On the complexity of unique quantum witnesses and quantum approximate counting

• URL: http://arxiv.org/abs/2410.23811v2
• Date: Thu, 18 Sep 2025 04:58:28 GMT
• Title: On the complexity of unique quantum witnesses and quantum approximate counting
• Authors: Anurag Anshu, Jonas Haferkamp, Yeongwoo Hwang, Quynh T. Nguyen,
• Abstract summary: We show a quantum oracle separation between $mathsfBQPmathsfUniqueQMA$ and $mathsfQMA$.<n>We then ask a natural question; what structural properties of the local Hamiltonian problem can we exploit?<n>We introduce a physically motivated candidate by showing that the ground energy of local Hamiltonians can be estimated through a $mathsfUniqueQMA$ protocol.
• Score: 2.4475760284813255
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the long-standing open question on the power of unique witnesses in quantum protocols, which asks if $\textsf{UniqueQMA}$, a variant of $\textsf{QMA}$ whose accepting witness space is 1-dimensional, contains $\mathsf{QMA}$ under quantum reductions. This work rules out any black-box reduction from $\mathsf{QMA}$ to $\mathsf{UniqueQMA}$ by showing a quantum oracle separation between $\mathsf{BQP}^\mathsf{UniqueQMA}$ and $\mathsf{QMA}$. This provides a contrast to the classical case, where the Valiant-Vazirani theorem shows a black-box randomized reduction from $\mathsf{UniqueNP}$ to $\mathsf{NP}$, and suggests the need for studying the structure of the ground space of local Hamiltonians in distilling a potential unique witness. Via similar techniques, we show, relative to a quantum oracle, that $\mathsf{QMA}^\mathsf{QMA}$ cannot decide quantum approximate counting, ruling out a quantum analogue of Stockmeyer's algorithm in the black-box setting. We then ask a natural question; what structural properties of the local Hamiltonian problem can we exploit? We introduce a physically motivated candidate by showing that the ground energy of local Hamiltonians that satisfy a computational variant of the eigenstate thermalization hypothesis (ETH) can be estimated through a $\mathsf{UniqueQMA}$ protocol. Our protocol can be viewed as a quantum expander test in a low energy subspace of the Hamiltonian and verifies a unique entangled state across two copies of the subspace. This allows us to conclude that if $\mathsf{UniqueQMA}$ is not equivalent to $\mathsf{QMA}$, then $\mathsf{QMA}$-hard Hamiltonians must violate ETH under adversarial perturbations. This also serves as evidence that chaotic local Hamiltonians, such as the SYK model may be computationally simpler than general local Hamiltonians.

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