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DOI: 10.4230/LIPIcs.ITCS.2026.103

A General Framework for Low Soundness Homomorphism Testing

Authors: Tushant Mittal and Sourya Roy

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

Abstract

We introduce a general framework to design and analyze algorithms for the problem of testing homomorphisms between finite groups in the low-soundness regime. In this regime, we give the first constant-query tests for various families of groups. These include tests for: (i) homomorphisms between arbitrary cyclic groups, (ii) homomorphisms between any finite group and ℤ_p, (iii) automorphisms of dihedral and symmetric groups, (iv) inner automorphisms of non-abelian finite simple groups and extraspecial groups, and (v) testing linear characters of GL_n(F_q), and finite-dimensional Lie algebras over F_q. We also recover the result of Kiwi [TCS'03] for testing homomorphisms between F_qⁿ and F_q. Prior to this work, such tests were only known for abelian groups with a constant maximal order (such as F_qⁿ). No tests were known for non-abelian groups. As an additional corollary, our framework gives combinatorial list decoding bounds for cyclic groups with list size dependence of O(ε^{-2}) (for agreement parameter ε). This improves upon the currently best-known bound of O(ε^{-105}) due to Dinur, Grigorescu, Kopparty, and Sudan [STOC'08], and Guo and Sudan [RANDOM'14].

Cite as

Tushant Mittal and Sourya Roy. A General Framework for Low Soundness Homomorphism Testing. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 103:1-103:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{mittal_et_al:LIPIcs.ITCS.2026.103,
  author =	{Mittal, Tushant and Roy, Sourya},
  title =	{{A General Framework for Low Soundness Homomorphism Testing}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{103:1--103:18},
  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.2026.103},
  URN =		{urn:nbn:de:0030-drops-253901},
  doi =		{10.4230/LIPIcs.ITCS.2026.103},
  annote =	{Keywords: Property Testing, Coding Theory}
}

Document

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.

Cite as

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}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.83

The Pure-State Consistency of Local Density Matrices Problem: In PSPACE and Complete for a Class Between QMA and QMA(2)

Authors: Jonas Kamminga and Dorian Rudolph

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

Abstract

In this work we investigate the computational complexity of the pure consistency of local density matrices (PureCLDM) and pure N-representability (Pure-N-Representability; analog of PureCLDM for bosonic or fermionic systems) problems. In these problems the input is a set of reduced density matrices and the task is to determine whether there exists a global pure state consistent with these reduced density matrices. While mixed CLDM, i.e. where the global state can be mixed, was proven to be QMA-complete by Broadbent and Grilo [JoC 2022], almost nothing was known about the complexity of the pure version. Before our work the best upper and lower bounds were QMA(2) and QMA. Our contribution to the understanding of these problems is twofold. Firstly, we define a pure state analogue of the complexity class QMA^+ of Aharanov and Regev [FOCS 2003], which we call PureSuperQMA. We prove that both pure-N-Representability and PureCLDM are complete for this new class. Along the way we supplement Broadbent and Grilo by proving hardness for 2-qubit reduced density matrices and showing that mixed N-Representability is QMA-complete. Secondly, we improve the upper bound on PureCLDM. Using methods from algebraic geometry, we prove that PureSuperQMA ⊆ PSPACE. Our methods, and the PSPACE upper bound, are also valid for PureCLDM with exponential or even perfect precision, hence precisePureCLDM is not preciseQMA(2) = NEXP-complete, unless PSPACE = NEXP. We view this as evidence for a negative answer to the longstanding open question whether PureCLDM is QMA(2)-complete. The techniques we develop for our PSPACE upper bound are quite general. We are able to use them for various applications: from proving PSPACE upper bounds on other quantum problems to giving an efficient parallel (NC) algorithm for (non-convex) quadratically constrained quadratic programs with few constraints.

Cite as

Jonas Kamminga and Dorian Rudolph. The Pure-State Consistency of Local Density Matrices Problem: In PSPACE and Complete for a Class Between QMA and QMA(2). In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 83:1-83:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kamminga_et_al:LIPIcs.ITCS.2026.83,
  author =	{Kamminga, Jonas and Rudolph, Dorian},
  title =	{{The Pure-State Consistency of Local Density Matrices Problem: In PSPACE and Complete for a Class Between QMA and QMA(2)}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{83:1--83:23},
  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.2026.83},
  URN =		{urn:nbn:de:0030-drops-253701},
  doi =		{10.4230/LIPIcs.ITCS.2026.83},
  annote =	{Keywords: Quantum Complexity Theory, PSPACE, QMA(2), Consistency of Local Density Matrices, Polynomial Optimization}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.56

The Hardness of Learning Quantum Circuits and Its Cryptographic Applications

Authors: Bill Fefferman, Soumik Ghosh, Makrand Sinha, and Henry Yuen

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

Abstract

We show that concrete hardness assumptions about learning or cloning the output state of a random quantum circuit can be used as the foundation for secure quantum cryptography. In particular, under these assumptions we construct secure one-way state generators (OWSGs), digital signature schemes, quantum bit commitments, and private key encryption schemes. We also discuss evidence for these hardness assumptions by analyzing the best-known quantum learning algorithms, as well as proving black-box lower bounds for cloning and learning given state preparation oracles. Our random circuit-based constructions provide concrete instantiations of quantum cryptographic primitives whose security do not depend on the existence of one-way functions. The use of random circuits in our constructions also opens the door to {NISQ-friendly quantum cryptography}. We discuss noise tolerant versions of our OWSG and digital signature constructions which can potentially be implementable on noisy quantum computers connected by a quantum network. On the other hand, they are still secure against {noiseless} quantum adversaries, raising the intriguing possibility of a useful implementation of an end-to-end cryptographic protocol on near-term quantum computers. Finally, our explorations suggest that the rich interconnections between learning theory and cryptography in classical theoretical computer science also extend to the quantum setting.

Cite as

Bill Fefferman, Soumik Ghosh, Makrand Sinha, and Henry Yuen. The Hardness of Learning Quantum Circuits and Its Cryptographic Applications. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 56:1-56:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{fefferman_et_al:LIPIcs.ITCS.2026.56,
  author =	{Fefferman, Bill and Ghosh, Soumik and Sinha, Makrand and Yuen, Henry},
  title =	{{The Hardness of Learning Quantum Circuits and Its Cryptographic Applications}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{56:1--56:21},
  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.2026.56},
  URN =		{urn:nbn:de:0030-drops-253431},
  doi =		{10.4230/LIPIcs.ITCS.2026.56},
  annote =	{Keywords: quantum learning, quantum circuits, cryptographic hardness, one-way state generators}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.61

Random Unitaries in Constant (Quantum) Time

Authors: Ben Foxman, Natalie Parham, Francisca Vasconcelos, and Henry Yuen

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

Abstract

Random unitaries are a central object of study in quantum information, with applications to quantum computation, quantum many-body physics, and quantum cryptography. Recent work has constructed unitary designs and pseudorandom unitaries (PRUs) using Θ(log log n)-depth unitary circuits with two-qubit gates. In this work, we show that unitary designs and PRUs can be efficiently constructed in several well-studied models of constant-time quantum computation (i.e., the time complexity on the quantum computer is independent of the system size). These models are constant-depth circuits augmented with certain nonlocal operations, such as (a) many-qubit TOFFOLI gates, (b) many-qubit FANOUT gates, or (c) mid-circuit measurements with classical feedforward control. Recent advances in quantum computing hardware suggest experimental feasibility of these models in the near future. Our results demonstrate that unitary designs and PRUs can be constructed in much weaker circuit models than previously thought. Furthermore, our construction of PRUs in constant-depth with many-qubit TOFFOLI gates shows that, under cryptographic assumptions, there is no polynomial-time learning algorithm for the circuit class QAC⁰. Finally, our results suggest a new approach towards proving that PARITY is not computable in QAC⁰, a long-standing question in quantum complexity theory.

Cite as

Ben Foxman, Natalie Parham, Francisca Vasconcelos, and Henry Yuen. Random Unitaries in Constant (Quantum) Time. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 61:1-61:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{foxman_et_al:LIPIcs.ITCS.2026.61,
  author =	{Foxman, Ben and Parham, Natalie and Vasconcelos, Francisca and Yuen, Henry},
  title =	{{Random Unitaries in Constant (Quantum) Time}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{61:1--61:25},
  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.2026.61},
  URN =		{urn:nbn:de:0030-drops-253481},
  doi =		{10.4230/LIPIcs.ITCS.2026.61},
  annote =	{Keywords: Quantum Information, Pseudorandomness, Circuit Complexity}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.57

Anti-Concentration for the Unitary Haar Measure and Applications to Random Quantum Circuits

Authors: Bill Fefferman, Soumik Ghosh, and Wei Zhan

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

Abstract

We prove a Carbery-Wright style anti-concentration inequality for the unitary Haar measure, by showing that the probability of a polynomial in the entries of a random unitary falling into an ε range is at most a polynomial in ε. Using it, we show that the scrambling speed of a random quantum circuit is lower bounded: Namely, every input qubit has an influence that is at least inverse exponential in depth, on any output qubit touched by its lightcone. Our result on scrambling speed works with high probability over the choice of a circuit from an ensemble, as opposed to just working in expectation. As an application, we give the first polynomial-time algorithm for learning log-depth random quantum circuits with Haar random gates up to polynomially small diamond distance, given oracle access to the circuit. Other applications of this new scrambling speed lower bound include: - An optimal Ω(log ε^{-1}) depth lower bound for ε-approximate unitary designs on any circuit architecture; - A polynomial-time quantum algorithm that computes the depth of a bounded-depth circuit, given oracle access to the circuit. Our learning and depth-testing algorithms apply to architectures defined over any geometric dimension, and can be generalized to a wide class of architectures with good lightcone properties.

Cite as

Bill Fefferman, Soumik Ghosh, and Wei Zhan. Anti-Concentration for the Unitary Haar Measure and Applications to Random Quantum Circuits. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 57:1-57:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{fefferman_et_al:LIPIcs.ITCS.2026.57,
  author =	{Fefferman, Bill and Ghosh, Soumik and Zhan, Wei},
  title =	{{Anti-Concentration for the Unitary Haar Measure and Applications to Random Quantum Circuits}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{57:1--57:24},
  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.2026.57},
  URN =		{urn:nbn:de:0030-drops-253443},
  doi =		{10.4230/LIPIcs.ITCS.2026.57},
  annote =	{Keywords: Haar measure, anti-concentration, random quanytum circuit, learning}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.24

Unitary Complexity and the Uhlmann Transformation Problem

Authors: John Bostanci, Yuval Efron, Tony Metger, Alexander Poremba, Luowen Qian, and Henry Yuen

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

Abstract

State transformation problems such as compressing quantum information or breaking quantum commitments are fundamental quantum tasks. However, their computational difficulty cannot easily be characterized using traditional complexity theory, which focuses on tasks with classical inputs and outputs. To study the complexity of such state transformation tasks, we introduce a framework for unitary synthesis problems, including notions of reductions and unitary complexity classes. We use this framework to study the complexity of transforming one entangled state into another via local operations. We formalize this as the Uhlmann Transformation Problem, an algorithmic version of Uhlmann’s theorem. Then, we prove structural results relating the complexity of the Uhlmann Transformation Problem, polynomial space quantum computation, and zero knowledge protocols. The Uhlmann Transformation Problem allows us to characterize the complexity of a variety of tasks in quantum information processing, including decoding noisy quantum channels, breaking falsifiable quantum cryptographic assumptions, implementing optimal prover strategies in quantum interactive proofs, and decoding the Hawking radiation of black holes. Our framework for unitary complexity thus provides new avenues for studying the computational complexity of many natural quantum information processing tasks.

Cite as

John Bostanci, Yuval Efron, Tony Metger, Alexander Poremba, Luowen Qian, and Henry Yuen. Unitary Complexity and the Uhlmann Transformation Problem. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 24:1-24:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bostanci_et_al:LIPIcs.ITCS.2026.24,
  author =	{Bostanci, John and Efron, Yuval and Metger, Tony and Poremba, Alexander and Qian, Luowen and Yuen, Henry},
  title =	{{Unitary Complexity and the Uhlmann Transformation Problem}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{24:1--24:17},
  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.2026.24},
  URN =		{urn:nbn:de:0030-drops-253111},
  doi =		{10.4230/LIPIcs.ITCS.2026.24},
  annote =	{Keywords: Uhlmann’s theorem, unitary complexity theory}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.26

Local Transformations of Bipartite Entanglement Are Rigid

Authors: John Bostanci, Tony Metger, and Henry Yuen

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

Abstract

Uhlmann’s theorem is a fundamental result in quantum information theory that quantifies the optimal overlap between two bipartite pure states after applying local unitary operations (called Uhlmann transformations). We show that optimal Uhlmann transformations are rigid - in other words, they must be unique up to some well-characterized degrees of freedom. This rigidity is also robust: Uhlmann transformations achieving near-optimal overlaps must be close to the unique optimal transformation (again, up to well-characterized degrees of freedom). We describe two applications of our robust rigidity theorem: (a) we obtain better interactive proofs for synthesizing Uhlmann transformations and (b) we obtain a simple, alternative proof of the Gowers-Hatami theorem on the stability of approximate representations of finite groups.

Cite as

John Bostanci, Tony Metger, and Henry Yuen. Local Transformations of Bipartite Entanglement Are Rigid. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 26:1-26:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bostanci_et_al:LIPIcs.ITCS.2026.26,
  author =	{Bostanci, John and Metger, Tony and Yuen, Henry},
  title =	{{Local Transformations of Bipartite Entanglement Are Rigid}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{26:1--26:8},
  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.2026.26},
  URN =		{urn:nbn:de:0030-drops-253138},
  doi =		{10.4230/LIPIcs.ITCS.2026.26},
  annote =	{Keywords: Uhlmann’s theorem, quantum entanglement, stability theorems}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.70

Unconditional Pseudorandomness Against Shallow Quantum Circuits

Authors: Soumik Ghosh, Sathyawageeswar Subramanian, and Wei Zhan

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

Abstract

Quantum computational pseudorandomness has emerged as a fundamental notion that spans connections to complexity theory, cryptography and fundamental physics. However, all known constructions of efficient quantum-secure pseudorandom objects rely on complexity theoretic assumptions. In this work, we establish the first unconditionally secure efficient pseudorandom constructions against shallow-depth quantum circuit classes. We prove that: - Any quantum state 2-design yields unconditional pseudorandomness against both QNC⁰ circuits with arbitrarily many ancillae and AC⁰∘QNC⁰ circuits with nearly linear ancillae. - Random phased subspace states, where the phases are picked using a 4-wise independent function, are unconditionally pseudoentangled against the above circuit classes. - Any unitary 2-design yields unconditionally secure parallel-query pseudorandom unitaries against geometrically local QNC⁰ adversaries, even with limited AC⁰ postprocessing. Our results stand in stark contrast to the standard guarantee of the 2-design property, which only ensures that they cannot be distinguished from Haar random ensembles using two copies or queries. Our work demonstrates that quantum computational pseudorandomness can be achieved unconditionally for natural classes of restricted adversaries, opening new directions in quantum complexity theory.

Cite as

Soumik Ghosh, Sathyawageeswar Subramanian, and Wei Zhan. Unconditional Pseudorandomness Against Shallow Quantum Circuits. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 70:1-70:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{ghosh_et_al:LIPIcs.ITCS.2026.70,
  author =	{Ghosh, Soumik and Subramanian, Sathyawageeswar and Zhan, Wei},
  title =	{{Unconditional Pseudorandomness Against Shallow Quantum Circuits}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{70:1--70:25},
  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.2026.70},
  URN =		{urn:nbn:de:0030-drops-253578},
  doi =		{10.4230/LIPIcs.ITCS.2026.70},
  annote =	{Keywords: quantum pseudorandomness, shallow quantum circuits, pseudorandomness, t-designs}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.108

The Learning Stabilizers with Noise Problem

Authors: Alexander Poremba, Yihui Quek, and Peter Shor

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

Abstract

Random classical codes have good error correcting properties, and yet they are notoriously hard to decode in practice. Despite many decades of extensive study, the fastest known algorithms still run in exponential time. The Learning Parity with Noise (LPN) problem, which can be seen as the task of decoding a random linear code in the presence of noise, has thus emerged as a prominent hardness assumption with numerous applications in both cryptography and learning theory. Is there a natural quantum analog of the LPN problem? In this work, we introduce the Learning Stabilizers with Noise (LSN) problem, the task of decoding a random stabilizer code in the presence of local depolarizing noise. We give both polynomial-time and exponential-time quantum algorithms for solving LSN in various depolarizing noise regimes, ranging from extremely low noise, to low constant noise rates, and even higher noise rates up to a threshold. Next, we provide concrete evidence that LSN is hard. First, we show that LSN includes LPN as a special case, which suggests that it is at least as hard as its classical counterpart. Second, we prove worst-case to average-case reductions for variants of LSN. We then ask: what is the computational complexity of solving LSN? Because the task features quantum inputs, its complexity cannot be characterized by traditional complexity classes. Instead, we show that the LSN problem lies in a recently introduced (distributional and oracle) unitary synthesis class. Finally, we identify several applications of our LSN assumption, ranging from the construction of quantum bit commitment schemes to the computational limitations of learning from quantum data.

Cite as

Alexander Poremba, Yihui Quek, and Peter Shor. The Learning Stabilizers with Noise Problem. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 108:1-108:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{poremba_et_al:LIPIcs.ITCS.2026.108,
  author =	{Poremba, Alexander and Quek, Yihui and Shor, Peter},
  title =	{{The Learning Stabilizers with Noise Problem}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{108:1--108:19},
  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.2026.108},
  URN =		{urn:nbn:de:0030-drops-253950},
  doi =		{10.4230/LIPIcs.ITCS.2026.108},
  annote =	{Keywords: Random quantum stabilizer codes, average-case hardness}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.109

Cloning Games, Black Holes and Cryptography

Authors: Alexander Poremba, Seyoon Ragavan, and Vinod Vaikuntanathan

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

Abstract

In this work, we introduce a new toolkit for analyzing cloning games, a notion that captures stronger and more quantitative versions of the celebrated quantum no-cloning theorem. This framework allows us to analyze a new cloning game based on binary phase states. Our results provide evidence that these games may be able to overcome important limitations of previous candidates based on BB84 states and subspace coset states: in a model where the adversaries are restricted to making a single oracle query, we show that the binary phase variant is t-copy secure when t = o(n/log n). Moreover, for constant t, we obtain the first optimal bounds of O(2^{-n}), asymptotically matching the value attained by a trivial adversarial strategy. We also show a worst-case to average-case reduction which allows us to show the same quantitative results for the new and natural notion of Haar cloning games. Our analytic toolkit, which we believe will find further applications, is based on binary subtypes and uses novel bounds on the operator norms of block-wise tensor products of matrices. To illustrate the effectiveness of these new techniques, we present two applications: first, in black-hole physics, where our asymptotically optimal bound offers quantitative insights into information scrambling in idealized models of black holes; and second, in unclonable cryptography, where we (a) construct succinct unclonable encryption schemes from the existence of pseudorandom unitaries, and (b) propose and provide evidence for the security of multi-copy unclonable encryption schemes.

Cite as

Alexander Poremba, Seyoon Ragavan, and Vinod Vaikuntanathan. Cloning Games, Black Holes and Cryptography. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 109:1-109:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{poremba_et_al:LIPIcs.ITCS.2026.109,
  author =	{Poremba, Alexander and Ragavan, Seyoon and Vaikuntanathan, Vinod},
  title =	{{Cloning Games, Black Holes and Cryptography}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{109:1--109:21},
  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.2026.109},
  URN =		{urn:nbn:de:0030-drops-253961},
  doi =		{10.4230/LIPIcs.ITCS.2026.109},
  annote =	{Keywords: Unclonable cryptography, quantum pseudorandomness, black hole physics}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.34

Testing Classical Properties from Quantum Data

Authors: Matthias C. Caro, Preksha Naik, and Joseph Slote

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

Abstract

Many properties of Boolean functions can be tested far more efficiently than the function itself can be learned. However, this dramatic advantage often disappears when testers are limited to random samples of f instead of adaptively chosen queries to f. In this work we investigate the quantum version of this restriction: quantum algorithms that test properties of a Boolean function f solely from copies of either the function state |f⟩∝ ∑_x|x,f(x)⟩ or the phase state |(-1)^f⟩∝ ∑_x (-1)^{f(x)}|x⟩. Quantum advantage in testing from data. For monotonicity, symmetry, and triangle-freeness, we show passive quantum testers are unboundedly or super-polynomially better than their classical passive testing counterparts. They are competitive with classic query-based testers in each case. Inadequacy of Fourier sampling. Our new testers use techniques beyond quantum Fourier sampling, and it turns out this is necessary: we show a certain class of bent functions can be tested from 𝒪(1) function states but has a sample complexity lower bound of 2^{Ω(n)} for any tester relying exclusively on Fourier and classical samples. Classical queries vs. quantum data. Our passive quantum testers are competitive with classical query-based testers, but this isn't universal: we exhibit a testing problem that can be solved from 𝒪(1) classical queries but requires Ω(2^{n/2}) function state copies. The Forrelation problem provides a separation of the same magnitude in the opposite direction, so we conclude that quantum data and classical queries are "maximally incomparable" resources for testing. Towards lower bounds. We also begin the study of lower bounds for testing from quantum data. For quantum monotonicity testing, we prove that the ensembles of [Goldreich et al., 2000; Black, 2024], which give exponential lower bounds for classical sample-based testing, do not yield any nontrivial lower bounds for testing from quantum data. New insights specific to quantum data will be required for proving copy complexity lower bounds for testing in this model.

Cite as

Matthias C. Caro, Preksha Naik, and Joseph Slote. Testing Classical Properties from Quantum Data. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 34:1-34:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{caro_et_al:LIPIcs.ITCS.2026.34,
  author =	{Caro, Matthias C. and Naik, Preksha and Slote, Joseph},
  title =	{{Testing Classical Properties from Quantum Data}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{34:1--34:26},
  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.2026.34},
  URN =		{urn:nbn:de:0030-drops-253213},
  doi =		{10.4230/LIPIcs.ITCS.2026.34},
  annote =	{Keywords: Quantum Property Testing, Quantum Data, Boolean Functions}
}

Document

RANDOM

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.59

Testing Tensor Products of Algebraic Codes

Authors: Sumegha Garg, Madhu Sudan, and Gabriel Wu

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract

Motivated by recent advances in locally testable codes and quantum LDPCs based on robust testability of tensor product codes, we explore the local testability of tensor products of (an abstraction of) algebraic geometry codes. Such codes are parameterized by, in addition to standard parameters such as block length n and dimension k, their genus g. We show that the tensor product of two algebraic geometry codes is robustly locally testable provided n = Ω((k+g)²). Apart from Reed-Solomon codes, this seems to be the first explicit family of two-wise tensor codes of high dual distance that is robustly locally testable by the natural test that measures the expected distance of a random row/column from the underlying code.

Cite as

Sumegha Garg, Madhu Sudan, and Gabriel Wu. Testing Tensor Products of Algebraic Codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 59:1-59:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{garg_et_al:LIPIcs.APPROX/RANDOM.2025.59,
  author =	{Garg, Sumegha and Sudan, Madhu and Wu, Gabriel},
  title =	{{Testing Tensor Products of Algebraic Codes}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{59:1--59:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.59},
  URN =		{urn:nbn:de:0030-drops-244254},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.59},
  annote =	{Keywords: Algebraic geometry codes, Robust testability, Tensor products of codes}
}

Document

DOI: 10.4230/LIPIcs.TQC.2025.9

Efficient Quantum Pseudorandomness from Hamiltonian Phase States

Authors: John Bostanci, Jonas Haferkamp, Dominik Hangleiter, and Alexander Poremba

Published in: LIPIcs, Volume 350, 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025)

Abstract

Quantum pseudorandomness has found applications in many areas of quantum information, ranging from entanglement theory, to models of scrambling phenomena in chaotic quantum systems, and, more recently, in the foundations of quantum cryptography. Kretschmer (TQC '21) showed that both pseudorandom states and pseudorandom unitaries exist even in a world without classical one-way functions. To this day, however, all known constructions require classical cryptographic building blocks which are themselves synonymous with the existence of one-way functions, and which are also challenging to implement on realistic quantum hardware. In this work, we seek to make progress on both of these fronts simultaneously - by decoupling quantum pseudorandomness from classical cryptography altogether. We introduce a quantum hardness assumption called the Hamiltonian Phase State (HPS) problem, which is the task of decoding output states of a random instantaneous quantum polynomial-time (IQP) circuit. Hamiltonian phase states can be generated very efficiently using only Hadamard gates, single-qubit Z rotations and CNOT circuits. We show that the hardness of our problem reduces to a worst-case version of the problem, and we provide evidence that our assumption is plausibly fully quantum; meaning, it cannot be used to construct one-way functions. We also show information-theoretic hardness when only few copies of HPS are available by proving an approximate t-design property of our ensemble. Finally, we show that our HPS assumption and its variants allow us to efficiently construct many pseudorandom quantum primitives, ranging from pseudorandom states, to quantum pseudoentanglement, to pseudorandom unitaries, and even primitives such as public-key encryption with quantum keys.

Cite as

John Bostanci, Jonas Haferkamp, Dominik Hangleiter, and Alexander Poremba. Efficient Quantum Pseudorandomness from Hamiltonian Phase States. In 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 350, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bostanci_et_al:LIPIcs.TQC.2025.9,
  author =	{Bostanci, John and Haferkamp, Jonas and Hangleiter, Dominik and Poremba, Alexander},
  title =	{{Efficient Quantum Pseudorandomness from Hamiltonian Phase States}},
  booktitle =	{20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025)},
  pages =	{9:1--9:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-392-8},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{350},
  editor =	{Fefferman, Bill},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2025.9},
  URN =		{urn:nbn:de:0030-drops-240586},
  doi =		{10.4230/LIPIcs.TQC.2025.9},
  annote =	{Keywords: Quantum pseudorandomness, quantum phase states, quantum cryptography}
}

Document

DOI: 10.4230/LIPIcs.TQC.2025.10

Hamiltonian Locality Testing via Trotterized Postselection

Authors: John Kallaugher and Daniel Liang

Published in: LIPIcs, Volume 350, 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025)

Abstract

The (tolerant) Hamiltonian locality testing problem, introduced in [Bluhm, Caro, Oufkir '24], is to determine whether a Hamiltonian H is ε₁-close to being k-local (i.e. can be written as the sum of weight-k Pauli operators) or ε₂-far from any k-local Hamiltonian, given access to its time evolution operator and using as little total evolution time as possible, with distance typically defined by the normalized Frobenius norm. We give the tightest known bounds for this problem, proving an O(√(ε₂/((ε₂-ε₁)⁵)) evolution time upper bound and an Ω(1/(ε₂-ε₁)) lower bound. Our algorithm does not require reverse time evolution or controlled application of the time evolution operator, although our lower bound applies to algorithms using either tool. Furthermore, we show that if we are allowed reverse time evolution, this lower bound is tight, giving a matching O(1/(ε₂-ε₁)) evolution time algorithm.

Cite as

John Kallaugher and Daniel Liang. Hamiltonian Locality Testing via Trotterized Postselection. In 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 350, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kallaugher_et_al:LIPIcs.TQC.2025.10,
  author =	{Kallaugher, John and Liang, Daniel},
  title =	{{Hamiltonian Locality Testing via Trotterized Postselection}},
  booktitle =	{20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025)},
  pages =	{10:1--10:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-392-8},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{350},
  editor =	{Fefferman, Bill},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2025.10},
  URN =		{urn:nbn:de:0030-drops-240593},
  doi =		{10.4230/LIPIcs.TQC.2025.10},
  annote =	{Keywords: quantum algorithms, property testing, hamiltonians}
}

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