Characterizing the intersection of QMA and coQMA

TL;DR

This paper explores the relationships between quantum complexity classes, showing that certain functional analogues of QMA and coQMA are equivalent to TFQMA, and discusses implications for the inclusions among these classes.

Contribution

It introduces alternative definitions of QMA∩coQMA and establishes their equivalence to TFQMA, providing insights into the structure of quantum complexity classes.

Findings

F(QMA∩coQMA) equals TFQMA

If TFQMA equals FBQP, then QMA∩coQMA equals BQP

QMA∩coQMA equals QMA under certain reductions

Abstract

We show that the functional analogue of QMA$\cap$coQMA, denoted F(QMA$\cap$coQMA), equals the complexity class Total Functional QMA (TFQMA). To prove this we need to introduce alternative definitions of QMA$\cap$coQMA in terms of a single quantum verification procedure. We show that if TFQMA equals the functional analogue of BQP (FBQP), then QMA$\cap$coQMA = BQP. We show that if there is a QMA complete problem that (robustly) reduces to a problem in TFQMA, then QMA$\cap$coQMA = QMA. These results provide strong evidence that the inclusions FBQP$\subseteq$TFQMA$\subseteq$FQMA are strict, since otherwise the corresponding inclusions in BQP$\subseteq$QMA$\cap$coQMA$\subseteq$QMA would become equalities.