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On the Complexity of Unique Quantum Witnesses and Quantum Approximate Counting

Authors Anurag Anshu , Jonas Haferkamp , Yeongwoo Hwang , Quynh T. Nguyen

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ACM Subject Classification
  • Theory of computation → Quantum computation theory
Keywords
  • Quantum complexity
  • approximate counting
  • Valiant-Vazirani
  • eigenstate thermalization hypothesis

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Abstract

We study the long-standing open question on the power of unique witnesses in quantum protocols, which asks if UniqueQMA, a variant of QMA whose accepting witness space is 1-dimensional, contains QMA under quantum reductions.
This work rules out any black-box reduction from QMA to UniqueQMA by showing a quantum oracle separation between BQP^UniqueQMA and QMA. This provides a contrast to the classical case, where the Valiant-Vazirani theorem shows a black-box randomized reduction from UniqueNP to 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 QMA^QMA cannot decide quantum approximate counting, ruling out a quantum analogue of Stockmeyer’s algorithm in the black-box setting. Our results employ a subspace reflection oracle, previously considered in [Scott Aaronson and Greg Kuperberg, 2007; Scott Aaronson et al., 2020; She and Yuen, 2023], but we introduce new tools which allow us to exploit the unique witness constraint. We also show a strong "polarization" behavior of QMA circuits, which could be of independent interest in studying quantum polynomial hierarchies.
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 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 UniqueQMA is not equivalent to QMA, then QMA-hard Hamiltonians must violate ETH under adversarial perturbations (more accurately, further assuming the quantum PCP conjecture if ETH only applies to extensive energy subspaces). Under the same assumption, this also serves as evidence that chaotic local Hamiltonians, such as the SYK model may be computationally simpler than general local Hamiltonians.

Cite As Get BibTex

Anurag Anshu, Jonas Haferkamp, Yeongwoo Hwang, and Quynh T. Nguyen. On the Complexity of Unique Quantum Witnesses and Quantum Approximate Counting. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.10

Author Details

Anurag Anshu
  • Harvard University, Cambridge, MA, USA
Jonas Haferkamp
  • Saarland University, Saarbrücken, Germany
Yeongwoo Hwang
  • Harvard University, Cambridge, MA, USA
Quynh T. Nguyen
  • Harvard University, Cambridge, MA, USA

Funding

This work was done in part while the authors were visiting the Simons Institute for the Theory of Computing, supported by DOE QSA grant number FP00010905. AA and QTN acknowledge support through the NSF Award No. 2238836.
  • Anshu, Anurag: AA acknowledges support through the NSF award QCIS-FF: Quantum Computing & Information Science Faculty Fellow at Harvard University (NSF 2013303).
  • Haferkamp, Jonas: JH acknowledges support through the Harvard Quantum Initiative postdoctoral fellowship.
  • Hwang, Yeongwoo: YH is supported by the National Science Foundation Graduate Research Fellowship under Grant No. 2140743. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors(s) and do not necessarily reflect the views of the National Science Foundation.
  • Nguyen, Quynh T.: QTN acknowledges support through the Harvard Quantum Initiative PhD fellowship.

Acknowledgements

In an earlier version of this paper, the oracle separation between UniqueQMA and QMA was given via a quantum oracle. We thank Chinmay Nirkhe and Anand Natarajan for pointing out that our proof did not in fact need quantumness in this case. This version has been updated to provide a classical oracle separation. We thank Soonwon Choi, Sam Garratt, Yingfei Gu, Rahul Jain and Shivaji Sondhi for insightful discussions, and especially thank Soonwon Choi for sharing an example which shows that the computational ETH assumption is incorrect if the Gaussian prescription is applied at high energies (we only applying it at low energies).

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