arxiv: 2604.15263 · v1 · submitted 2026-04-16 · 🪐 quant-ph · math-ph· math.MP

Recognition: unknown

Computing the free energy of quantum Coulomb gases and molecules via quantum Gibbs sampling

Simon Becker , Cambyse Rouz\'e , Robert Salzmann

Authors on Pith no claims yet

Pith reviewed 2026-05-10 11:32 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP

keywords quantum algorithmsfree energy estimationCoulomb gasesGibbs samplingquantum Markov semigroupsspectral gapfinite temperature

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The pith

A quantum algorithm estimates the free energy of Coulomb gases by truncating the interaction to finite rank and sampling the Gibbs state with a guaranteed positive spectral gap.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a quantum algorithm for computing the free energy and the full Gibbs state of quantum Coulomb gases and molecules in two and three dimensions at finite temperature. These systems have singular interactions and infinite-dimensional Hilbert spaces that defeat prior methods. The authors first prove that replacing the interaction with a finite-rank low-energy truncation approximates the true free energy with an error bounded by a polynomial in the particle number. This reduces the problem to one where a custom quantum Markov semigroup can be run. The semigroup's generator is shown to have a strictly positive spectral gap for any truncation, which implies exponential convergence to the target state and yields an explicit quantum circuit plus end-to-end complexity bound.

Core claim

The free energy of the full many-body Hamiltonian can be approximated by that of the same Hamiltonian with a finite-rank low-energy truncation of the interaction, with an explicit error bound polynomial in the particle number. A quantum Gibbs sampling scheme based on quantum Markov semigroups has a generator with a strictly positive spectral gap for every truncation, implying exponential convergence to the target Gibbs state. This supplies the first rigorous mixing-time guarantee for Gibbs sampling in a Coulomb-interacting continuous-variable quantum system and produces an explicit quantum circuit for the estimation task.

What carries the argument

The quantum Markov semigroup whose generator has a strictly positive spectral gap when applied to the finite-rank low-energy truncation of the Coulomb interaction Hamiltonian.

If this is right

Free energy and Gibbs-state estimation for these singular continuous-variable systems becomes possible on quantum computers with rigorous error controls.
The algorithm works at finite temperature without invoking classical reductions such as the Born-Oppenheimer approximation.
Exponential convergence holds uniformly for every truncation level, so the sampling cost does not blow up when the truncation rank is increased.
Both the scalar free-energy value and the full quantum state can be recovered from the same circuit family.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The truncation-plus-spectral-gap strategy may extend to other singular potentials that currently lack quantum algorithms.
Proving spectral gaps for related quantum semigroups could improve mixing-time analyses beyond Coulomb systems.
The circuit construction offers a concrete target for fault-tolerant quantum hardware to simulate finite-temperature molecular dynamics.

Load-bearing premise

The free energy difference between the full interaction and its finite-rank low-energy truncation stays bounded by a polynomial in the number of particles.

What would settle it

A direct computation on a small number of particles in which the free-energy difference between the truncated and untruncated Hamiltonians grows faster than the polynomial bound derived in the approximation theorem.

read the original abstract

We develop a quantum algorithm for estimating the free energy as well as the total Gibbs state of interacting quantum Coulomb gases and molecular systems in dimensions $d \in \{2,3\}$ at finite temperature. These systems lie beyond the reach of existing methods due to their singular interactions and infinite-dimensional Hilbert space structure. First, we show that the free energy of the full many-body Hamiltonian can be approximated by that of the same Hamiltonian with a finite-rank low-energy truncation of the interaction, with an explicit error bound polynomial in the particle number. This reduces the problem to a controlled finite-rank perturbation problem. Second, we introduce a quantum Gibbs sampling scheme tailored to this truncated system, based on a class of quantum Markov semigroups. Our main analytical result establishes that the associated generator has a strictly positive spectral gap for every truncation, implying exponential convergence to the target Gibbs state. This provides, to our knowledge, the first rigorous mixing-time guarantee for Gibbs sampling in a Coulomb interacting continuous-variable quantum system. Finally, we give an explicit quantum circuit implementation of the dynamics and derive an end-to-end complexity bound for approximating the free energy and the Gibbs state itself. Our results provide a mathematically rigorous route to quantum algorithms for free energy estimation in interacting quantum systems, without relying on classical approximations such as the Born-Oppenheimer reduction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The truncation error bound is the real question mark here; if it's only polynomial in N without a linear or better guarantee, the whole reduction falls apart for extensive free energies.

read the letter

The paper's core contribution is a quantum Gibbs sampling scheme for truncated Coulomb interactions in 2D and 3D, with a proven strictly positive spectral gap for every finite-rank truncation. That gives the first rigorous exponential mixing time for these singular continuous-variable systems, which is genuinely new and not just a re-packaging of existing quantum Markov chain results. They also supply an explicit quantum circuit implementation and an end-to-end complexity bound, which is useful even if the constants are large. The truncation reduction itself is presented cleanly as an independent analytical step with an explicit polynomial-in-N error bound on the free energy difference. That step is what lets them move from the infinite-dimensional problem to something they can actually sample. The writing is direct and the claims are stated with error bounds rather than hand-waving. The main soft spot is exactly the one the stress-test flags: free energy is extensive, so an error that grows like N^2 or higher swamps the quantity you are trying to estimate. The abstract only says “polynomial in the particle number” without giving the degree or prefactors, so it is impossible to tell from the summary whether the bound is actually useful or merely formal. If the full proof shows something like O(N log N) or better with reasonable constants, the reduction works; otherwise the end-to-end guarantee is empty. No other red flags in the outline—no circularity, no hidden fitting, and the spectral-gap claim is independent of the truncation details. This is squarely for people working on quantum algorithms for quantum chemistry and many-body systems who care about rigorous mixing times rather than heuristics. It is worth sending to a serious referee who can check the truncation error degree and the circuit depth numbers, because the mixing-time result is a real technical step even if the overall complexity ends up impractical.

Referee Report

2 major / 1 minor

Summary. The paper develops a quantum algorithm for estimating the free energy and the Gibbs state of interacting quantum Coulomb gases and molecules in d=2,3 at finite temperature. It reduces the infinite-dimensional problem to a finite-rank low-energy truncation of the interaction Hamiltonian, for which an explicit polynomial-in-N error bound on the free energy is claimed. A tailored quantum Gibbs sampling scheme based on quantum Markov semigroups is introduced, with a proof that the generator has a strictly positive spectral gap for every truncation (implying exponential convergence). An explicit quantum circuit implementation is given, together with end-to-end complexity bounds. The work avoids classical approximations such as Born-Oppenheimer.

Significance. If the truncation error is controlled at a sub-extensive scale, the result supplies the first rigorous mixing-time guarantee for Gibbs sampling in a Coulomb-interacting continuous-variable quantum system and a mathematically rigorous route to quantum algorithms for free energy estimation in singular interacting systems. The spectral-gap analysis for the truncated generator is a notable technical strength.

major comments (2)

[Abstract] Abstract (truncation error bound): the free-energy approximation error is stated only as 'polynomial in the particle number' without the explicit degree or prefactors. Because the free energy is extensive (Θ(N)), any bound of degree ≥2 produces an error that grows faster than the target quantity and cannot be removed by the subsequent Gibbs sampling step on the truncated system. This reduction is load-bearing for the end-to-end guarantee; the precise scaling must be stated and shown to be o(N) (or at worst O(N) with a sufficiently small constant) relative to the desired additive precision ε.
[Main analytical result on the generator] Spectral gap result (main analytical theorem): while a strictly positive gap is claimed for every truncation, the explicit dependence of the gap lower bound on the truncation rank, inverse temperature, and particle number must be derived to convert the exponential convergence into a concrete mixing-time bound and thence into the stated circuit complexity. If the gap deteriorates with the truncation parameters, the end-to-end scaling could be affected.

minor comments (1)

[Notation and definitions] The notation for the finite-rank projection and the precise definition of the low-energy truncation operator should be introduced with an equation number in the main text rather than only in the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address each major comment point by point below, indicating the revisions we will make to strengthen the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract (truncation error bound): the free-energy approximation error is stated only as 'polynomial in the particle number' without the explicit degree or prefactors. Because the free energy is extensive (Θ(N)), any bound of degree ≥2 produces an error that grows faster than the target quantity and cannot be removed by the subsequent Gibbs sampling step on the truncated system. This reduction is load-bearing for the end-to-end guarantee; the precise scaling must be stated and shown to be o(N) (or at worst O(N) with a sufficiently small constant) relative to the desired additive precision ε.

Authors: We agree that the abstract would benefit from greater precision on this point. The main text derives an explicit polynomial bound on the free-energy truncation error (including degree and prefactors) and separates this error from the subsequent sampling error on the truncated system. We will revise the abstract to state the explicit scaling and add a short remark confirming that the error is o(N) relative to the extensive free energy, so that it remains compatible with any target additive precision ε. This clarification will be added without altering the technical claims. revision: yes
Referee: [Main analytical result on the generator] Spectral gap result (main analytical theorem): while a strictly positive gap is claimed for every truncation, the explicit dependence of the gap lower bound on the truncation rank, inverse temperature, and particle number must be derived to convert the exponential convergence into a concrete mixing-time bound and thence into the stated circuit complexity. If the gap deteriorates with the truncation parameters, the end-to-end scaling could be affected.

Authors: The main theorem asserts a strictly positive spectral gap for every finite truncation, with the proof deriving a concrete lower bound whose dependence on truncation rank, inverse temperature, and particle number is already present in the appendix. We will revise the theorem statement to display this explicit dependence and update the end-to-end complexity section to convert it directly into a mixing-time bound. This will make the circuit complexity fully transparent and address any possible deterioration of the gap with the truncation parameters. revision: yes

Circularity Check

0 steps flagged

No circularity: independent analytical bounds and spectral-gap proof

full rationale

The derivation proceeds in two distinct analytical steps that do not reduce to each other or to fitted inputs. First, an explicit polynomial-in-N error bound is established between the free energy of the full Coulomb Hamiltonian and its finite-rank low-energy truncation; this is a direct approximation result that reduces the infinite-dimensional problem to a finite-rank perturbation without invoking the subsequent sampling procedure. Second, a quantum Markov semigroup is defined on the truncated system and a strictly positive spectral gap is proved for its generator, yielding exponential convergence to the Gibbs state. Both results are presented as self-contained mathematical contributions that build on standard quantum Markov chain theory rather than on self-citations, self-definitions, or renaming of known empirical patterns. The end-to-end complexity bound follows from composing these independent pieces with an explicit circuit implementation; no equation equates a claimed prediction to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard mathematical assumptions from quantum information and functional analysis, with no free parameters or invented entities introduced; the truncation error bound and spectral gap are derived rather than postulated.

axioms (2)

domain assumption The generator of the quantum Markov semigroup for the truncated system has a strictly positive spectral gap
Invoked as the main analytical result establishing exponential convergence; proven for every truncation in the paper.
domain assumption Finite-rank low-energy truncation approximates the full Hamiltonian with polynomial-in-N error
Key reduction step allowing finite-dimensional treatment; stated with explicit bound in the abstract.

pith-pipeline@v0.9.0 · 5540 in / 1401 out tokens · 89983 ms · 2026-05-10T11:32:10.180571+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

49 extracted references · 9 canonical work pages · 1 internal anchor

[1]

6, 1043–1065

Gernot Akemann and Sung-Soo Byun,The high temperature crossover for general 2d coulomb gases, Journal of Statistical Physics175(2019), no. 6, 1043–1065

2019
[2]

Ryan Babbush, Dominic W Berry, Jarrod R McClean, and Hartmut Neven,Quantum simulation of chemistry with sublinear scaling in basis size, npj Quantum Information5(2019), no. 1, 92

2019
[3]

Simon Becker, Cambyse Rouz´ e, and Robert Salzmann,Quantum Gibbs sampling in infinite dimensions: Generation, mixing times and circuit implementation, arXiv:quant-ph/2604.01192, 2026

work page arXiv 2026
[4]

,Simulating thermal properties of bose-hubbard models on a quantum computer, arXiv:quant- ph/2604.06077, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
[5]

Fran¸ cois Bolley, Djalil Chafa¨ ı, and Joaqu´ ın Fontbona,Dynamics of a planar coulomb gas, The Annals of Applied Probability28(2018), no. 5

2018
[6]

Bourgade and G

P. Bourgade and G. Dubach,The distribution of overlaps between eigenvectors of ginibre matrices, Prob- ability Theory and Related Fields177(2019), no. 1–2, 397–464

2019
[7]

2, 211–232

Eric Canc` es, Fran¸ cois Castella, Philippe Chartier, Erwan Faou, Claude Le Bris, Fr´ ed´ eric Legoll, and Gabriel Turinici,Long-time averaging for integrable hamiltonian dynamics, Numerische Mathematik100 (2005), no. 2, 211–232

2005
[8]

Djalil Chafa¨ ı,Aspects of coulomb gases, arXiv preprint arXiv:2108.10653 (2021)

work page arXiv 2021
[9]

3, 692–714

Djalil Chafa¨ ı and Gr´ egoire Ferr´ e,Simulating Coulomb and log-gases with hybrid Monte Carlo algorithms, Journal of Statistical Physics174(2019), no. 3, 692–714

2019
[10]

1, 181–220

Djalil Chafa¨ ı, Gr´ egoire Ferr´ e, and Gabriel Stoltz,Coulomb gases under constraint: some theoretical and numerical results, SIAM Journal on Mathematical Analysis53(2021), no. 1, 181–220

2021
[11]

219–246, Springer International Publishing, 2020

Djalil Chafa¨ ı and Joseph Lehec,On poincar´ e and logarithmic sobolev inequalities for a class of singular gibbs measures, p. 219–246, Springer International Publishing, 2020

2020
[12]

Chi-Fang Chen, Michael J Kastoryano, Fernando GSL Brand˜ ao, and Andr´ as Gily´ en,Quantum thermal state preparation, arXiv preprint arXiv:2303.18224 (2023)

work page arXiv 2023
[13]

Chi-Fang Chen, Michael J Kastoryano, and Andr´ as Gily´ en,An efficient and exact noncommutative quan- tum gibbs sampler, arXiv preprint arXiv:2311.09207 (2023)

work page arXiv 2023
[14]

Martin Lindsay,Spectral Analysis and Feller Property for Quantum Ornstein-Uhlenbeck Semigroups, Communications in Mathematical Physics210(2000), no

Fabio Cipriani, Franco Fagnola, and J. Martin Lindsay,Spectral Analysis and Feller Property for Quantum Ornstein-Uhlenbeck Semigroups, Communications in Mathematical Physics210(2000), no. 1, 85–105

2000
[15]

Derezi´ nski, V

J. Derezi´ nski, V. Jakˇ si´ c, and C.-A. Pillet,Perturbation theory of w*-dynamics, liouvilleans and kms-states, Reviews in Mathematical Physics15(2003), no. 05, 447–489

2003
[16]

Zhiyan Ding, Bowen Li, and Lin Lin,Efficient quantum gibbs samplers with kubo–martin–schwinger de- tailed balance condition, Communications in Mathematical Physics406(2025), no. 3, 67

2025
[17]

Manh Hong Duong and Hung Dang Nguyen,Asymptotic analysis for the generalized langevin equation with singular potentials, Journal of Nonlinear Science34(2024), no. 4

2024
[18]

Dmitry A Fedorov, Matthew J Otten, Stephen K Gray, and Yuri Alexeev,Ab initio molecular dynamics on quantum computers, The Journal of Chemical Physics154(2021), no. 16

2021
[19]

Jeffrey Galkowski and Maciej Zworski,Localised davies generators for unbounded operators, 2026

2026
[20]

Andr´ as Gily´ en, Chi-Fang Chen, Joao F Doriguello, and Michael J Kastoryano,Quantum generalizations of glauber and metropolis dynamics, arXiv preprint arXiv:2405.20322 (2024)

work page arXiv 2024
[21]

Paul Gondolf, Tim M¨ obus, and Cambyse Rouz´ e,Energy preserving evolutions over bosonic systems, Quantum8(2024), 1551

2024
[22]

Jakob G¨ unther et al.,How to use quantum computers for biomolecular free energies, arXiv preprint (2025)

2025
[23]

Po-Wei Huang, Gregory Boyd, Gian-Luca R Anselmetti, Matthias Degroote, Nikolaj Moll, Raffaele San- tagati, Michael Streif, Benjamin Ries, Daniel Marti-Dafcik, Hamza Jnane, et al.,Fullqubit alchemist: Quantum algorithm for alchemical free energy calculations, arXiv preprint arXiv:2508.16719 (2025)

work page arXiv 2025
[24]

GIBBS SAMPLING FOR COULOMB QUANTUM GASES 47

Shi Jin, Nana Liu, and Yue Yu,Time complexity analysis of quantum algorithms via linear representations for nonlinear ordinary and partial differential equations, Journal of Computational Physics487(2023), 112149. GIBBS SAMPLING FOR COULOMB QUANTUM GASES 47

2023
[25]

4, 043102

Ilon Joseph,Koopman–von neumann approach to quantum simulation of nonlinear classical dynamics, Physical Review Research2(2020), no. 4, 043102

2020
[26]

1, 185–207

Ivan Kassal, James D Whitfield, Alejandro Perdomo-Ortiz, Man-Hong Yung, and Al´ an Aspuru-Guzik, Simulating chemistry using quantum computers, Annual review of physical chemistry62(2011), no. 1, 185–207

2011
[27]

5, 315–318

Bernard O Koopman,Hamiltonian systems and transformation in hilbert space, Proceedings of the Na- tional Academy of Sciences17(1931), no. 5, 315–318

1931
[28]

Qiong-Gui Lin,Anisotropic harmonic oscillator in a static electromagnetic field, Communications in The- oretical Physics38(2002), 667–674, arXiv:quant-ph/0212038v2

work page arXiv 2002
[29]

2, 675–699

Yulong Lu and Jonathan C Mattingly,Geometric ergodicity of langevin dynamics with coulomb interac- tions, Nonlinearity33(2020), no. 2, 675–699

2020
[30]

Malrieu,Logarithmic sobolev inequalities for some nonlinear pde’s, Stochastic Processes and their Applications95(2001), no

F. Malrieu,Logarithmic sobolev inequalities for some nonlinear pde’s, Stochastic Processes and their Applications95(2001), no. 1, 109–132

2001
[31]

Madushanka Manathunga, Andreas W G¨ otz, and Kenneth M Merz Jr,Computer-aided drug design, quantum-mechanical methods for biological problems, Current opinion in structural biology75(2022), 102417

2022
[32]

42–95, Springer Berlin Heidelberg, 1996

Sylvie M´ el´ eard,Asymptotic behaviour of some interacting particle systems; mckean-vlasov and boltzmann models, p. 42–95, Springer Berlin Heidelberg, 1996

1996
[33]

Nielsen and Isaac L

Michael A. Nielsen and Isaac L. Chuang,Quantum computation and quantum information: 10th anniver- sary edition, Cambridge University Press, 2010

2010
[34]

4, 043210

Thomas E O’Brien, Michael Streif, Nicholas C Rubin, Raffaele Santagati, Yuan Su, William J Huggins, Joshua J Goings, Nikolaj Moll, Elica Kyoseva, Matthias Degroote, et al.,Efficient quantum computation of molecular forces and other energy gradients, Physical Review Research4(2022), no. 4, 043210

2022
[35]

26, 260511

Pauline J Ollitrault, Guglielmo Mazzola, and Ivano Tavernelli,Nonadiabatic molecular quantum dynamics with quantum computers, Physical Review Letters125(2020), no. 26, 260511

2020
[36]

Thomas E O’Brien, Bruno Senjean, Ramiro Sagastizabal, Xavier Bonet-Monroig, Alicja Dutkiewicz, Francesco Buda, Leonardo DiCarlo, and Lucas Visscher,Calculating energy derivatives for quantum chem- istry on a quantum computer, npj Quantum Information5(2019), no. 1, 113

2019
[37]

Patrick Rall, Changpeng Wang, and Pawel Wocjan,Thermal state preparation via rounding promises, Quantum7(2023), 1132

2023
[38]

2, 117–130

Benjamin Ries, Karl Normak, R Gregor Weiß, Salom´ e Rieder, Em´ ılia P Barros, Candide Champion, Gerhard K¨ onig, and Sereina Riniker,Relative free-energy calculations for scaffold hopping-type transfor- mations with an automated re-eds sampling procedure, Journal of Computer-Aided Molecular Design36 (2022), no. 2, 117–130

2022
[39]

Mathias Rousset, Gabriel Stoltz, and Tony Lelievre,Free energy computations: a mathematical perspective, World Scientific, 2010

2010
[40]

4, 549–557

Raffaele Santagati, Alan Aspuru-Guzik, Ryan Babbush, Matthias Degroote, Leticia Gonz´ alez, Elica Kyo- seva, Nikolaj Moll, Markus Oppel, Robert M Parrish, Nicholas C Rubin, et al.,Drug design on quantum computers, Nature Physics20(2024), no. 4, 549–557

2024
[41]

Konrad Schm¨ udgen,Unbounded self-adjoint operators on hilbert space, Springer Netherlands, 2012

2012
[42]

1, 010343

Sophia Simon, Raffaele Santagati, Matthias Degroote, Nikolaj Moll, Michael Streif, and Nathan Wiebe, Improved precision scaling for simulating coupled quantum-classical dynamics, PRX Quantum5(2024), no. 1, 010343

2024
[43]

Samuel Slezak, Matteo Scandi, ´Alvaro M Alhambra, Daniel Stilck Fran¸ ca, and Cambyse Rouz´ e, Polynomial-time thermalization and gibbs sampling from system-bath couplings, arXiv preprint arXiv:2601.16154 (2026)

work page arXiv 2026
[44]

1, 013125

Igor O Sokolov, Panagiotis Kl Barkoutsos, Lukas Moeller, Philippe Suchsland, Guglielmo Mazzola, and Ivano Tavernelli,Microcanonical and finite-temperature ab initio molecular dynamics simulations on quan- tum computers, Physical Review Research3(2021), no. 1, 013125. 48 SIMON BECKER, CAMBYSE ROUZ ´E, AND ROBERT SALZMANN

2021
[45]

Mark Steudtner, Sam Morley-Short, William Pol, Sukin Sim, Cristian L Cortes, Matthias Loipersberger, Robert M Parrish, Matthias Degroote, Nikolaj Moll, Raffaele Santagati, et al.,Fault-tolerant quantum computation of molecular observables, Quantum7(2023), 1164

2023
[46]

4, 040332

Yuan Su, Dominic W Berry, Nathan Wiebe, Nicholas Rubin, and Ryan Babbush,Fault-tolerant quantum simulations of chemistry in first quantization, PRX Quantum2(2021), no. 4, 040332

2021
[47]

165–251, Springer Berlin Heidelberg, 1991

Alain-Sol Sznitman,Topics in propagation of chaos, p. 165–251, Springer Berlin Heidelberg, 1991

1991
[48]

Mark E Tuckerman and Glenn J Martyna,Understanding modern molecular dynamics: Techniques and applications, 2000, pp. 159–178

2000
[49]

2, 67–74

Huafeng Xu,The slow but steady rise of binding free energy calculations in drug discovery, Journal of Computer-Aided Molecular Design37(2023), no. 2, 67–74. Email address:simon.becker@unibocconi.it Uni Bocconi, 20136, Milan, Italy Email address:cambyse.rouze@inria.fr Inria, T´el´ecom Paris – LTCI, Institut Polytechnique de Paris, 91120 Palaiseau, France E...

2023