Oracle separation of QMA and QCMA with bounded adaptivity
Shalev Ben-David, Srijita Kundu
TL;DR
This work establishes a bona-fide oracle separation between QMA and QCMA under bounded adaptivity in oracle queries, by adapting a Yamakawa-Zhandry–based relational framework to a standard oracle model with a tight round bound. The authors introduce the slipperiness property for relations to control how much of the oracle can be learned with limited witness information, and they leverage dense-distribution and hybrid arguments to iteratively eliminate query rounds while preserving behavior on a large set of oracles. They construct a function family solvable by 1-round QMA but not by any bounded-round QCMA algorithm, and they define an oracle that separates the two classes for sublogarithmic rounds, accompanied by a diagonalization argument. The results illuminate the nuanced power difference between quantum and classical proofs in a query setting and suggest directions toward stronger, fully general oracle separations via strengthened slipperiness and related coding-theoretic methods. The work also discusses potential extensions toward oracle separations for BQP with quantum vs classical advice and outlines key obstacles and conjectures for achieving fully general separations without adaptivity constraints.
Abstract
We give an oracle separation between QMA and QCMA for quantum algorithms that have bounded adaptivity in their oracle queries; that is, the number of rounds of oracle calls is small, though each round may involve polynomially many queries in parallel. Our oracle construction is a simplified version of the construction used recently by Li, Liu, Pelecanos, and Yamakawa (2023), who showed an oracle separation between QMA and QCMA when the quantum algorithms are only allowed to access the oracle classically. To prove our results, we introduce a property of relations called \emph{slipperiness}, which may be useful for getting a fully general classical oracle separation between QMA and QCMA.