 
Summary: On Perfect Completeness for QMA
Scott Aaronson
MIT
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
QMA = QMA1. 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 ZQEXP.
1 Introduction
The complexity class MA (MerlinArthur) was introduced by Babai [4] in 1985. Intuitively, MA
is a probabilistic version of NP; it contains all problems for which an omniscient wizard Merlin
can convince a probabilistic polynomialtime verifier Arthur of a "yes" answer, by a oneround
protocol in which Merlin sends Arthur a purported proof z, and then Arthur checks z. In the
usual definition, if the answer to the problem is "yes" then there should exist a string z that makes
Arthur accept with probability at least 2/3 (this property is called completeness), while if the
answer is "no" then no z should make Arthur accept with probability more than 1/3 (this property
is called soundness).
One of the first questions people asked about MA was whether it can be made to have perfect
