than the Cauchy-Schwarz inequality, which we are able to tensorize to get that $\mathsf{val}((Ψ')^{\otimes n})$ vanishes exponentially fast in $n$.