A distribution testing oracle separation between QMA and QCMA

TL;DR

This paper constructs a classical randomized oracle to distinguish between QMA and QCMA, advancing understanding of quantum complexity classes by providing a separation that previously required quantum oracles.

Contribution

It introduces a classical oracle separation between QMA and QCMA, moving beyond prior quantum oracle-based separations and addressing a key open question in quantum complexity theory.

Findings

Classical randomized oracle separates QMA and QCMA

The separating problem involves graph distribution properties

Advances understanding of quantum complexity class distinctions

Abstract

It is a long-standing open question in quantum complexity theory whether the definition of $\textit{non-deterministic}$ quantum computation requires quantum witnesses $(\textsf{QMA})$ or if classical witnesses suffice $(\textsf{QCMA})$. We make progress on this question by constructing a randomized classical oracle separating the respective computational complexity classes. Previous separations [Aaronson-Kuperberg (CCC'07), Fefferman-Kimmel (MFCS'18)] required a quantum unitary oracle. The separating problem is deciding whether a distribution supported on regular un-directed graphs either consists of multiple connected components (yes instances) or consists of one expanding connected component (no instances) where the graph is given in an adjacency-list format by the oracle. Therefore, the oracle is a distribution over $n$-bit boolean functions.