Quantum Arthur-Merlin Games

TL;DR

This paper explores quantum Arthur-Merlin games, demonstrating error reduction techniques, equivalences with quantum interactive proofs, and implications for quantum complexity classes like QMA, BQP, and PP.

Contribution

It introduces exponential error reduction for one-message quantum Arthur-Merlin games and establishes their equivalence with quantum interactive proof systems.

Findings

Exponential error reduction in QMA without increasing message length

Logarithmic quantum certificates do not surpass BQP

QMA is contained within PP

Abstract

This paper studies quantum Arthur-Merlin games, which are Arthur-Merlin games in which Arthur and Merlin can perform quantum computations and Merlin can send Arthur quantum information. As in the classical case, messages from Arthur to Merlin are restricted to be strings of uniformly generated random bits. It is proved that for one-message quantum Arthur-Merlin games, which correspond to the complexity class QMA, completeness and soundness errors can be reduced exponentially without increasing the length of Merlin's message. Previous constructions for reducing error required a polynomial increase in the length of Merlin's message. Applications of this fact include a proof that logarithmic length quantum certificates yield no increase in power over BQP and a simple proof that QMA is contained in PP. Other facts that are proved include the equivalence of three (or more) message quantum Arthur-Merlin games with ordinary quantum interactive proof systems and some basic properties concerning two-message quantum Arthur-Merlin games.