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On the Quantum Chromatic Gap

Lorenzo Ciardo

TL;DR

This work investigates the gap between quantum and classical graph chromatic numbers, proving an unbounded separation conditional on a strengthened pseudo-telepathy conjecture for $d$-to-$1$ label-cover CSPs. The authors develop a quantum adjunction theory for Pultr functors to ensure that standard CSP reductions preserve quantum completeness, and they construct a reduction chain (including Dinur’s, line-digraph, and GS-type steps) that transfers pseudo-telepathy from CSPs to graph colouring. A parallel path via $3$XOR games shows how strong pseudo-telepathy can be amplified to yield fixed-color quantum strategies with arbitrarily large classical colourings, culminating in a conditional $3$-colour quantum strategy with large classical chromatic number. Collectively, the results bridge quantum information with CSP hardness, offering conditional yet arguably optimal pathways to realize unbounded quantum-classical gaps in graph colouring and linking pseudo-telepathy to approximation hardness frameworks.

Abstract

The largest known gap between quantum and classical chromatic number of graphs, obtained via quantum protocols for colouring Hadamard graphs based on the Deutsch--Jozsa algorithm and the quantum Fourier transform, is exponential. We put forth a quantum pseudo-telepathy version of Khot's $d$-to-$1$ Games Conjecture and prove that, conditional to its validity, the gap is unbounded: There exist graphs whose quantum chromatic number is $3$ and whose classical chromatic number is arbitrarily large. Furthermore, we show that the existence of a certain form of pseudo-telepathic XOR games would imply the conjecture and, thus, the unboundedness of the quantum chromatic gap. As two technical steps of our proof that might be of independent interest, we establish a quantum adjunction theorem for Pultr functors between categories of relational structures, and we prove that the Dinur--Khot--Kindler--Minzer--Safra reduction, recently used for proving the $2$-to-$2$ Games Theorem, is quantum complete.

On the Quantum Chromatic Gap

TL;DR

This work investigates the gap between quantum and classical graph chromatic numbers, proving an unbounded separation conditional on a strengthened pseudo-telepathy conjecture for -to- label-cover CSPs. The authors develop a quantum adjunction theory for Pultr functors to ensure that standard CSP reductions preserve quantum completeness, and they construct a reduction chain (including Dinur’s, line-digraph, and GS-type steps) that transfers pseudo-telepathy from CSPs to graph colouring. A parallel path via XOR games shows how strong pseudo-telepathy can be amplified to yield fixed-color quantum strategies with arbitrarily large classical colourings, culminating in a conditional -colour quantum strategy with large classical chromatic number. Collectively, the results bridge quantum information with CSP hardness, offering conditional yet arguably optimal pathways to realize unbounded quantum-classical gaps in graph colouring and linking pseudo-telepathy to approximation hardness frameworks.

Abstract

The largest known gap between quantum and classical chromatic number of graphs, obtained via quantum protocols for colouring Hadamard graphs based on the Deutsch--Jozsa algorithm and the quantum Fourier transform, is exponential. We put forth a quantum pseudo-telepathy version of Khot's -to- Games Conjecture and prove that, conditional to its validity, the gap is unbounded: There exist graphs whose quantum chromatic number is and whose classical chromatic number is arbitrarily large. Furthermore, we show that the existence of a certain form of pseudo-telepathic XOR games would imply the conjecture and, thus, the unboundedness of the quantum chromatic gap. As two technical steps of our proof that might be of independent interest, we establish a quantum adjunction theorem for Pultr functors between categories of relational structures, and we prove that the Dinur--Khot--Kindler--Minzer--Safra reduction, recently used for proving the -to- Games Theorem, is quantum complete.

Paper Structure

This paper contains 29 sections, 31 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Quantum pseudo-telepathy
  3. Correspondences
  4. CSP approximation
  5. Contributions
  6. Related work
  7. Overview of results and techniques
  8. The source of pseudo-telepathy
  9. Classical setting
  10. Quantum setting
  11. Quantum adjunction of Pultr functors
  12. The quantum chromatic gap via $d$-to-$1$ games
  13. The quantum chromatic gap via $3$XOR games
  14. Classical and quantum CSPs
  15. Predicate satisfiability
  16. ...and 14 more sections

Key Result

Theorem 4

Let $\mathfrak{T}$ be a $(\rho,\tau)$-Pultr template. For any $\tau$-structure $\mathbf{X}$ and any $\rho$-structure $\mathbf{Y}$ it holds that $\Lambda\mathbf{X}\to\mathbf{Y}$ if and only if $\mathbf{X}\to\Gamma\mathbf{Y}$.

Theorems & Definitions (94)

  • Conjecture 1: $d$-to-$1$ Conjecture, Khot02stoc
  • Conjecture 2
  • Example 3
  • Theorem 4: pultr_adjoints
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9: abramsky2017quantum
  • Corollary 10
  • ...and 84 more