Summary: On Perfect Completeness for QMA
Scott Aaronson #
Whether the class QMA (Quantum Merlin Arthur) is equal to QMA 1 , or QMA with one
sided error, has been an open problem for years. This note helps to explain why the problem
is di#cult, by using ideas from real analysis to give a ``quantum oracle'' relative to which
QMA #= QMA 1 . 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.
The complexity class MA (MerlinArthur) was introduced by Babai  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