A note about a partial no-go theorem for quantum PCP

TL;DR

This paper presents a partial no-go theorem for the quantum PCP conjecture, showing how the promise gap diminishes as the system becomes more commuting, which may inform future research on the conjecture.

Contribution

It introduces a bound on the promise gap in quantum PCP based on the system's non-commutativity, advancing understanding of the conjecture's limitations.

Findings

Promise gap shrinks with increased commutativity

Provides an upper bound on the promise gap for quantum PCP

Suggests a potential critical point for quantum PCP validity

Abstract

This is not a disproof of the quantum PCP conjecture! In this note we use perturbation on the commuting Hamiltonian problem on a graph, based on results by Bravyi and Vyalyi, to provide a partial no-go theorem for quantum PCP. Specifically, we derive an upper bound on how large the promise gap can be for the quantum PCP still to hold, as a function of the non-commuteness of the system. As the system becomes more and more commuting, the maximal promise gap shrinks. We view these results as possibly a preliminary step towards disproving the quantum PCP conjecture. A different way to view these results is actually as indications that a critical point exists, beyond which quantum PCP indeed holds; in any case, we hope that these results will lead to progress on this important open problem.