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

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.