En Route to a Standard QMA1 vs. QCMA Oracle Separation

TL;DR

This paper establishes oracle separations between quantum complexity classes QMA1 and QCMA under various conditions, advancing understanding of quantum witness power and oracle models.

Contribution

It constructs classical oracle separations between QMA1 and QCMA with limited verifier rounds and queries, and derandomizes previous oracle separations.

Findings

Constructed an oracle where QMA1 is not in QCMA with polynomial rounds and queries.

Derandomized the permutation-oracle separation between QMA1 and QCMA.

Showed separation results depend on the size of the spectral gap in quantum classes.

Abstract

We study the power of quantum witnesses under perfect completeness. We construct a classical oracle relative to which a language lies in $\mathsf{QMA}_1$ but not in $\mathsf{QCMA}$ when the $\mathsf{QCMA}$ verifier is only allowed polynomially many adaptive rounds and exponentially many parallel queries per round. Additionally, we derandomize the permutation-oracle separation of Fefferman and Kimmel, obtaining an in-place oracle separation between $\mathsf{QMA}_1$ and $\mathsf{QCMA}$. Furthermore, we focus on $\mathsf{QCMA}$ and $\mathsf{QMA}$ with an exponentially small gap, where we show a separation assuming the gap is fixed, but not when it may be arbitrarily small. Finally, we derive consequences for approximate ground-state preparation from sparse Hamiltonian oracle access, including a bounded-adaptivity frustration-free variant.