Simulating Thermal Properties of Bose-Hubbard Models on a Quantum Computer

Simon Becker, Cambyse Rouzé, Robert Salzmann

Apr 7, 2026arXiv:2604.06077v1

quant-phmath-ph

#59of 3346·Quantum Physics

#59 of 3346 · Quantum Physics

Tournament Score

1571±36

10501750

77%

Win Rate

Wins

Losses

Matches

Rating

8/ 10

Significance8.5

Rigor9

Novelty8.5

Clarity7.5

Abstract

While recent advances have established efficient quantum algorithms for preparing Gibbs states of finite-dimensional systems, comparable complexity results for bosonic and other infinite-dimensional models remain unexplored. We introduce the first general rigorous Gibbs sampling framework for bosonic many-body systems, showing that physically relevant bosonic models admit gapped dissipative generators, enabling efficient preparation of thermal states. Although our results hold for broad classes of models, we illustrate them using Bose-Hubbard Hamiltonians, both within and beyond the mean-field regime. In both cases, we show that the associated dissipative generators maintain a positive spectral gap, thereby implying exponential convergence to the thermal state. Our argument in the multi-mode case is based on a finite-rank reduction of the dissipative dynamics, which allows us to control the generator via compact perturbations and deduce the discreteness of the spectrum and the stability of the gap. We apply our results to provide efficient preparation of the corresponding Gibbs state on qubit hardware, and by that a quantum algorithm to compute thermal properties of the associated model. This provides the first mathematically controlled route to Gibbs sampling in infinite-dimensional systems, with implications for quantum simulation, thermalization, and many-body complexity, where quantum advantages may arise.

AI Impact Assessments

(3 models)

Scientific Impact Assessment

1. Core Contribution

This paper establishes the first mathematically rigorous framework for quantum Gibbs sampling in infinite-dimensional (bosonic) quantum systems. The central achievement is proving that physically relevant bosonic models—specifically Bose-Hubbard Hamiltonians—admit dissipative generators with positive spectral gaps, implying exponential convergence to thermal equilibrium. This is a genuine conceptual advance: prior work on quantum Gibbs sampling was confined to finite-dimensional systems (spin chains, fermionic lattices), and extending these results to bosonic systems with unbounded operators and infinite-dimensional Fock spaces required fundamentally new techniques.

The paper addresses three regimes: (i) the mean-field Bose-Hubbard model (Theorem III.1), (ii) a superfluid-phase regularization (finite-rank perturbation of a quadratic Hamiltonian), and (iii) a Mott-insulator regularization (finite-rank perturbation of a number-diagonal Hamiltonian). For all three, positive spectral gaps are established (Theorem III.3), and end-to-end quantum algorithms with explicit runtime guarantees are provided (Theorems V.1, V.2).

2. Methodological Rigor

The mathematical machinery is substantial and carefully developed. The key technical innovation is a finite-rank reduction strategy: by decomposing interacting bosonic Hamiltonians as exactly solvable references (Gaussian or number-diagonal) plus finite-rank perturbations, the authors control the Lindbladian generator through compact perturbation theory. Specifically:

For Gaussian references (superfluid phase), the perturbation only modifies dressed jump operators on a finite-dimensional block (Lemma B.1-B.3), preserving compact resolvents and hence discrete spectra (Theorem B.4).

For number-diagonal references (Mott-insulator phase), the generator decomposes as an anticommutator with diverging eigenvalues plus a compact remainder (Lemma B.5, Corollary B.7-B.8), again yielding discrete spectrum.

For the mean-field model, a Dirichlet-form perturbation analysis (Lemma B.9) combined with detailed eigenvector perturbation theory (Lemma C.4) establishes gap stability for small superfluid order parameter.

The proofs are thorough, with the appendices providing complete details spanning ~30 pages. The approximation lemmas (Lemma III.2, D.1, D.2) rigorously justify that the regularized models approximate the full Bose-Hubbard Gibbs state with truncation levels scaling as $M' = \Omega(n + \log(1/\varepsilon))$ .

3. Potential Impact

Quantum algorithms for bosonic systems: This opens a new direction for quantum simulation. The end-to-end runtime analysis (Theorem V.1) shows that Gibbs states can be prepared with polynomial qubit overhead and circuit depth scaling as $\tilde{O}(\lambda_2^{-1} \text{poly}(n, \log(1/\varepsilon)))$ , where $\lambda_2$ is the spectral gap. The free energy estimation algorithm (Theorem V.2) provides a concrete thermodynamic application.

Quantum advantage candidates: The paper persuasively argues that bosonic systems may be natural settings for quantum advantage in thermal simulation. Unlike finite-dimensional systems where classical algorithms often match quantum ones, bosonic systems resist classical techniques: SDP relaxations fail, cluster expansions face unboundedness issues, and entanglement can persist at arbitrarily high temperatures. This positions the work at an important frontier.

Mathematical physics: The spectral gap analysis for Lindbladians in infinite dimensions, particularly the compact perturbation framework, contributes tools that should generalize beyond Bose-Hubbard to other continuous-variable models.

4. Timeliness & Relevance

The paper is exceptionally timely. Quantum Gibbs sampling has seen an explosion of activity (2023-2025), with rapid progress on finite-dimensional systems. Simultaneously, a complexity-theoretic framework for bosonic computation is emerging. This work bridges these two directions. The Bose-Hubbard model is experimentally relevant (optical lattices, cold atoms), making the results potentially testable.

5. Strengths & Limitations

Key Strengths:

First-of-its-kind result: no prior rigorous Gibbs sampling guarantees existed for interacting infinite-dimensional systems.

The finite-rank perturbation approach is elegant and broadly applicable.

Complete end-to-end analysis from spectral gap to circuit complexity.

Clear identification of the mathematical obstacles unique to bosonic systems.

Notable Limitations:

Gap scaling with system size is uncontrolled: The spectral gap

\lambda_2

is shown to be positive but its dependence on

n

(number of modes) is not quantified. This is explicitly acknowledged as the key open problem. Without this, the algorithm's practical efficiency cannot be assessed—the runtime could be exponential in

n

if the gap closes.

Regularization requirement: The multi-mode results apply to regularized Bose-Hubbard models (

H_{SF}

H_{MI}

), not directly to the full

H_{BH}

. While Lemma III.2 shows these approximate

H_{BH}

, the gap analysis does not extend to the thermodynamic limit.

Mean-field result limited to small $|\psi|$ : Theorem III.1 applies only for sufficiently small superfluid order parameter, excluding the deep superfluid regime.

No explicit numerical benchmarks: The paper is purely analytical with no numerical demonstrations of convergence rates.

Filter function constraints: The choice of filter function matters critically—some choices provably yield gapless samplers, and the Metropolis-type filter requires working in frequency domain, complicating implementation.

Overall Assessment

This is a foundational theoretical contribution that opens a new chapter in quantum Gibbs sampling by extending the framework to infinite-dimensional systems. The mathematical depth is impressive, and the physical motivation is compelling. The main limitation—lack of quantitative gap bounds scaling with system size—prevents immediate algorithmic impact but defines the critical open problem for the field. The work should stimulate significant follow-up research at the intersection of quantum algorithms, mathematical physics, and many-body bosonic systems.

Rating:8/ 10

Significance 8.5Rigor 9Novelty 8.5Clarity 7.5

Generated Apr 8, 2026

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