Short Multi-Prover Quantum Proofs for SAT without Entangled Measurements

TL;DR

This paper presents a simplified BellQMA protocol for 3-SAT that uses unentangled measurements, achieving constant soundness gap with efficient quantum proofs, and addresses open questions in quantum proof complexity.

Contribution

It introduces a new BellQMA protocol for 3-SAT with constant soundness gap using unentangled measurements, simplifying previous entangled measurement protocols.

Findings

Achieves BellQMA proofs of satisfiability for 3-SAT with O(√m) polylog(m) Merlins.

Each Merlin sends O(log m) qubits to Arthur.

Protocol's analysis is simpler than previous entangled measurement protocols.

Abstract

BellQMA protocols are a subclass of multi-prover quantum Merlin-Arthur protocols in which the verifier is restricted to perform nonadaptive,unentangled measurements on the quantum states received from each Merlin. In this paper, we prove that m-clause 3-SAT instances have BellQMA proofs of satisfiability with constant soundness gap, in which O(sqrt(m)polylog(m)) Merlins each send O(log m) qubits to Arthur. Our result answers a question of Aaronson et al., who gave a protocol with similar parameters that used entangled measurements; the analysis of our protocol is significantly simpler than that of Aaronson et al. Our result also complements recent work of Brandao, Christandl, and Yard, who showed upper bounds on the power of multiprover quantum proofs with unentangled but adaptive (LOCC) measurements.