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arXiv:2606.15985 · Physical Review Research

Quantum-enhanced Markov chain Monte Carlo sampling to model Lagrangian tracer dispersion in turbulent boundary layer

Fabian Schindler, Jörg Schumacher

physics.flu-dynquant-phSubmitted Jun 14, 2026• Announced Jun 16, 2026• 0 citations
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Abstract

We present a quantum-enhanced Markov chain Monte Carlo (QE-MCMC) method to sample turbulent acceleration vectors from a joint target distribution that depends on all three components and height to model the transport and dispersion of massless Lagrangian tracer particles in two turbulent shear flows. A homogeneous shear flow, characterized by a uniform shear rate S, is considered as the starting point. Secondly, a turbulent boundary layer, which forms in both halves of a plane turbulent channel flow at friction Reynolds number Re_tau = 1000, is considered, where the mean shear rate S(y) varies with distance from the wall y. In this hybrid quantum-classical method, the proposal distribution Q for the first of two Metropolis-Hastings sampling substeps is constructed by a parametric quantum circuit. The algorithm generates synthetic tracer particle tracks. The resulting scaling laws for tracer-particle pair dispersion, a central quantity to probe turbulent mixing from a Lagrangian perspective, agree with a stochastic transport model consisting of coupled Langevin equations and with the classical MCMC counterpart. Differently from the classical sampling method, QE-MCMC uses a tempered target distribution. Due to the height dependence of the tracer dynamics in turbulent channel flow, an effective height-weighted spectral gap between the first and second eigenvalue of the Markov-chain transition matrix is introduced. The latter is found to significantly exceed the one of classical MCMC when sampling from a multivariate distribution with cross-correlations at the highest qubit numbers and thus resolutions. Consequently, our results support the applicability of this one-shot algorithm as a generative Lagrangian quantum-computing module, possibly embedded in a complex fluid-flow problem. Our module is found to work reliably for a relatively small number of qubits per spatial dimension of Nq <= 6.

DOI
10.1103/5qcg-fhk1

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We present a quantum-enhanced Markov chain Monte Carlo (QE-MCMC) method to sample turbulent acceleration vectors from a joint target distribution that depends on all three components and height to model the transport and dispersion of massless Lagrangian tracer particles in two turbulent shear flows. A homogeneous shear flow, characterized by a uniform shear rate S, is considered as the starting point. Secondly, a turbulent boundary layer, which forms in both halves of a plane turbulent channel flow at friction Reynolds number Re_tau = 1000, is considered, where the mean shear rate S(y) varies with distance from the wall y. In this hybrid quantum-classical method, the proposal distribution Q for the first of two Metropolis-Hastings sampling substeps is constructed by a parametric quantum circuit. The algorithm generates synthetic tracer particle tracks. The resulting scaling laws for tracer-particle pair dispersion, a central quantity to probe turbulent mixing from a Lagrangian perspective, agree with a stochastic transport model consisting of coupled Langevin equations and with the classical MCMC counterpart. Differently from the classical sampling method, QE-MCMC uses a tempered target distribution. Due to the height dependence of the tracer dynamics in turbulent channel flow, an effective height-weighted spectral gap between the first and second eigenvalue of the Markov-chain transition matrix is introduced. The latter is found to significantly exceed the one of classical MCMC when sampling from a multivariate distribution with cross-correlations at the highest qubit numbers and thus resolutions. Consequently, our results support the applicability of this one-shot algorithm as a generative Lagrangian quantum-computing module, possibly embedded in a complex fluid-flow problem. Our module is found to work reliably for a relatively small number of qubits per spatial dimension of Nq <= 6.
We present a quantum-assisted generative algorithm for synthetic tracks of Lagrangian tracer particles in a simple turbulent shear flow. The parallelism, a result of the underlying tensor product structure of an n-qubit quantum register, and the sampling properties of quantum algorithms are used to build and optimize a parametric quantum circuit, which generates a quantum state that corresponds to the joint probability density function of the classical turbulent velocity components, p(u1′,u2′,u3′). Velocity samples are drawn by one-shot measurements on the quantum circuit. The hybrid quantum-classical algorithm is validated with two classical methods, a standard stochastic Lagrangian model and a classical sampling scheme in the form of a Markov-chain Monte Carlo approach. In detail, we consider a homogeneous turbulent shear flow with a constant shear rate S as a proof of concept for which the velocity fluctuations are Gaussian. The generation of the joint probability density function is also tested on a real quantum device, the 20-qubit IQM Resonance quantum computing platform for cases of up to 10 qubits. Our study paves the way to applications of Lagrangian small-scale parameterizations of turbulent transport in complex turbulent flows with quantum computers. A quantum advantage is possible by the efficient reconstruction of high-dimensional probability density functions, which can be sampled efficiently.
Quantum computation is expected to accelerate certain computational tasks over classical counterparts. Its most primitive advantage is its ability to sample from classically intractable probability distributions. A promising approach to make use of this fact is the so-called quantum-enhanced Markov-chain Monte Carlo (qe-MCMC) method [D. Layden , ], which uses outputs from quantum circuits as the proposal distributions. In this paper, we propose the use of a quantum alternating operator ansatz (QAOA) for qe-MCMC and provide a strategy to optimize its parameters to improve convergence speed while keeping its depth shallow. The proposed QAOA-type circuit is designed to satisfy the specific constraint which qe-MCMC requires with arbitrary parameters. Through our extensive numerical analysis, we find a correlation in a certain parameter range between an experimentally measurable value, acceptance rate of MCMC, and the spectral gap of the MCMC transition matrix, which determines the convergence speed. This allows us to optimize the parameter in the QAOA circuit and achieve quadratic speedup in convergence. Since MCMC is used in various areas such as statistical physics and machine learning, this paper represents an important step toward realizing practical quantum advantage with currently available quantum computers through qe-MCMC. Published by the American Physical Society 2024
In this study, we propose quantum annealing-enhanced Markov Chain Monte Carlo (QAEMCMC), where QA is integrated into the MCMC subroutine. QA efficiently explores low-energy configurations and overcomes local minima, enabling the generation of proposal states with a high acceptance probability. We benchmark QAEMCMC for the Sherrington-Kirkpatrick model and demonstrate its superior performance over the classical MCMC method. Our results reveal larger spectral gaps, faster convergence of energy observables, and reduced total variation distance between the empirical and target distributions. QAEMCMC accelerates MCMC and provides an efficient method for complex systems, paving the way for scalable quantum-assisted sampling strategies.
Probability density function (PDF) methods have been very useful in describing many physical aspects of turbulent mixing. In applications of these methods, modeled PDF transport equations are commonly simulated via classical Monte Carlo techniques, which provide estimates of moments of the PDF at arbitrary accuracy. In this work, recently developed techniques in quantum computing and quantum enhanced measurements (quantum metrology) are used to construct a quantum algorithm that accelerates the computation of such estimates. This quantum algorithm provides a quadratic speedup over classical Monte Carlo methods in terms of the number of repetitions needed to achieve the desired precision. This paper illustrates the power of this algorithm by considering a binary scalar mixing process modeled by means of the coalescence/dispersion (C/D) closure. The equation is first simulated using classical Monte Carlo methods, where error estimates for the computation of central moments are provided. Then the quantum algorithm for this problem is simulated by sampling from the same probability distribution as that of the output of a quantum computer, and it is shown that significantly fewer resources are required to achieve the same precision. The results demonstrate potential applications of future quantum computers for simulation of turbulent mixing, and large classes of related problems.
Detailed balance underlies conventional Markov-chain Monte Carlo (MCMC) algorithms. Yet in classical systems, breaking detailed balance generates irreversible probability currents and can accelerate sampling. Whether irreversibility can similarly enhance quantum MCMC remains an intriguing question. Here we show that irreversibility provides a new route to improving the recent quantum-enhanced MCMC (QEMC), which combines quantum proposals with classical acceptance. By introducing state-dependent proposals that break detailed balance while preserving the target stationary distribution, we develop an irreversible quantum-enhanced Monte Carlo (IQEMC). Guided by Landau-Zener transitions, IQEMC promotes large energy descents from high-energy states while maintaining stable transitions near low-energy states. On spin-glass benchmarks, IQEMC outperforms QEMC without increasing computational complexity and, unlike the annealing baseline, exhibits a spectral gap that increases with system size and annealing speed. These results establish irreversibility as a physically grounded mechanism for enhancing quantum MCMC.
One of the open challenges in quantum computing is to find meaningful and practical methods to leverage quantum computation to accelerate classical machine-learning workflows. A ubiquitous problem in machine-learning workflows is sampling from probability distributions that we only have access to via their log probability. To this end, we extend the well-known Hamiltonian Monte Carlo (HMC) method for Markov chain Monte Carlo (MCMC) sampling to leverage quantum computation in a hybrid manner as a proposal function. Our new algorithm, Quantum Dynamical Hamiltonian Monte Carlo (QD-HMC), replaces the classical symplectic integration proposal step with simulations of quantum-coherent continuous-space dynamics on digital or analog quantum computers. We show that QD-HMC maintains key characteristics of HMC, such as maintaining the detailed balanced condition with momentum inversion, while also having the potential for polynomial speedups over its classical counterpart in certain scenarios. As sampling is a core subroutine in many forms of probabilistic inference, and MCMC in continuously parametrized spaces covers a large class of potential applications, this work widens the areas of applicability of quantum devices. Published by the American Physical Society 2024