En Route to a Standard QMA1 vs. QCMA Oracle Separation

2026-04-29

AI Summary

This work investigates the computational power gap between the quantum witness class QMA₁ and the class QCMA under perfect completeness, particularly whether they can be separated in the oracle model. By constructing classical oracles and derandomized permutation oracles, and combining adaptive query complexity analysis with sparse Hamiltonian models, the study achieves the first separation between QMA₁ and a restricted adaptive variant of QCMA relative to a standard classical oracle. It further establishes an in-place oracle separation without relying on randomness. The results highlight the critical role of exponentially small spectral gaps in enabling such separations and yield new implications for the approximate ground state preparation of frustration-free Hamiltonians.

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.

Problem

• QMA1
• QCMA
• oracle separation
• quantum witnesses
• complexity classes

Innovation

• QMA1 vs. QCMA separation
• classical oracle
• derandomization
• in-place oracle
• sparse Hamiltonian