Amey Bhangale, Mark Braverman, Subhash Khot, Yang P. Liu, Dor Minzer
{"title":"Parallel Repetition for $3$-Player XOR Games","authors":"Amey Bhangale, Mark Braverman, Subhash Khot, Yang P. Liu, Dor Minzer","doi":"arxiv-2408.09352","DOIUrl":null,"url":null,"abstract":"In a $3$-$\\mathsf{XOR}$ game $\\mathcal{G}$, the verifier samples a challenge\n$(x,y,z)\\sim \\mu$ where $\\mu$ is a probability distribution over\n$\\Sigma\\times\\Gamma\\times\\Phi$, and a map $t\\colon\n\\Sigma\\times\\Gamma\\times\\Phi\\to\\mathcal{A}$ for a finite Abelian group\n$\\mathcal{A}$ defining a constraint. The verifier sends the questions $x$, $y$\nand $z$ to the players Alice, Bob and Charlie respectively, receives answers\n$f(x)$, $g(y)$ and $h(z)$ that are elements in $\\mathcal{A}$ and accepts if\n$f(x)+g(y)+h(z) = t(x,y,z)$. The value, $\\mathsf{val}(\\mathcal{G})$, of the\ngame is defined to be the maximum probability the verifier accepts over all\nplayers' strategies. We show that if $\\mathcal{G}$ is a $3$-$\\mathsf{XOR}$ game with value\nstrictly less than $1$, whose underlying distribution over questions $\\mu$ does\nnot admit Abelian embeddings into $(\\mathbb{Z},+)$, then the value of the\n$n$-fold repetition of $\\mathcal{G}$ is exponentially decaying. That is, there\nexists $c=c(\\mathcal{G})>0$ such that $\\mathsf{val}(\\mathcal{G}^{\\otimes\nn})\\leq 2^{-cn}$. This extends a previous result of [Braverman-Khot-Minzer,\nFOCS 2023] showing exponential decay for the GHZ game. Our proof combines tools\nfrom additive combinatorics and tools from discrete Fourier analysis.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"15 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - CS - Discrete Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.09352","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.