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DOI: 10.4230/LIPIcs.ITCS.2026.10

On the Complexity of Unique Quantum Witnesses and Quantum Approximate Counting

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

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

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.

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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)

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@InProceedings{anshu_et_al:LIPIcs.ITCS.2026.10,
  author =	{Anshu, Anurag and Haferkamp, Jonas and Hwang, Yeongwoo and Nguyen, Quynh T.},
  title =	{{On the Complexity of Unique Quantum Witnesses and Quantum Approximate Counting}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{10:1--10:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.10},
  URN =		{urn:nbn:de:0030-drops-252978},
  doi =		{10.4230/LIPIcs.ITCS.2026.10},
  annote =	{Keywords: Quantum complexity, approximate counting, Valiant-Vazirani, eigenstate thermalization hypothesis}
}

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DOI: 10.4230/LIPIcs.ITCS.2026.2

Oracle Separations for the Quantum-Classical Polynomial Hierarchy

Authors: Avantika Agarwal and Shalev Ben{-}David

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract

We study the quantum-classical polynomial hierarchy, QCPH, which is the class of languages solvable by a constant number of alternating classical quantifiers followed by a quantum verifier. Our main result is that QCPH is infinite relative to a random oracle (previously, this was not even known relative to any oracle). We further prove that higher levels of PH are not contained in lower levels of QCPH relative to a random oracle; this is a strengthening of the somewhat recent result that PH is infinite relative to a random oracle (Rossman, Servedio, and Tan 2016). The oracle separation requires lower bounding a certain type of low-depth alternating circuit with some quantum gates. To establish this, we give a new switching lemma for quantum algorithms which may be of independent interest. Our lemma says that for any d, if we apply a random restriction to a function f with quantum query complexity Q(f) ≤ n^{1/3}, the restricted function becomes exponentially close (in terms of d) to a depth-d decision tree. Our switching lemma works even in a "worst-case" sense, in that only the indices to be restricted are random; the values they are restricted to are chosen adversarially. Moreover, the switching lemma also works for polynomial degree in place of quantum query complexity.

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Avantika Agarwal and Shalev Ben-David. Oracle Separations for the Quantum-Classical Polynomial Hierarchy. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 2:1-2:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{agarwal_et_al:LIPIcs.ITCS.2026.2,
  author =	{Agarwal, Avantika and Ben\{-\}David, Shalev},
  title =	{{Oracle Separations for the Quantum-Classical Polynomial Hierarchy}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{2:1--2:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.2},
  URN =		{urn:nbn:de:0030-drops-252893},
  doi =		{10.4230/LIPIcs.ITCS.2026.2},
  annote =	{Keywords: Switching Lemma, Polynomial Hierarchy, Approximate Degree, Random Oracles, Query Complexity, Quantum Computing}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.72

Forrelation Is Extremally Hard

Authors: Uma Girish and Rocco Servedio

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract

The Forrelation problem is a central problem that demonstrates an exponential separation between quantum and classical capabilities. In this problem, given query access to n-bit Boolean functions f and g, the goal is to estimate the Forrelation function forr(f,g), which measures the correlation between g and the Fourier transform of f. In this work we provide a new linear algebraic perspective on the Forrelation problem, as opposed to prior analytic approaches. We establish a connection between the Forrelation problem and bent Boolean functions and through this connection, analyze an extremal version of the Forrelation problem where the goal is to distinguish between extremal instances of Forrelation, namely (f,g) with forr(f,g) = 1 and forr(f,g) = -1. We show that this problem can be solved with one quantum query and success probability one, yet requires Ω̃(2^{n/4}) classical randomized queries, even for algorithms with a one-third failure probability, highlighting the remarkable power of one exact quantum query. We also study a restricted variant of this problem where the inputs f,g are computable by small classical circuits and show classical hardness under cryptographic assumptions.

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Uma Girish and Rocco Servedio. Forrelation Is Extremally Hard. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 72:1-72:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{girish_et_al:LIPIcs.ITCS.2026.72,
  author =	{Girish, Uma and Servedio, Rocco},
  title =	{{Forrelation Is Extremally Hard}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{72:1--72:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.72},
  URN =		{urn:nbn:de:0030-drops-253594},
  doi =		{10.4230/LIPIcs.ITCS.2026.72},
  annote =	{Keywords: Forrelation, exact quantum, query complexity}
}

Document

DOI: 10.4230/LIPIcs.CCC.2022.20

The Acrobatics of BQP

Authors: Scott Aaronson, DeVon Ingram, and William Kretschmer

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

Abstract

One can fix the randomness used by a randomized algorithm, but there is no analogous notion of fixing the quantumness used by a quantum algorithm. Underscoring this fundamental difference, we show that, in the black-box setting, the behavior of quantum polynomial-time (BQP) can be remarkably decoupled from that of classical complexity classes like NP. Specifically: - There exists an oracle relative to which NP^{BQP} ⊄ BQP^{PH}, resolving a 2005 problem of Fortnow. As a corollary, there exists an oracle relative to which 𝖯 = NP but BQP ≠ QCMA. - Conversely, there exists an oracle relative to which BQP^{NP} ⊄ PH^{BQP}. - Relative to a random oracle, PP is not contained in the "QMA hierarchy" QMA^{QMA^{QMA^{⋯}}}. - Relative to a random oracle, Σ_{k+1}^𝖯 ⊄ BQP^{Σ_k^𝖯} for every k. - There exists an oracle relative to which BQP = P^#P and yet PH is infinite. (By contrast, relative to all oracles, if NP ⊆ BPP, then PH collapses.) - There exists an oracle relative to which 𝖯 = NP ≠ BQP = 𝖯^#P. To achieve these results, we build on the 2018 achievement by Raz and Tal of an oracle relative to which BQP ⊄ PH, and associated results about the Forrelation problem. We also introduce new tools that might be of independent interest. These include a "quantum-aware" version of the random restriction method, a concentration theorem for the block sensitivity of AC⁰ circuits, and a (provable) analogue of the Aaronson-Ambainis Conjecture for sparse oracles.

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Scott Aaronson, DeVon Ingram, and William Kretschmer. The Acrobatics of BQP. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{aaronson_et_al:LIPIcs.CCC.2022.20,
  author =	{Aaronson, Scott and Ingram, DeVon and Kretschmer, William},
  title =	{{The Acrobatics of BQP}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{20:1--20:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.20},
  URN =		{urn:nbn:de:0030-drops-165820},
  doi =		{10.4230/LIPIcs.CCC.2022.20},
  annote =	{Keywords: BQP, Forrelation, oracle separations, Polynomial Hierarchy, query complexity}
}

Document

Invited Talk

DOI: 10.4230/LIPIcs.FSTTCS.2021.1

BQP After 28 Years (Invited Talk)

Authors: Scott Aaronson

Published in: LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

Abstract

I will discuss the now-ancient question of where BQP, Bounded-Error Quantum Polynomial-Time, fits in among classical complexity classes. After reviewing some basics from the 90s, I will discuss the Forrelation problem that I introduced in 2009 to yield an oracle separation between BQP and PH, and the dramatic completion of that program by Ran Raz and Avishay Tal in 2018. I will then discuss very recent work, with William Kretschmer and DeVon Ingram, which leverages the Raz-Tal theorem, along with a new "quantum-aware" random restriction method, to obtain results that illustrate just how differently BQP can behave from BPP. These include oracles relative to which NP^{BQP} ̸ ⊂ BQP^{PH} - solving a 2005 open problem of Lance Fortnow - and conversely, relative to which BQP^{NP} ̸ ⊂ PH^{BQP}; an oracle relative to which 𝖯 = NP and yet BQP ≠ QCMA; an oracle relative to which NP ⊆ BQP yet PH is infinite; an oracle relative to which 𝖯 = NP≠ BQP = PP; and an oracle relative to which PP = PostBQP ̸ ⊂ QMA^{QMA^{…}}. By popular demand, I will also speculate about the status of BQP in the unrelativized world.

Cite as

Scott Aaronson. BQP After 28 Years (Invited Talk). In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, p. 1:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{aaronson:LIPIcs.FSTTCS.2021.1,
  author =	{Aaronson, Scott},
  title =	{{BQP After 28 Years}},
  booktitle =	{41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
  pages =	{1:1--1:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-215-0},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{213},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.1},
  URN =		{urn:nbn:de:0030-drops-155124},
  doi =		{10.4230/LIPIcs.FSTTCS.2021.1},
  annote =	{Keywords: quantum computing, complexity theory, oracle separations, circuit lower bounds}
}

5 Search Results for "Ingram, DeVon"

On the Complexity of Unique Quantum Witnesses and Quantum Approximate Counting

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Oracle Separations for the Quantum-Classical Polynomial Hierarchy

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Forrelation Is Extremally Hard

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The Acrobatics of BQP

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BQP After 28 Years (Invited Talk)

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