NP vs QMA_log(2)

TL;DR

This paper demonstrates that the NP-complete problem 3-SAT can be efficiently verified within the quantum Merlin-Arthur framework with two unentangled logarithmic-size proofs, achieving a polynomially small gap.

Contribution

It establishes a new quantum verification protocol for 3-SAT with a polynomially small gap, advancing understanding of the power of QMA with logarithmic witnesses.

Findings

3-SAT admits a QMA_log(2) protocol with gap 1/n^{3+ε}

Improves previous results by achieving a polynomially small gap

Shows potential for NP problems within quantum proof systems

Abstract

Although it is believed unlikely that $\NP$-hard problems admit efficient quantum algorithms, it has been shown that a quantum verifier can solve $\NP$-complete problems given a "short" quantum proof; more precisely, $\NP\subseteq \QMA_{\log}(2)$ where $\QMA_{\log}(2)$ denotes the class of quantum Merlin-Arthur games in which there are two unentangled provers who send two logarithmic size quantum witnesses to the verifier. The inclusion $\NP\subseteq \QMA_{\log}(2)$ has been proved by Blier and Tapp by stating a quantum Merlin-Arthur protocol for 3-coloring with perfect completeness and gap $\frac{1}{24n^6}$. Moreover, Aaronson {\it et al.} have shown the above inclusion with a constant gap by considering $\widetilde{O}(\sqrt{n})$ witnesses of logarithmic size. However, we still do not know if $\QMA_{\log}(2)$ with a constant gap contains $\NP$. In this paper, we show that 3-SAT admits a $\QMA_{\log}(2)$ protocol with the gap $\frac{1}{n^{3+\epsilon}}$ for every constant $\epsilon>0$.