Classical vs. quantum satisfiability in linear constraint systems modulo an integer

TL;DR

This paper investigates the limitations of quantum solutions in linear constraint systems modulo an integer, proving that certain expected satisfiability gaps cannot be achieved with specific observables, and characterizing conditions for quantum solutions.

Contribution

It provides a proof that tensor product observables cannot produce a satisfiability gap and extends the characterization of quantum solutions to arbitrary moduli d, including a no-go theorem for phase-commuting systems.

Findings

No satisfiability gap using tensor product of generalized Pauli observables in odd dimensions.

Characterization of quantum solutions extends to arbitrary d with minor modifications.

Phase-commutation prevents quantum solutions in odd d, ruling out certain quantum satisfiability gaps.

Abstract

A system of linear constraints can be unsatisfiable and yet admit a solution in the form of quantum observables whose correlated outcomes satisfy the constraints. Recently, it has been claimed that such a satisfiability gap can be demonstrated using tensor products of generalized Pauli observables in odd dimensions. We provide an explicit proof that no quantum-classical satisfiability gap in any linear constraint system can be achieved using these observables. We prove a few other results for linear constraint systems modulo d > 2. We show that a characterization of the existence of quantum solutions when d is prime, due to Cleve et al, holds with a small modification for arbitrary d. We identify a key property of some linear constraint systems, called phase-commutation, and give a no-go theorem for the existence of quantum solutions to constraint systems for odd d whenever phase-commutation is present. As a consequence, all natural generalizations of the Peres-Mermin magic square and pentagram to odd prime d do not exhibit a satisfiability gap.