On the complexity of unique quantum witnesses and quantum approximate counting

TL;DR

This paper investigates the limitations of unique quantum witnesses in quantum complexity classes, showing separations from QMA and exploring the structure of local Hamiltonians under physical hypotheses, with implications for quantum computational hardness.

Contribution

It establishes quantum oracle separations between QMA and UniqueQMA, and introduces a protocol for estimating ground energy of Hamiltonians satisfying ETH, linking complexity to physical properties.

Findings

No black-box reduction from QMA to UniqueQMA exists.

Quantum oracle shows QMA^QMA cannot decide quantum approximate counting.

UniqueQMA protocols can estimate ground energy of ETH-satisfying Hamiltonians.

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.