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Classical Simulation of Quantum CSP Strategies

Demian Banakh, Lorenzo Ciardo, Marcin Kozik, Jan Tułowiecki

TL;DR

The paper studies two-prover CSP games to quantify how much classical communication is needed to perfectly simulate any perfect quantum strategy. It develops a combinatorial characterization of classical strategies with extra communication and a geometric rounding technique for quantum measurements, proving that a finite classical channel suffices with size depending only on the shared-space dimension $d$ and the CSP template. In the digraph case, the authors give explicit bounds $k = d+ obreak obreak obreak obreak + obreak obreak$ for Alice-to-Bob and an exponential bound for Bob-to-Alice, and they extend the results to general CSPs via a pattern-based framework, establishing a bounded quantum-classical chromatic gap for fixed $d$ with $ ext{α}_d=(2+o(1))^{d-1}$. These results illuminate the boundary between quantum and classical resources in CSP contexts and connect finite-communication simulations to sphere-covering geometry, with implications for CSP complexity and quantum information theory.

Abstract

We prove that any perfect quantum strategy for the two-prover game encoding a constraint satisfaction problem (CSP) can be simulated via a perfect classical strategy with an extra classical communication channel, whose size depends only on $(i)$ the size of the shared quantum system used in the quantum strategy, and $(ii)$ structural parameters of the CSP template. The result is obtained via a combinatorial characterisation of perfect classical strategies with extra communication channels and a geometric rounding procedure for the projection-valued measurements involved in quantum strategies. A key intermediate step of our proof is to establish that the gap between the classical chromatic number of graphs and its quantum variant is bounded when the quantum strategy involves shared quantum information of bounded size.

Classical Simulation of Quantum CSP Strategies

TL;DR

The paper studies two-prover CSP games to quantify how much classical communication is needed to perfectly simulate any perfect quantum strategy. It develops a combinatorial characterization of classical strategies with extra communication and a geometric rounding technique for quantum measurements, proving that a finite classical channel suffices with size depending only on the shared-space dimension and the CSP template. In the digraph case, the authors give explicit bounds for Alice-to-Bob and an exponential bound for Bob-to-Alice, and they extend the results to general CSPs via a pattern-based framework, establishing a bounded quantum-classical chromatic gap for fixed with . These results illuminate the boundary between quantum and classical resources in CSP contexts and connect finite-communication simulations to sphere-covering geometry, with implications for CSP complexity and quantum information theory.

Abstract

We prove that any perfect quantum strategy for the two-prover game encoding a constraint satisfaction problem (CSP) can be simulated via a perfect classical strategy with an extra classical communication channel, whose size depends only on the size of the shared quantum system used in the quantum strategy, and structural parameters of the CSP template. The result is obtained via a combinatorial characterisation of perfect classical strategies with extra communication channels and a geometric rounding procedure for the projection-valued measurements involved in quantum strategies. A key intermediate step of our proof is to establish that the gap between the classical chromatic number of graphs and its quantum variant is bounded when the quantum strategy involves shared quantum information of bounded size.

Paper Structure

This paper contains 17 sections, 30 theorems, 23 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related work
  3. Games and strategies
  4. Classical strategies---no communication
  5. Quantum strategies---no communication
  6. Classical strategies---Alice messages Bob
  7. Classical strategies---Bob messages Alice
  8. Capturing classical strategies with extra communication
  9. Alice messages Bob
  10. Bob messages Alice
  11. Quantum vs. classical strategies
  12. Binary case
  13. General case
  14. The quantum vs. classical chromatic gap
  15. Further directions
  16. ...and 2 more sections

Key Result

Theorem 2

$\mathop{\mathrm{CSP}}\nolimits^\mathbb{H}(\mathbf{Y})=\mathop{\mathrm{CSP}}\nolimits(\mathbf{Y}^\mathbb{H})$ for each $\sigma$-structure $\mathbf{Y}$ and each Hilbert space $\mathbb{H}$.

Figures (1)

  • Figure 1: The $\mathbf{X},\mathbf{Y}$ game.

Theorems & Definitions (58)

  • Theorem 2: abramsky2017quantum
  • Theorem 3
  • Example 4
  • Theorem 5
  • Definition 6
  • Theorem 7
  • proof
  • Example 8
  • Theorem 9
  • Lemma 10
  • ...and 48 more