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LIPIcs, Volume 389
21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026)
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- published at: 2026-08-25
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-439-0
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Complete Volume
LIPIcs, Volume 389, TQC 2026, Complete Volume
Abstract
LIPIcs, Volume 389, TQC 2026, Complete Volume
Cite as
21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 389, pp. 1-170, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@Proceedings{arnon_et_al:LIPIcs.TQC.2026,
title = {{LIPIcs, Volume 389, TQC 2026, Complete Volume}},
booktitle = {21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026)},
pages = {1--170},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-439-0},
ISSN = {1868-8969},
year = {2026},
volume = {389},
editor = {Arnon, Rotem and Harrow, Aram W.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2026},
URN = {urn:nbn:de:0030-drops-276925},
doi = {10.4230/LIPIcs.TQC.2026},
annote = {Keywords: LIPIcs, Volume 389, TQC 2026, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 389, pp. 0:i-0:xii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{arnon_et_al:LIPIcs.TQC.2026.0,
author = {Arnon, Rotem and Harrow, Aram W.},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026)},
pages = {0:i--0:xii},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-439-0},
ISSN = {1868-8969},
year = {2026},
volume = {389},
editor = {Arnon, Rotem and Harrow, Aram W.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2026.0},
URN = {urn:nbn:de:0030-drops-276915},
doi = {10.4230/LIPIcs.TQC.2026.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
A Sharp Computational Phase Transition for the Partition Function of the Transverse-Field Ising Model
Abstract
We study the problem of approximating the partition function of the transverse-field Ising model (TFIM), a widely studied quantum many-body model with important applications in quantum simulation and quantum annealing. Despite its fundamental importance, the algorithmic landscape for computing the TFIM partition function has remained poorly understood beyond restricted parameter regimes.
We provide a precise characterization of the temperature regimes in which efficient approximation is possible, establishing a sharp computational phase transition. Let J denote the symmetric interaction matrix and Δ(J) = λ_{max}(J)-λ_{min}(J) denote its spectral width. We show that, for all inverse temperatures β ∈ [0,1/Δ(J)], there exists an efficient classical randomized algorithm that approximates the partition function tr(e^{-β H}) to within an arbitrarily small multiplicative factor. To obtain this result, we apply the standard Trotter decomposition to map the quantum model to a classical spin system, and then leverage new techniques in Markov chain analysis to derive an efficient algorithm that samples from and computes the partition function of the resulting distribution. This temperature threshold is tight: for β > 1/Δ(J), we show that there exists interaction matrix J for which approximating the partition function, even within an exponential factor, is NP-hard and thus is unlikely to admit an efficient classical or quantum algorithm.
Cite as
Alistair Sinclair and Thuy-Duong Vuong. A Sharp Computational Phase Transition for the Partition Function of the Transverse-Field Ising Model. In 21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 389, pp. 1:1-1:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{sinclair_et_al:LIPIcs.TQC.2026.1,
author = {Sinclair, Alistair and Vuong, Thuy-Duong},
title = {{A Sharp Computational Phase Transition for the Partition Function of the Transverse-Field Ising Model}},
booktitle = {21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026)},
pages = {1:1--1:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-439-0},
ISSN = {1868-8969},
year = {2026},
volume = {389},
editor = {Arnon, Rotem and Harrow, Aram W.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2026.1},
URN = {urn:nbn:de:0030-drops-272986},
doi = {10.4230/LIPIcs.TQC.2026.1},
annote = {Keywords: Computational phase transition, partition function, transverse-field Ising, sampling, counting}
}
Document
Characterization of Permutation Gates in the Third Level of the Clifford Hierarchy
Abstract
The Clifford hierarchy is a fundamental structure in quantum computation whose mathematical properties are not fully understood. In this work, we characterize permutation gates - unitaries which permute the 2ⁿ basis states - in the third level of the hierarchy. We prove that any permutation gate in the third level must be a product of Toffoli gates in what we define as staircase form, up to left and right multiplications by Clifford permutations. We then present necessary and sufficient conditions for a staircase form permutation gate to be in the third level of the Clifford hierarchy. As a corollary, we construct a family of non-semi-Clifford permutation gates {U_k}_{k ≥ 3} in staircase form such that each U_k is in the third level but its inverse is not in the k-th level.
Cite as
Zhiyang He, Luke Robitaille, and Xinyu Tan. Characterization of Permutation Gates in the Third Level of the Clifford Hierarchy. In 21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 389, pp. 2:1-2:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{he_et_al:LIPIcs.TQC.2026.2,
author = {He, Zhiyang and Robitaille, Luke and Tan, Xinyu},
title = {{Characterization of Permutation Gates in the Third Level of the Clifford Hierarchy}},
booktitle = {21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026)},
pages = {2:1--2:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-439-0},
ISSN = {1868-8969},
year = {2026},
volume = {389},
editor = {Arnon, Rotem and Harrow, Aram W.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2026.2},
URN = {urn:nbn:de:0030-drops-272997},
doi = {10.4230/LIPIcs.TQC.2026.2},
annote = {Keywords: Quantum fault-tolerance, Clifford hierarchy, permutation gates}
}
Document
Quantum Search with Generalized Wildcards
Abstract
In the "search with wildcards" problem [Ambainis, Montanaro, Quantum Inf. Comput.'14], one’s goal is to learn an unknown bit-string x ∈ {-1,1}ⁿ. An algorithm may, at unit cost, test equality of any subset of the hidden string with a string of its choice. Ambainis and Montanaro showed a quantum algorithm of cost O(√n log n) and a near-matching lower bound of Ω(√n). Belovs [Comput. Comp.'15] subsequently showed a tight O(√n) upper bound.
We consider a natural generalization of this problem, parametrized by a subset Q ⊆ 2^{[n]}, where an algorithm may test whether x_S = b for an arbitrary S ∈ Q and b ∈ {-1,1}^S of its choice, at unit cost. We show the following:
- For all k ∈ [n], when Q is the collection of all sets of size at most k, the quantum query complexity is Θ(n/√k). In particular when k = n, this corresponds to the standard search with wildcards setting. This recovers and generalizes the tight characterization of Belovs, and Ambainis and Montanaro, using completely different techniques.
- When Q is the collection of contiguous blocks, the quantum query complexity is Θ̃(n).
- When Q is the collection of prefixes, the quantum query complexity is Θ(n). All of these results are derived using a framework that we develop. We apply a symmetry reduction to the primal version of the negative-weight adversary bound, and show that the quantum query complexity of learning x is characterized, up to a constant factor, by a particular optimization program, which can be succinctly described as follows: `maximize over all odd functions f : {-1,1}ⁿ → ℝ the ratio of the maximum value of f to the maximum (over T ∈ Q) standard deviation of f on a subcube whose free variables are exactly T.'
To the best of our knowledge, ours is the first work to use the primal version of the negative-weight adversary bound (which is a maximization program typically used to show lower bounds) to show new quantum query upper bounds without explicitly resorting to SDP duality.
Cite as
Arjan Cornelissen, Nikhil S. Mande, Subhasree Patro, Nithish Raja, and Swagato Sanyal. Quantum Search with Generalized Wildcards. In 21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 389, pp. 3:1-3:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cornelissen_et_al:LIPIcs.TQC.2026.3,
author = {Cornelissen, Arjan and Mande, Nikhil S. and Patro, Subhasree and Raja, Nithish and Sanyal, Swagato},
title = {{Quantum Search with Generalized Wildcards}},
booktitle = {21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026)},
pages = {3:1--3:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-439-0},
ISSN = {1868-8969},
year = {2026},
volume = {389},
editor = {Arnon, Rotem and Harrow, Aram W.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2026.3},
URN = {urn:nbn:de:0030-drops-273007},
doi = {10.4230/LIPIcs.TQC.2026.3},
annote = {Keywords: quantum algorithms, quantum query complexity, adversary bound, symmetry reduction, substring queries}
}
Document
Provable Speedups in Convex Optimization via Quantum Dynamics
Abstract
This work investigates the possibility of quantum speedups for continuous optimization through quantum Hamiltonian simulation. We establish the first rigorous query complexity bounds for unconstrained convex optimization via digital quantum annealing, based on the non-adiabatic Quantum Hamiltonian Descent (QHD) framework. In the process, we derive rigorous resource estimates for digital quantum simulation of Schrödinger operators depending only on input simulation parameters, given black-box evaluation access to separable, Lipschitz continuous potential b(t) f(x). We apply these simulation bounds to assess the complexity of high-dimensional convex optimization.
Our annealing schedule achieves arbitrarily fast convergence rates in the evolution time, with computational time determined solely by the cost of discretization. We show that a G-Lipschitz convex function can be optimized to an error of ε with 𝒪̃(d^{1.5} G² R²/ε²) queries, given a starting point that is Euclidean distance R from optimal. Under reasonable assumptions such as the complexity of simulating Schrödinger operators, we show that Ω̃(d/ε²) queries are necessary. This suggests QHD does not offer improvements over classical methods in the noiseless zeroth order setting.
However, we show that the QHD algorithm can tolerate 𝒪̃(ε³ /d^{1.5} G² R²) noise in function evaluation, and as a result, provides a super-quadratic query advantage over the best existing noise-tolerant classical algorithms in the high-dimensional setting. We leverage this to design a quantum algorithm for stochastic convex optimization that offers a super-quadratic speedup over all known classical and quantum algorithms in this regime. To our knowledge, these results represent the first rigorous quantum speedups for convex optimization obtained through a dynamical algorithm.
Cite as
Shouvanik Chakrabarti, Dylan Herman, Jacob Watkins, Enrico Fontana, Brandon Augustino, Junhyung Lyle Kim, and Marco Pistoia. Provable Speedups in Convex Optimization via Quantum Dynamics. In 21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 389, pp. 4:1-4:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chakrabarti_et_al:LIPIcs.TQC.2026.4,
author = {Chakrabarti, Shouvanik and Herman, Dylan and Watkins, Jacob and Fontana, Enrico and Augustino, Brandon and Kim, Junhyung Lyle and Pistoia, Marco},
title = {{Provable Speedups in Convex Optimization via Quantum Dynamics}},
booktitle = {21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026)},
pages = {4:1--4:25},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-439-0},
ISSN = {1868-8969},
year = {2026},
volume = {389},
editor = {Arnon, Rotem and Harrow, Aram W.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2026.4},
URN = {urn:nbn:de:0030-drops-273011},
doi = {10.4230/LIPIcs.TQC.2026.4},
annote = {Keywords: Convex optimization, Hamiltonian simulation, zeroth order}
}
Document
Quantum Merlin-Arthur with an Internally Separable Proof
Abstract
While the role of entanglement in quantum proof systems has been extensively studied, the computational power of unentanglement remains poorly understood. Since entanglement admits many inequivalent multipartite structures, it is natural to ask how more fine-grained structural promises affect computational power. In this work we investigate a mild promise: each proof is internally separable, meaning that after tracing out one register, a designated constant-size subsystem is separable from the rest - even though the overall proof may still be entangled across every bipartition. We prove a qualitative jump from one proof to two: with one internally separable proof, the resulting class is contained in EXP (even allowing an inverse-exponential completeness–soundness gap), whereas with two unentangled internally separable proofs, the class equals NEXP at constant gap. Notably, in the NEXP construction, the second proof is used solely to implement a SWAP-based purity test.
Cite as
Roozbeh Bassirian, Bill Fefferman, Itai Leigh, Kunal Marwaha, and Pei Wu. Quantum Merlin-Arthur with an Internally Separable Proof. In 21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 389, pp. 5:1-5:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bassirian_et_al:LIPIcs.TQC.2026.5,
author = {Bassirian, Roozbeh and Fefferman, Bill and Leigh, Itai and Marwaha, Kunal and Wu, Pei},
title = {{Quantum Merlin-Arthur with an Internally Separable Proof}},
booktitle = {21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026)},
pages = {5:1--5:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-439-0},
ISSN = {1868-8969},
year = {2026},
volume = {389},
editor = {Arnon, Rotem and Harrow, Aram W.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2026.5},
URN = {urn:nbn:de:0030-drops-273022},
doi = {10.4230/LIPIcs.TQC.2026.5},
annote = {Keywords: entanglement structures, unentanglement, quantum complexity, QMA(2), NEXP}
}
Document
Limitations of Decoded Quantum Interferometry for MaxCut
Abstract
Decoded Quantum Interferometry (DQI) is a framework for approximating special kinds of discrete optimization problems that relies on problem structure in a way that sets it apart from other classical or quantum approaches. We show that any family of instances of MaxCut on which DQI attains a nontrivial asymptotic approximation guarantee is also solvable exactly in classical polynomial time. We include a streamlined exposition of DQI tailored for MaxCut that relies on elementary graph theory instead of coding theory to motivate and explain the algorithm.
Cite as
Ojas Parekh. Limitations of Decoded Quantum Interferometry for MaxCut. In 21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 389, pp. 6:1-6:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{parekh:LIPIcs.TQC.2026.6,
author = {Parekh, Ojas},
title = {{Limitations of Decoded Quantum Interferometry for MaxCut}},
booktitle = {21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026)},
pages = {6:1--6:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-439-0},
ISSN = {1868-8969},
year = {2026},
volume = {389},
editor = {Arnon, Rotem and Harrow, Aram W.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2026.6},
URN = {urn:nbn:de:0030-drops-273038},
doi = {10.4230/LIPIcs.TQC.2026.6},
annote = {Keywords: Decoded Quantum Interferometry, MaxCut, quantum approximation algorithms}
}
Document
On the Complexity of the Circuit Width Problem
Abstract
Montanaro associated to every quantum circuit over the gate set {H,Z,CZ,CCZ} a degree-three polynomial over 𝔽₂, and showed that the strong simulation cost of the circuit is governed by the minimum number of qubit wires needed to realize that polynomial. We study this minimum-width problem. We prove that deciding whether a degree-three polynomial has width at most k is NP-complete. The reduction is parsimonious enough to imply that, for every α < 49/48, the corresponding gap version is NP-hard. We also prove the same inapproximability threshold for degree-two polynomials by a twin-copy replacement of the cubic constraints. On the positive side, we give a nondeterministic search algorithm using only O(klog(en/k)) nondeterministic bits, and a deterministic fixed-parameter algorithm running in k^{6k+o(k)}n+O(m) time on an input with n symbols and m monomials.
Cite as
Zhengfeng Ji, Yinchen Liu, and Zhe'ou Zhou. On the Complexity of the Circuit Width Problem. In 21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 389, pp. 7:1-7:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ji_et_al:LIPIcs.TQC.2026.7,
author = {Ji, Zhengfeng and Liu, Yinchen and Zhou, Zhe'ou},
title = {{On the Complexity of the Circuit Width Problem}},
booktitle = {21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026)},
pages = {7:1--7:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-439-0},
ISSN = {1868-8969},
year = {2026},
volume = {389},
editor = {Arnon, Rotem and Harrow, Aram W.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2026.7},
URN = {urn:nbn:de:0030-drops-273045},
doi = {10.4230/LIPIcs.TQC.2026.7},
annote = {Keywords: IQP circuits, circuit width, NP-completeness, approximation hardness, parameterized algorithms}
}
Document
Quantum Speedups for Sampling and Non-Convex Optimization with Stochastic Oracles
Abstract
We present quantum speedups for sampling from distributions of the form π∝ e^{-f} on ℝ^d. We consider two stochastic oracle models: a stochastic gradient oracle, where f = 1/n∑_{i = 1}ⁿ f_i and component gradients {∇ f_i}_{i ∈ [n]} are available, and a stochastic evaluation oracle, where only noisy values of f are available. Our framework accelerates classical stochastic Langevin Monte Carlo (LMC) and Hamiltonian Monte Carlo (HMC) algorithms by replacing stochastic gradient estimators with variance-controlled quantum mean estimation and gradient estimation subroutines. Unlike quantum walk based approaches, our algorithms do not require reversibility or exact gradients, and they preserve the structure of the underlying Markov chain. In the finite-sum setting, quantum mean estimation combined with classical variance-reduction techniques improves the stochastic gradient-query complexity for the approximate sampling task. In the stochastic zeroth-order setting, we develop gradient estimators robust to noisy function evaluations, yielding improved evaluation complexity for LMC and HMC. These results apply to strongly log-concave and/or non-log-concave distributions satisfying a log-Sobolev inequality, with convergence guarantees in Wasserstein distance and Kullback-Leibler divergence. We also show that faster sampling methods lead to quantum speedups for optimization, including for non-smooth and approximately convex objectives.
Cite as
Guneykan Ozgul, Xiantao Li, Mehrdad Mahdavi, and Chunhao Wang. Quantum Speedups for Sampling and Non-Convex Optimization with Stochastic Oracles. In 21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 389, pp. 8:1-8:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ozgul_et_al:LIPIcs.TQC.2026.8,
author = {Ozgul, Guneykan and Li, Xiantao and Mahdavi, Mehrdad and Wang, Chunhao},
title = {{Quantum Speedups for Sampling and Non-Convex Optimization with Stochastic Oracles}},
booktitle = {21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026)},
pages = {8:1--8:25},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-439-0},
ISSN = {1868-8969},
year = {2026},
volume = {389},
editor = {Arnon, Rotem and Harrow, Aram W.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2026.8},
URN = {urn:nbn:de:0030-drops-273053},
doi = {10.4230/LIPIcs.TQC.2026.8},
annote = {Keywords: Quantum algorithms, sampling, Langevin Monte Carlo, Hamiltonian Monte Carlo, stochastic oracles, non-convex optimization}
}