Stronger Methods of Making Quantum Interactive Proofs Perfectly Complete

TL;DR

This paper introduces advanced techniques to achieve perfect completeness in quantum interactive proofs, demonstrating that QMA problems can have two-message proof systems with perfect completeness and reducing message complexity.

Contribution

It proves QMA has two-message quantum interactive proofs with perfect completeness and improves message complexity bounds for m-message quantum proofs.

Findings

QMA is included in QIP_1(2), the class with two-message perfect completeness proofs.

Any m-message quantum proof system can be transformed into an (m+1)-message system with perfect completeness.

The results provide the first nontrivial upper bound for QMA in terms of quantum interactive proofs.

Abstract

This paper presents stronger methods of achieving perfect completeness in quantum interactive proofs. First, it is proved that any problem in QMA has a two-message quantum interactive proof system of perfect completeness with constant soundness error, where the verifier has only to send a constant number of halves of EPR pairs. This in particular implies that the class QMA is necessarily included by the class QIP_1(2) of problems having two-message quantum interactive proofs of perfect completeness, which gives the first nontrivial upper bound for QMA in terms of quantum interactive proofs. It is also proved that any problem having an $m$-message quantum interactive proof system necessarily has an $(m+1)$-message quantum interactive proof system of perfect completeness. This improves the previous result due to Kitaev and Watrous, where the resulting system of perfect completeness requires $m+2$ messages if not using the parallelization result.