On Perfect Completeness for QMA

TL;DR

This paper explores the longstanding open problem of whether QMA equals QMA1, using real analysis to construct a quantum oracle that separates the classes, revealing complexities in quantum relativization.

Contribution

It introduces a quantum oracle relative to which QMA and QMA1 differ, highlighting the difficulty of proving their equivalence and uncovering non-relativizing facts about quantum complexity classes.

Findings

Existence of a quantum oracle separating QMA and QMA1

Identification of classically relativizing but quantumly non-relativizing facts

Insight into the complexity of proving perfect completeness in QMA

Abstract

Whether the class QMA (Quantum Merlin Arthur) is equal to QMA1, or QMA with one-sided error, has been an open problem for years. This note helps to explain why the problem is difficult, by using ideas from real analysis to give a "quantum oracle" relative to which they are different. As a byproduct, we find that there are facts about quantum complexity classes that are classically relativizing but not quantumly relativizing, among them such "trivial" containments as BQP in ZQEXP.