5 Search Results for "Weggemans, Jordi"

Document
DOI: 10.4230/LIPIcs.ESA.2025.73
Classical Algorithms for Constant Approximation of the Ground State Energy of Local Hamiltonians

Authors: François Le Gall

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
We construct classical algorithms computing an approximation of the ground state energy of an arbitrary k-local Hamiltonian acting on n qubits. We first consider the setting where a good "guiding state" is available, which is the main setting where quantum algorithms are expected to achieve an exponential speedup over classical methods. We show that a constant approximation (i.e., an approximation with constant relative accuracy) of the ground state energy can be computed classically in poly (1/χ,n) time and poly(n) space, where χ denotes the overlap between the guiding state and the ground state (as in prior works in dequantization, we assume sample-and-query access to the guiding state). This gives a significant improvement over the recent classical algorithm by Gharibian and Le Gall (SICOMP 2023), and matches (up to a polynomial overhead) both the time and space complexities of quantum algorithms for constant approximation of the ground state energy. We also obtain classical algorithms for higher-precision approximation. For the setting where no guided state is given (i.e., the standard version of the local Hamiltonian problem), we obtain a classical algorithm computing a constant approximation of the ground state energy in 2^O(n) time and poly(n) space. To our knowledge, before this work it was unknown how to classically achieve these bounds simultaneously, even for constant approximation. We also discuss complexity-theoretic aspects of our results.
Cite as

François Le Gall. Classical Algorithms for Constant Approximation of the Ground State Energy of Local Hamiltonians. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 73:1-73:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{legall:LIPIcs.ESA.2025.73,
  author =	{Le Gall, Fran\c{c}ois},
  title =	{{Classical Algorithms for Constant Approximation of the Ground State Energy of Local Hamiltonians}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{73:1--73:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.73},
  URN =		{urn:nbn:de:0030-drops-245419},
  doi =		{10.4230/LIPIcs.ESA.2025.73},
  annote =	{Keywords: approximation algorithms, quantum computing, dequantization}
}
Document
DOI: 10.4230/LIPIcs.TQC.2025.6
Quantum SAT Problems with Finite Sets of Projectors Are Complete for a Plethora of Classes

Authors: Ricardo Rivera Cardoso, Alex Meiburg, and Daniel Nagaj

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

Abstract
Previously, all known variants of the Quantum Satisfiability (QSAT) problem - consisting of determining whether a k-local (k-body) Hamiltonian is frustration-free - could be classified as being either in 𝖯; or complete for NP, MA, or QMA₁. Here, we present new qubit variants of this problem that are complete for BQP₁, coRP, QCMA, PI(coRP,NP), PI(BQP₁,NP), PI(BQP₁,MA), SoPU(coRP,NP), SoPU(BQP₁,NP), and SoPU(BQP₁,MA). Our result implies that a complete classification of quantum constraint satisfaction problems (QCSPs), analogous to Schaefer’s dichotomy theorem for classical CSPs, must either include these 13 classes, or otherwise show that some are equal. Additionally, our result showcases two new types of QSAT problems that can be decided efficiently, as well as the first nontrivial BQP₁-complete problem. We first construct QSAT problems on qudits that are complete for BQP₁, coRP, and QCMA. These are made by restricting the finite set of Hamiltonians to consist of elements similar to H_{init}, H_{prop}, and H_{out}, seen in the circuit-to-Hamiltonian transformation. Usually, these are used to demonstrate hardness of QSAT and Local Hamiltonian problems, and so our proofs of hardness are simple. The difficulty lies in ensuring that all Hamiltonians generated with these three elements can be decided in their respective classes. For this, we build our Hamiltonian terms with high-dimensional data and clock qudits, ternary logic, and either monogamy of entanglement or specific clock encodings. We then show how to express these problems in terms of qubits, by proving that any QCSP can be reduced to a qubit problem while maintaining the same complexity - something not believed possible classically. The remaining six problems are obtained by considering "sums" and "products" of some of the QSAT problems mentioned here. Before this work, the QSAT problems generated in this way resulted in complete problems for PI and SoPU classes that were trivially equal to NP, MA, or QMA₁. We thus commence the study of these new and seemingly nontrivial classes. While [Meiburg, 2021] first sought to prove completeness for coRP, BQP₁, and QCMA, we note that those constructions are flawed. Here, we rework them, provide correct proofs, and obtain improvements on the required qudit dimensionality.
Cite as

Ricardo Rivera Cardoso, Alex Meiburg, and Daniel Nagaj. Quantum SAT Problems with Finite Sets of Projectors Are Complete for a Plethora of Classes. In 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 350, pp. 6:1-6:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{riveracardoso_et_al:LIPIcs.TQC.2025.6,
  author =	{Rivera Cardoso, Ricardo and Meiburg, Alex and Nagaj, Daniel},
  title =	{{Quantum SAT Problems with Finite Sets of Projectors Are Complete for a Plethora of Classes}},
  booktitle =	{20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025)},
  pages =	{6:1--6:24},
  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.2025.6},
  URN =		{urn:nbn:de:0030-drops-240557},
  doi =		{10.4230/LIPIcs.TQC.2025.6},
  annote =	{Keywords: Quantum complexity theory, quantum satisfiability, circuit-to-Hamiltonian, pairwise union of classes, pairwise intersection of classes}
}
Document
DOI: 10.4230/LIPIcs.TQC.2024.10
Guidable Local Hamiltonian Problems with Implications to Heuristic Ansatz State Preparation and the Quantum PCP Conjecture

Authors: Jordi Weggemans, Marten Folkertsma, and Chris Cade

Published in: LIPIcs, Volume 310, 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024)

Abstract
We study "Merlinized" versions of the recently defined Guided Local Hamiltonian problem, which we call "Guidable Local Hamiltonian" problems. Unlike their guided counterparts, these problems do not have a guiding state provided as a part of the input, but merely come with the promise that one exists. We consider in particular two classes of guiding states: those that can be prepared efficiently by a quantum circuit; and those belonging to a class of quantum states we call classically evaluatable, for which it is possible to efficiently compute expectation values of local observables classically. We show that guidable local Hamiltonian problems for both classes of guiding states are QCMA-complete in the inverse-polynomial precision setting, but lie within NP (or NqP) in the constant precision regime when the guiding state is classically evaluatable. Our completeness results show that, from a complexity-theoretic perspective, classical Ansätze selected by classical heuristics are just as powerful as quantum Ansätze prepared by quantum heuristics, as long as one has access to quantum phase estimation. In relation to the quantum PCP conjecture, we (i) define a complexity class capturing quantum-classical probabilistically checkable proof systems and show that it is contained in BQP^NP[1] for constant proof queries; (ii) give a no-go result on "dequantizing" the known quantum reduction which maps a QPCP-verification circuit to a local Hamiltonian with constant promise gap; (iii) give several no-go results for the existence of quantum gap amplification procedures that preserve certain ground state properties; and (iv) propose two conjectures that can be viewed as stronger versions of the NLTS theorem. Finally, we show that many of our results can be directly modified to obtain similar results for the class MA.
Cite as

Jordi Weggemans, Marten Folkertsma, and Chris Cade. Guidable Local Hamiltonian Problems with Implications to Heuristic Ansatz State Preparation and the Quantum PCP Conjecture. In 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 310, pp. 10:1-10:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{weggemans_et_al:LIPIcs.TQC.2024.10,
  author =	{Weggemans, Jordi and Folkertsma, Marten and Cade, Chris},
  title =	{{Guidable Local Hamiltonian Problems with Implications to Heuristic Ansatz State Preparation and the Quantum PCP Conjecture}},
  booktitle =	{19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024)},
  pages =	{10:1--10:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-328-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{310},
  editor =	{Magniez, Fr\'{e}d\'{e}ric and Grilo, Alex Bredariol},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2024.10},
  URN =		{urn:nbn:de:0030-drops-206804},
  doi =		{10.4230/LIPIcs.TQC.2024.10},
  annote =	{Keywords: Quantum complexity theory, local Hamiltonian problem, quantum state ansatzes, QCMA, quantum PCP conjecture}
}
Document
DOI: 10.4230/LIPIcs.ESA.2023.81
Tight Bounds for Quantum Phase Estimation and Related Problems

Authors: Nikhil S. Mande and Ronald de Wolf

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Abstract
Phase estimation, due to Kitaev [arXiv'95], is one of the most fundamental subroutines in quantum computing, used in Shor’s factoring algorithm, optimization algorithms, quantum chemistry algorithms, and many others. In the basic scenario, one is given black-box access to a unitary U, and an eigenstate |ψ⟩ of U with unknown eigenvalue e^{iθ}, and the task is to estimate the eigenphase θ within ±δ, with high probability. The repeated application of U and U^{-1} is typically the most expensive part of phase estimation, so for us the cost of an algorithm will be that number of applications. Motivated by the "guided Hamiltonian problem" in quantum chemistry, we tightly characterize the cost of several variants of phase estimation where we are no longer given an arbitrary eigenstate, but are required to estimate the maximum eigenphase of U, aided by advice in the form of states (or a unitary preparing those states) which are promised to have at least a certain overlap γ with the top eigenspace. We give algorithms and matching lower bounds (up to logarithmic factors) for all ranges of parameters. We show a crossover point below which advice is not helpful: o(1/γ²) copies of the advice state (or o(1/γ) applications of an advice-preparing unitary) are not significantly better than having no advice at all. We also show that having knowledge of the eigenbasis of U does not significantly reduce cost. Our upper bounds use the subroutine of generalized maximum-finding of van Apeldoorn, Gilyén, Gribling, and de Wolf [Quantum'20], the state-based Hamiltonian simulation of Lloyd, Mohseni, and Rebentrost [Nature Physics'13], and several other techniques. Our lower bounds follow by reductions from a fractional version of the Boolean OR function with advice, which we lower bound by a simple modification of the adversary method of Ambainis [JCSS'02]. As an immediate consequence we also obtain a lower bound on the complexity of the Unitary recurrence time problem, matching an upper bound of She and Yuen [ITCS'23] and resolving an open question posed by them. Lastly, we study how efficiently one can reduce the error probability in the basic phase-estimation scenario. We show that an algorithm solving phase estimation to precision δ with error probability at most ε must have cost Ω(1/δ log(1/ε)), matching the obvious way to error-reduce the basic constant-error-probability phase estimation algorithm. This contrasts with some other scenarios in quantum computing (e.g. search) where error-reduction costs only a factor O(√{log(1/ε)}). Our lower bound technique uses a variant of the polynomial method with trigonometric polynomials.
Cite as

Nikhil S. Mande and Ronald de Wolf. Tight Bounds for Quantum Phase Estimation and Related Problems. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 81:1-81:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{mande_et_al:LIPIcs.ESA.2023.81,
  author =	{Mande, Nikhil S. and de Wolf, Ronald},
  title =	{{Tight Bounds for Quantum Phase Estimation and Related Problems}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{81:1--81:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.81},
  URN =		{urn:nbn:de:0030-drops-187346},
  doi =		{10.4230/LIPIcs.ESA.2023.81},
  annote =	{Keywords: Phase estimation, quantum computing}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.32
Improved Hardness Results for the Guided Local Hamiltonian Problem

Authors: Chris Cade, Marten Folkertsma, Sevag Gharibian, Ryu Hayakawa, François Le Gall, Tomoyuki Morimae, and Jordi Weggemans

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
Estimating the ground state energy of a local Hamiltonian is a central problem in quantum chemistry. In order to further investigate its complexity and the potential of quantum algorithms for quantum chemistry, Gharibian and Le Gall (STOC 2022) recently introduced the guided local Hamiltonian problem (GLH), which is a variant of the local Hamiltonian problem where an approximation of a ground state (which is called a guiding state) is given as an additional input. Gharibian and Le Gall showed quantum advantage (more precisely, BQP-completeness) for GLH with 6-local Hamiltonians when the guiding state has fidelity (inverse-polynomially) close to 1/2 with a ground state. In this paper, we optimally improve both the locality and the fidelity parameter: we show that the BQP-completeness persists even with 2-local Hamiltonians, and even when the guiding state has fidelity (inverse-polynomially) close to 1 with a ground state. Moreover, we show that the BQP-completeness also holds for 2-local physically motivated Hamiltonians on a 2D square lattice or a 2D triangular lattice. Beyond the hardness of estimating the ground state energy, we also show BQP-hardness persists when considering estimating energies of excited states of these Hamiltonians instead. Those make further steps towards establishing practical quantum advantage in quantum chemistry.
Cite as

Chris Cade, Marten Folkertsma, Sevag Gharibian, Ryu Hayakawa, François Le Gall, Tomoyuki Morimae, and Jordi Weggemans. Improved Hardness Results for the Guided Local Hamiltonian Problem. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 32:1-32:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cade_et_al:LIPIcs.ICALP.2023.32,
  author =	{Cade, Chris and Folkertsma, Marten and Gharibian, Sevag and Hayakawa, Ryu and Le Gall, Fran\c{c}ois and Morimae, Tomoyuki and Weggemans, Jordi},
  title =	{{Improved Hardness Results for the Guided Local Hamiltonian Problem}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{32:1--32:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.32},
  URN =		{urn:nbn:de:0030-drops-180840},
  doi =		{10.4230/LIPIcs.ICALP.2023.32},
  annote =	{Keywords: Quantum computing, Quantum advantage, Quantum Chemistry, Guided Local Hamiltonian Problem}
}
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