Quantum Algorithms for Gibbs Expectation of Non-log-concave and Heavy-tailed Distributions

Xinmiao Li, Jin-Peng Liu

Apr 1, 2026arXiv:2604.00656v1

quant-ph

#169of 3346·Quantum Physics

#169 of 3346 · Quantum Physics

Tournament Score

1536±28

10501750

67%

Win Rate

Wins

Losses

Matches

Rating

6.8/ 10

Significance7

Rigor7.5

Novelty6.5

Clarity6.5

Abstract

We establish a systematic framework of unbiased quantum sampling and estimation protocols for the classical Gibbs expectation. This framework generalizes existing approaches to the partition function estimation and has broader applications in various fields. We consider sampling and estimation for a wide class of non-log-concave distributions, particularly heavy-tailed ones, under relaxed assumptions beyond strong convexity, such as dissipativity. We develop an unbiased extension of quantum-accelerated multilevel Monte Carlo (QA-MLMC) to eliminate all biases from discretization and time truncation, together with introducing a change-of-measure approach and the Girsanov theorem via Radon-Nikodym derivatives. As a result, our approach achieves quantum complexity $\widetilde{\mathcal{O}}(ε^{-1})$ within error $ε$ , whereas the classical MLMC requires $\widetilde{\mathcal{O}}(ε^{-2})$ and existing quantum algorithms yield biased estimators under stronger assumptions. Furthermore, our unified framework enables unbiased quantum sampling and estimation for certain heavy-tailed distributions after transformation. We provide several concrete applications of our approach in statistics, machine learning, and finance, towards more practical scenarios of the quantum acceleration of stochastic processes.

AI Impact Assessments

(3 models)

Scientific Impact Assessment

Core Contribution

This paper develops a systematic framework for unbiased quantum sampling and estimation of classical Gibbs expectations $E_\pi[\phi]$ under significantly relaxed assumptions compared to prior work. The main novelties are threefold:

1. Unbiased QA-MLMC: The authors extend quantum-accelerated multilevel Monte Carlo (QA-MLMC) to produce fully unbiased estimators by eliminating both discretization bias and time-truncation bias. The key construction generalizes prior debiasing techniques from the $p = 1$ case to $p \in (0,2)$ , enabling conversion of biased estimators to unbiased ones without increasing asymptotic complexity (Theorem 4).

2. Relaxed assumptions via change of measure: By introducing a "spring coupling" term into the Langevin dynamics and applying the Girsanov theorem through Radon-Nikodym derivatives, the framework extends from requiring strong convexity (one-sided Lipschitz) to merely requiring dissipativity — a substantially weaker condition that accommodates non-log-concave potentials.

3. Heavy-tailed distribution handling: Through a carefully designed transformation map $h: \mathbb{R}^d \to \mathbb{R}^d$ that converts heavy-tailed distributions into light-tailed ones, the framework achieves the same $\tilde{O}(\epsilon^{-1})$ complexity for heavy-tailed distributions (symmetric stable, Student- $t$ ).

Methodological Rigor

The paper is technically thorough, with detailed proofs spanning extensive appendices. The strong error analysis (Appendix A) carefully tracks dependence on all relevant parameters — dimension $d$ , smoothness $L$ , spring coefficient $S$ , dissipativity constants, and step size $h$ . The moment bounds (Lemmas 14-17) are established for general $p \geq 1$ using Itô calculus, Burkholder-Davis-Gundy inequalities, and careful exponential weight arguments.

The multilevel variance analysis correctly identifies the critical condition $\beta \geq 2\gamma$ needed for quantum speedup (versus $\beta \geq \gamma$ classically), which necessitates $L$ -Hessian smoothness to achieve first-order strong convergence rather than the half-order from standard Euler-Maruyama. Without Hessian smoothness, the complexity degrades to $\tilde{O}(\epsilon^{-1.5})$ , which the authors honestly report.

One concern is the oracle model: the paper assumes idealized quantum oracles (gradient oracle $O_{\nabla f}$ , Radon-Nikodym derivative oracle $O_{R}$ , etc.) whose implementation costs on actual quantum hardware are not analyzed. The use of pseudo-random seeds to circumvent the no-cloning theorem for shared Brownian increments is practically reasonable but its rigorous justification could be strengthened.

The dimensional dependence analysis reveals $O (d)$ complexity in the one-sided Lipschitz setting and $O(d^{3/2})$ under dissipativity, under the assumption that initial values and $\|\nabla f(0)\|$ are $O (1)$ . This assumption, while explicitly stated, is strong and not always natural.

Potential Impact

The framework has broad applicability across several domains:

Bayesian inference: The concrete applications to Bayesian logistic regression and mixture modeling demonstrate relevance to machine learning practitioners.

Robust statistics: The correntropy loss example explicitly shows a case where one-sided Lipschitz fails but dissipativity holds — precisely the regime this paper addresses.

Financial mathematics: Heavy-tailed models (Student-

t

) and capped payoff structures are natural in risk management.

Statistical physics: Gibbs expectations with non-convex energy landscapes (oscillatory perturbations, radial nonconvex potentials) are physically motivated.

The paper's Table 1 clearly delineates the improvement over prior work: achieving unbiased $\tilde{O}(\epsilon^{-1})$ estimation under dissipativity versus biased $\tilde{O}(\epsilon^{-1})$ under stronger assumptions or classical $\tilde{O}(\epsilon^{-2})$ .

Timeliness & Relevance

This work addresses a genuine bottleneck in quantum algorithm design for stochastic processes. Most prior quantum sampling algorithms (notably Childs et al., NeurIPS 2022) required strong convexity, excluding important practical distributions. The extension to dissipative and heavy-tailed settings substantially broadens the applicability of quantum MCMC methods. Given the growing interest in both quantum advantage for practical problems and Langevin-based sampling in machine learning, this work is well-timed.

Strengths

Comprehensive framework: The paper systematically addresses multiple settings (one-sided Lipschitz, dissipative, heavy-tailed) with a unified approach, providing clear comparisons between them.

Concrete applications: Seven distinct application examples with verified assumptions add significant practical value.

Careful error tracking: The explicit dependence on all parameters (

d

L

S

\lambda

m

) enables practitioners to assess applicability.

Clean presentation of ideas: Figures 1-3 effectively communicate the different coupling strategies, and Figure 4 concisely summarizes the generalization.

Limitations

Oracle model abstraction: Implementation costs of quantum oracles (

O_{R}

O_{m}

O_{a}

) are ignored. The Radon-Nikodym derivative oracle in particular involves exponentials of accumulated quantities, which could be numerically unstable.

Isotropic assumption for heavy-tailed case: Assumption 1 requiring

f(x) = f(\|x\|)

substantially limits the heavy-tailed framework's applicability to anisotropic distributions common in practice.

No numerical experiments for quantum algorithms: While Figure 6 shows classical simulation results for the spring-coupled sampler, no quantum simulation results are provided.

Dimensional dependence: The

O(d^{3/2})

scaling in the dissipative case may limit practical advantage in high dimensions.

Strong smoothness requirements:

L

-Hessian smoothness is needed for the optimal

\tilde{O}(\epsilon^{-1})

rate; without it, the rate degrades, somewhat limiting the claimed relaxation of assumptions.

Overall Assessment

This is a solid theoretical contribution that meaningfully extends the reach of quantum-accelerated MLMC methods. The combination of unbiased estimation, relaxed structural assumptions, and heavy-tailed handling represents genuine progress. The work is technically dense but well-organized, with the theoretical developments properly supported by detailed proofs and illustrative examples.

Rating:6.8/ 10

Significance 7Rigor 7.5Novelty 6.5Clarity 6.5

Generated Apr 2, 2026

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