# Generalized XOR non-locality games with graph description on a square lattice

## TL;DR

This paper introduces a family of non-locality games based on square lattices on arbitrary surfaces, analyzing their classical and quantum values, and classifying them via graph representations and permutations.

## Contribution

It presents a novel class of non-locality games linked to lattice structures, with polynomial-time classical value computation for d=2 and a classification scheme for arbitrary parameters.

## Key findings

- Classical values are efficiently computable for d=2.
- Quantum values are unaffected by boundary conditions under certain circumstances.
- Games can be classified into equivalence classes based on graph and permutation symmetries.

## Abstract

We propose a family of non-locality unique games for 2 parties based on a square lattice on an arbitrary surface. We show that, due to structural similarities with error correction codes of Kitaev for fault tolerant quantum computation, the games have classical values computable in polynomial time for $d=2$ measurement outcomes. By representing games in their graph form, for arbitrary $d$ and underlying surface we provide their classification into equivalence classes with respect to relabeling of measurement outcomes, for a selected set of permutations which define the winning conditions. A case study of games with periodic boundary conditions is presented in order to verify their impact on classical and quantum values of the family of games. It suggests that quantum values suffer independently from presence of different winning conditions that can be imposed due to periodicity, as long as no local restrictions are in place.