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Topics in Non-local Games: Synchronous Algebras, Algebraic Graph Identities, and Quantum NP-hardness Reductions

Entong He

TL;DR

The paper investigates the algebraic foundations of synchronous non-local games and their quantum strategy hierarchies by linking game satisfaction to noncommutative $*$-algebras and NC real algebraic geometry. It develops both theoretical results, such as the equivalence of hereditary and $C^*$ subalgebras and bounds relating clique numbers to Lovász theta, and algorithmic tools, including Gröbner-basis approaches and hierarchical SDPs via noncommutative Nullstellensätze, to certify the nonexistence of perfect strategies. It also extends quantum NP-hardness reductions by constructing new gadget-based reductions from $3$-SAT$^*$ to $3$-Coloring$^*$ and to $Clique^*$, widening the landscape of quantum satisfiability problems. Overall, the work unifies algebraic, computational, and complexity-theoretic perspectives on quantum non-local games, providing both theoretical insights and practical tools for analyzing quantum strategies.

Abstract

We review the correspondence between synchronous games and their associated $*$-algebra. Building upon the work of (Helton et al., New York J. Math. 2017), we propose results on algebraic and locally commuting graph identities. Based on the noncommutative Nullstellensätze (Watts, Helton and Klep, Annales Henri Poincaré 2023), we build computational tools that check the non-existence of perfect $C^*$ and algebraic strategies of synchronous games using Gröbner basis methods and semidefinite programming. We prove the equivalence between the hereditary and $C^*$ models questioned in (Helton et al., New York J. Math. 2017). We also extend the quantum-version NP-hardness reduction $\texttt{3-SAT}^* \leq_p \texttt{3-Coloring}^*$ due to (Ji, arXiv 2013) by exhibiting another instance of such reduction $\texttt{3-SAT}^* \leq_p \texttt{Clique}^*$.

Topics in Non-local Games: Synchronous Algebras, Algebraic Graph Identities, and Quantum NP-hardness Reductions

TL;DR

The paper investigates the algebraic foundations of synchronous non-local games and their quantum strategy hierarchies by linking game satisfaction to noncommutative -algebras and NC real algebraic geometry. It develops both theoretical results, such as the equivalence of hereditary and subalgebras and bounds relating clique numbers to Lovász theta, and algorithmic tools, including Gröbner-basis approaches and hierarchical SDPs via noncommutative Nullstellensätze, to certify the nonexistence of perfect strategies. It also extends quantum NP-hardness reductions by constructing new gadget-based reductions from -SAT to -Coloring and to , widening the landscape of quantum satisfiability problems. Overall, the work unifies algebraic, computational, and complexity-theoretic perspectives on quantum non-local games, providing both theoretical insights and practical tools for analyzing quantum strategies.

Abstract

We review the correspondence between synchronous games and their associated -algebra. Building upon the work of (Helton et al., New York J. Math. 2017), we propose results on algebraic and locally commuting graph identities. Based on the noncommutative Nullstellensätze (Watts, Helton and Klep, Annales Henri Poincaré 2023), we build computational tools that check the non-existence of perfect and algebraic strategies of synchronous games using Gröbner basis methods and semidefinite programming. We prove the equivalence between the hereditary and models questioned in (Helton et al., New York J. Math. 2017). We also extend the quantum-version NP-hardness reduction due to (Ji, arXiv 2013) by exhibiting another instance of such reduction .
Paper Structure (9 sections, 23 theorems, 46 equations)

This paper contains 9 sections, 23 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notations
  3. Non-local games and synchronous games
  4. Basics of groups and $*$-algebras
  5. Synchronous algebras, and a hierarchy of strategies
  6. Algebraic and locally commuting clique number
  7. Noncommutative Nullstellensätze and its algorithmic implementation
  8. Hereditary synchronous subalgebra, and its equivalence to the $C^*$ subalgebra
  9. Quantum version of Boolean constraint systems and NP-hardness reductions

Key Result

Proposition 3.1

$G \overset{alg}{\longrightarrow} H$ and $H \overset{alg}{\longrightarrow} K$ implies $G \overset{alg}{\longrightarrow} K$ for any graphs $G, H, K$.

Theorems & Definitions (47)

  • Definition 3.1
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • proof
  • Remark 3.1
  • Theorem 3.5
  • ...and 37 more