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arxiv: 2606.15985 · v1 · pith:BXGP5TQ5new · submitted 2026-06-14 · ⚛️ physics.flu-dyn · quant-ph

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

Fabian Schindler , J\"org Schumacher This is my paper

Pith reviewed 2026-06-27 03:34 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn quant-ph
keywords quantum-enhanced MCMCLagrangian tracer dispersionturbulent boundary layerpair dispersion scalingMarkov chain spectral gaphybrid quantum-classical samplingturbulent acceleration vectors
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The pith

A quantum-enhanced MCMC samples height-dependent turbulent accelerations to generate Lagrangian tracer tracks whose pair dispersion matches classical methods.

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

The paper develops a hybrid quantum-classical Markov chain Monte Carlo method for sampling acceleration vectors from a joint distribution that varies with height in turbulent flows. Proposals come from a parametric quantum circuit in one Metropolis-Hastings step. Synthetic tracer particle tracks are produced, and their pair-dispersion scaling laws agree with both a stochastic Langevin model and standard classical MCMC. The method employs a tempered target distribution and measures an effective height-weighted spectral gap of the transition matrix, which is larger than in the classical case for multivariate distributions with cross-correlations at the highest qubit numbers tested.

Core claim

The QE-MCMC method generates synthetic tracer particle tracks in homogeneous shear flow and turbulent boundary layers. The resulting scaling laws for tracer-particle pair dispersion agree with a stochastic transport model of coupled Langevin equations and with classical MCMC. The effective height-weighted spectral gap between the first and second eigenvalue of the Markov-chain transition matrix significantly exceeds that of classical MCMC when sampling from multivariate distributions with cross-correlations at the highest qubit numbers.

What carries the argument

A parametric quantum circuit that constructs the proposal distribution for the first Metropolis-Hastings substep when sampling the joint target distribution over the three acceleration components that depends on height.

Load-bearing premise

The joint target distribution over the three acceleration components that depends on height can be usefully sampled via a tempered distribution whose proposal is generated by a parametric quantum circuit without introducing systematic biases that would alter the measured pair-dispersion statistics or the reported spectral-gap comparison.

What would settle it

Direct numerical comparison in which the pair-dispersion scaling laws from QE-MCMC tracks deviate from those of the coupled Langevin equations or classical MCMC, or in which the height-weighted spectral gap fails to exceed the classical value at high qubit numbers, would falsify the claims.

Figures

Figures reproduced from arXiv: 2606.15985 by Fabian Schindler, J\"org Schumacher.

Figure 1
Figure 1. Figure 1: FIG. 1. Principal sketch of the workflow of the Quantum-enhanced MCMC method for the synthetic generation of Lagrangian [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Quantum circuit of the distribution [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Comparison of dispersion curves computed using the solution of the stochastic differential equations ( [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Comparison of the spectral gap [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Contours of the streamwise velocity component [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Wall-normal mean profiles of essential turbulent properties plotted in wall units for 64 selected horizontal planes, [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Statistics of the velocity fluctuations in the turbulent channel flow at different heights. Panels (a,b) display data for [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Temporal evolution of two Lagrangian particles within a tracer pair over 10 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Lagrangian particle pair dispersion curves. We compare a classical stochastic model (SDE) with a classical [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Time traces of the three components of the velocity fluctuation vector field, [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Total particle pair dispersion curves for different initial seeding heights, which are indicated in the legend on top of [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Comparison of the effective spectral gap [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Dependence of the effective spectral gap width on [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
read the original 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.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a quantum-enhanced Markov chain Monte Carlo (QE-MCMC) method that uses a parametric quantum circuit to construct proposal distributions for sampling turbulent acceleration vectors from a joint, height-dependent target distribution. This generates synthetic Lagrangian tracer tracks in homogeneous shear flow and a turbulent boundary layer (Re_tau=1000 channel flow). The resulting tracer-pair dispersion scaling laws are reported to agree with those from coupled Langevin equations and from classical MCMC; QE-MCMC employs a tempered target and yields a larger effective height-weighted spectral gap (between the first and second eigenvalues of the transition matrix) than classical MCMC, especially at the highest qubit counts (Nq=6).

Significance. If the tempering step is shown not to bias the sampled statistics, the work would demonstrate a viable hybrid quantum-classical module for generating Lagrangian statistics in wall-bounded turbulence, with a concrete efficiency metric (spectral gap) that scales with problem dimensionality. The explicit comparison to both a stochastic model and classical MCMC provides a falsifiable benchmark that strengthens the applicability claim.

major comments (2)
  1. [Abstract/Methods] Abstract and Methods: QE-MCMC explicitly uses a tempered target distribution while the classical MCMC counterpart does not. No quantitative check is supplied that the tempering parameter (or schedule) leaves the joint acceleration statistics, cross-correlations, and height dependence unchanged; without this, the reported agreement in pair-dispersion exponents could be an artifact of the altered target rather than evidence that the quantum proposal faithfully samples the intended distribution.
  2. [Results] Results: The effective height-weighted spectral gap is computed on the tempered QE-MCMC transition matrix. Because the classical comparison chain is untempered, the claimed advantage at Nq=6 requires an explicit demonstration that the gap difference is not an artifact of the tempering choice or of the height-weighting procedure itself.
minor comments (2)
  1. [Abstract] Abstract: No error bars, circuit-ansatz specifications, or data-exclusion criteria are stated for the dispersion-law comparisons or the spectral-gap values.
  2. [Abstract] Abstract: The phrase 'significantly exceeds' for the spectral-gap advantage would be strengthened by reporting the actual numerical ratios or differences at each Nq.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract/Methods] Abstract and Methods: QE-MCMC explicitly uses a tempered target distribution while the classical MCMC counterpart does not. No quantitative check is supplied that the tempering parameter (or schedule) leaves the joint acceleration statistics, cross-correlations, and height dependence unchanged; without this, the reported agreement in pair-dispersion exponents could be an artifact of the altered target rather than evidence that the quantum proposal faithfully samples the intended distribution.

    Authors: We agree that an explicit verification is required. In the revised manuscript we will add classical MCMC runs that employ the identical tempered target used by QE-MCMC. These runs will be used to compare marginal distributions, cross-correlations and height dependence against the untempered classical results already presented, thereby quantifying any bias introduced by tempering. We will also report the sensitivity of the pair-dispersion exponents to the tempering schedule. This addition will confirm that the observed agreement is not an artifact. revision: yes

  2. Referee: [Results] Results: The effective height-weighted spectral gap is computed on the tempered QE-MCMC transition matrix. Because the classical comparison chain is untempered, the claimed advantage at Nq=6 requires an explicit demonstration that the gap difference is not an artifact of the tempering choice or of the height-weighting procedure itself.

    Authors: We acknowledge the need to isolate the effect of the quantum proposal from tempering. The revision will include the height-weighted spectral gap computed for the classical MCMC chain under the same tempered target as QE-MCMC. We will also present a short parametric study of the gap versus tempering strength to show that the reported advantage at Nq=6 is robust. These additions will remove the potential confounding factor while preserving the main conclusions. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external benchmarks and direct computation

full rationale

The abstract and description present QE-MCMC as a hybrid sampler whose outputs (pair-dispersion scaling laws and height-weighted spectral gap) are compared to an independent stochastic Langevin model and to classical MCMC. The spectral gap is extracted directly from the transition matrix eigenvalues of each sampler; the tempering difference is explicitly noted rather than hidden. No equations, self-citations, or fitted parameters are shown reducing the central claims to their own inputs by construction. The method is evaluated against external references (Langevin model, classical MCMC) that are not derived from the QE-MCMC run itself.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Ledger extracted from abstract only; concrete parameter values and circuit details are absent.

free parameters (2)
  • parameters of the parametric quantum circuit
    Proposal distribution Q is constructed by a parametric quantum circuit whose tunable parameters are not specified.
  • tempering parameter
    Method employs a tempered target distribution whose temperature-like parameter is not quantified.
axioms (2)
  • domain assumption Acceleration vectors are drawn from a joint target distribution that depends on all three components and on height above the wall.
    This distribution is the explicit sampling target for both flow cases.
  • standard math An effective height-weighted spectral gap can be defined from the first two eigenvalues of the Markov-chain transition matrix and used to compare mixing efficiency.
    This quantity is introduced to quantify the reported advantage over classical MCMC.

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discussion (0)

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Reference graph

Works this paper leans on

85 extracted references · 3 canonical work pages · 2 internal anchors

  1. [1]

    Flow configuration MCMC and QE-MCMC methods were first tested in a three-dimensional homogeneous turbulent shear flow, see also ref. [18]. Such a turbulent flow, which evolves far away from walls and boundaries, is commonly described by a decomposition of the velocity field into a streamwise mean flow and fluctuations which are given by u(x, t) =⟨u x⟩(y)e...

  2. [2]

    To assess this quantity, we compute the mean-squared relative displacementD(t) between pairs of Lagrangian particles, initialized with small differences in their initial conditions

    Lagrangian particle pair dispersion A key quantity of interest in the Lagrangian frame- work is the particle dispersion, which characterizes the spreading of particle trajectories due to turbulent mixing. To assess this quantity, we compute the mean-squared relative displacementD(t) between pairs of Lagrangian particles, initialized with small differences...

  3. [3]

    (5) forQ(a ∗|a(n))

    Optimal parameters of the MCMC method To ensure efficient sampling within the MCMC frame- work, a parameter study was conducted to determine the optimal proposal varianceσ prop for the acceleration fluc- tuations, see again Eq. (5) forQ(a ∗|a(n)). The proposal standard deviations are calibrated by matching the tem- poral correlation structure of the Marko...

  4. [4]

    Optimal parameters of the QE-MCMC method The quantum-enhanced MCMC introduces additional parameters that influence the structure and efficiency of the sampling process. In the present setting, QE-MCMC was applied to the distribution of the acceleration fluc- tuations using separate quantum proposals for the joint (a1,a 2) and individual spanwise (a 3) com...

    2048

  5. [5]

    Since the spectral gap governs the asymptotic convergence rate of a Markov chain in the classical sense (see Sec

    Spectral gap analysis To quantitatively assess the potential quantum advan- tage, we compared the spectral gapsδof the Markov matrices of the classical MCMC and quantum-enhanced MCMC algorithms. Since the spectral gap governs the asymptotic convergence rate of a Markov chain in the classical sense (see Sec. I), larger values ofδare typi- cally associated ...

  6. [6]

    Flow configuration The second application case is a turbulent channel flow (TCF), a three-dimensional Navier-Stokes flow between 11 FIG. 4. Comparison of the spectral gapδbetween classical MCMC and QE-MCMC as a function of the number of qubits per dimensionN q for the HSF case. Here, the number of layers isp= 10 in the quantum algorithm. two parallel plat...

  7. [7]

    The accelera- tion statistics will be inferred from the velocity fluctua- tion statistics as discussed in subsection III.A

    Height dependence of velocity fluctuation statistics We proceed similarly to the HSF case. The accelera- tion statistics will be inferred from the velocity fluctua- tion statistics as discussed in subsection III.A. Complex- ity is, however, significantly increased by the dependence of the turbulent fluctuations from the distance to the wall. Figure 6 show...

  8. [8]

    The definitions (40) 14 FIG

    Lagrangian particle pair dispersion Similarly to subsection III.A.2, we analyze the La- grangian particle pair dispersion. The definitions (40) 14 FIG. 8. Temporal evolution of two Lagrangian particles within a tracer pair over 10 5 steps which were generated by the QE-MCMC algorithm. They were initialized atX +(0) = [0,500, L + z /2]T and ˜X+(0) = [0,500...

  9. [9]

    Optimal parameters of the MCMC and QE-MCMC method For the turbulent channel flow case, we employ the same calibration strategies as in the HSF benchmark to determine the sampling parameters of both MCMC and QE-MCMC. Due to the height-dependent flow statistics, the classical MCMC proposal scales must be adapted lo- cally iny +, i.e.σ 12,prop =σ 12,prop(y+)...

    2048

  10. [10]

    Spectral gap analysis Due to the wall-normal inhomogeneity of the turbu- lent channel flow, the acceleration statistics and thus the Markov transition mechanism depend on the instanta- neous wall distance. Consequently, we consider a height- conditioned transition probability Π(y +) and define the local spectral gap asδ y(y+) = 1− |λ 2(Π(y+))|, where λ2 d...

  11. [11]

    Nevertheless, it is interesting to take a closer look at this free parameter in the present case, which does not stand for a real temperature as in the original canonical equilibrium ensemble in applications in condensed matter physics (cf. Eq. (2)). The role ofTcan be understood from two complementary perspectives. From an algorithmic point of view, the ...

  12. [12]

    Table I summarizes the run- times of the SDE, classical MCMC, and QE-MCMC al- gorithms for both caes, HSF and TCF flow

    Runtime comparison To further assess the practical relevance of the pro- posed approach, we compare their computational cost in terms of wall-clock time. Table I summarizes the run- times of the SDE, classical MCMC, and QE-MCMC al- gorithms for both caes, HSF and TCF flow. The results show that, on current classical hardware, the QE-MCMC algorithm is comp...

  13. [13]

    C. M. Bishop,Pattern recognition and machine learning (Springer, New York, 2006)

    2006

  14. [14]

    Metropolis, A

    N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equation of state calcula- tions by fast computing machines, J. Chem. Phys.21, 1087 (1953)

    1953

  15. [15]

    W. K. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika57, 97 (1970)

    1970

  16. [16]

    J. A. Vrugt, C. J. F. ter Braak, C. G. H. Diks, and G. Schoups, Hydrologic data assimilation using particle Markov chain Monte Carlo simulation: Theory, concepts and applications, Adv. Water Resour.51, 457 (2013)

    2013

  17. [17]

    Zuccarello, G

    F. Zuccarello, G. Bilotta, F. Cannav` o, A. Cappello, R. Guardo, and G. Ganci, A Markov Chain Monte Carlo approach for complex lava flow simulations driven by satellite-derived data, Environ. Model. Softw.188, 106426 (2025)

    2025

  18. [18]

    Cerezo, A

    M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Variational quantum algo- rithms, Nat. Rev. Phys.3, 625 (2021)

    2021

  19. [19]

    T. L. Patti, O. Shehab, K. Najafi, and S. F. Yelin, Markov chain Monte Carlo enhanced variational quantum algo- rithms, Quantum Sci. Technol.8, 015019 (2023)

    2023

  20. [20]

    Graham, K

    J. Graham, K. Kanov, X. I. A. Yang, M. K. Lee, N. Malaya, C. C. Lalescu, R. Burns, G. Eyink, A. Szalay, R. D. Moser, and C. Meneveau, A Web Services - acces- sible database of turbulent channel flow and its use for testing a new integral wall model for LES, J. Turb.17, 181 (2016)

    2016

  21. [21]

    Nielsen and I

    M. Nielsen and I. Chuang,Quantum Computation and Quantum Information: 10th Anniversary Edition(Cam- bridge University Press, 2011). 21

    2011

  22. [22]

    Arute, K

    F. Arute, K. Arya, and R. Babbush et al., Quantum supremacy using a programmable superconducting pro- cessor, Nature574, 505 (2019)

    2019

  23. [23]

    D. S. Wild, D. Sels, H. Pichler, C. Zanoci, and M. D. Lukin, Quantum sampling algorithms for near-term de- vices, Phys. Rev. Lett.127, 100504 (2021)

    2021

  24. [24]

    Layden, G

    D. Layden, G. Mazzola, R. V. Mishmash, M. Motta, P. Wocjan, J.-S. Kim, and S. Sheldon, Quantum- enhanced Markov-chain Monte Carlo, Nature619, 282 (2023)

    2023

  25. [25]

    D. A. Levin and Y. Peres,Markov chains and mixing times: Second edition(American Mathematical Society, Providence, RI, 2017)

    2017

  26. [26]

    Nakano, H

    Y. Nakano, H. Hakoshima, K. Mitarai, and K. Fujii, Markov-chain Monte Carlo method enhanced by a quan- tum alternating operator ansatz, Phys. Rev. Res.6, 033105 (2024)

    2024

  27. [27]

    Orfi and D

    A. Orfi and D. Sels, Bounding speedup of quantum- enhanced Markov chain Monte Carlo, Phys. Rev. A110, 052414 (2024)

    2024

  28. [28]

    Nakano, K

    Y. Nakano, K. N. Okada, and K. Fujii, Neural-network- assisted Monte Carlo sampling trained by quantum ap- proximate optimization algorithm, PRX Quantum7, 010338 (2026)

    2026

  29. [29]

    Ferguson and P

    S. Ferguson and P. Wallden, Quantum-enhanced Markov chain Monte Carlo for systems larger than a quantum computer, Phys. Rev. Res.7, 013231 (2025)

    2025

  30. [30]

    Ingelmann, F

    J. Ingelmann, F. Schindler, and J. Schumacher, Quantum-assisted Lagrangian tracer dispersion in tur- bulent shear flow, Phys. Fluids37, 085251 (2025)

    2025

  31. [31]

    J. Kim, P. Moin, and R. D. Moser, Turbulence statistics in fully developed channel flow at low Reynolds number, J. Fluid Mech.177, 133 (1987)

    1987

  32. [32]

    P. K. Yeung, Lagrangian investigations of turbulence, Annu. Rev. Fluid Mech.34, 115 (2002)

    2002

  33. [33]

    Toschi and E

    F. Toschi and E. Bodenschatz, Lagrangian properties of particles in turbulence, Annu. Rev. Fluid Mech.41, 375 (2009)

    2009

  34. [34]

    K. R. Sreenivasan and J. Schumacher, Lagrangian views on turbulent mixing of passive scalars, Phil. Trans. R. Soc. A368, 1561 (2010)

    2010

  35. [35]

    I. M. Mazzitelli, F. Fornarelli, A. S. Lanotte, and P. Oresta, Pair and multi-particle dispersion in numeri- cal simulations of convective boundary layer turbulence, Phys. Fluids26, 055110 (2014)

    2014

  36. [36]

    I. M. Mazzitelli, F. Toschi, and A. S. Lanotte, An accu- rate and efficient Lagrangian sub-grid model, Phys. Flu- ids26, 095101 (2014)

    2014

  37. [37]

    B. L. Sawford, Reynolds number effects in Lagrangian stochastic models of turbulent dispersion, Phys. Fluids A3, 1577 (1991)

    1991

  38. [38]

    Viggiano, J

    B. Viggiano, J. Friedrich, R. Volk, M. Bourgoin, R. Bayo` an Cal, and L. Chevillard, Modelling Lagrangian velocity and acceleration in turbulent flows as infinitely differentiable stochastic processes, J. Fluid Mech.900, A27 (2020)

    2020

  39. [39]

    F. T. M. Nieuwstadt, A large-eddy simulation of a line source in a convective atmospheric boundary layer – I. Dispersion characteristics, Atmos. Environ.26A, 485 (1992)

    1992

  40. [40]

    Dosio and J

    A. Dosio and J. Vil` a-Guerau de Arrelano, Statistics of absolute and relative dispersion in the atmospheric con- vective boundary layer: A large-eddy simulation study, J. Atmos. Sci.63, 1253 (2006)

    2006

  41. [41]

    A. F. Stein, R. R. Draxler, G. D. Rolph, B. J. B. Stunder, M. D. Cohen, and F. Ngan, NOAA’s HYSPLIT atmo- spheric transport and dispersion modeling system, Bull. Am. Meteor. Soc.96, 2059 (2015)

    2059

  42. [42]

    Hoffmann, G

    L. Hoffmann, G. G¨ unther, D. Li, O. Stein, X. Wu, S. Griessbach, Y. Heng, P. Konopka, R. M¨ uller, B. Vogel, and J. S. Wright, From ERA-Interim to ERA5: the con- siderable impact of ECMWF’s next-generation reanaly- sis on Lagrangian transport simulations, Atmos. Chem. Phys.19, 3097 (2019)

    2019

  43. [43]

    Hoffmann, T

    L. Hoffmann, T. R¨ oßler, S. Griessbach, Y. Heng, and O. Stein, Lagrangian transport simulations of volcanic sulfur dioxide emissions: Impact of meteorological data products, J. Geophys. Res. Atmos.121, 4651 (2016)

    2016

  44. [44]

    Lubasch, J

    M. Lubasch, J. Joo, P. Moinier, M. Kiffner, and D. Jaksch, Variational quantum algorithms for nonlinear problems, Phys. Rev. A101, 010301 (2020)

    2020

  45. [45]

    B. N. Todorova and R. de Steijl, Quantum algorithm for the collisionless Boltzmann equation, J. Comp. Phys. 409, 109347 (2020)

    2020

  46. [46]

    Gaitan, Finding flows in a Navier-Stokes fluid through quantum computing, npj Quantum Inf.6, 61 (2021)

    F. Gaitan, Finding flows in a Navier-Stokes fluid through quantum computing, npj Quantum Inf.6, 61 (2021)

    2021

  47. [47]

    Budinski, Quantum algorithm for the advec- tion–diffusion equation simulated with the Lattice Boltz- mann method, Quantum Inf

    L. Budinski, Quantum algorithm for the advec- tion–diffusion equation simulated with the Lattice Boltz- mann method, Quantum Inf. Process.20, 57 (2021)

    2021

  48. [48]

    Pfeffer, F

    P. Pfeffer, F. Heyder, and J. Schumacher, Hybrid quantum-classical reservoir computing of thermal con- vection flow, Phys. Rev. Res.4, 033176 (2022)

    2022

  49. [49]

    Pfeffer, F

    P. Pfeffer, F. Heyder, and J. Schumacher, Reduced-order modeling of two-dimensional turbulent Rayleigh-B´ enard flow by hybrid quantum-classical reservoir computing, Phys. Rev. Res.5, 043242 (2023)

    2023

  50. [50]

    S. S. Bharadwaj and K. R. Sreenivasan, Hybrid quantum algorithms for flow problems, Proc. Natl. Acad. Sci. USA 120, e2311014120 (2023)

    2023

  51. [51]

    Jaksch, P

    D. Jaksch, P. Givi, A. J. Daley, and T. Rung, Variational quantum algorithms for computational fluid dynamics, AIAA J.61, 1885 (2023)

    2023

  52. [52]

    Meng and Y

    Z. Meng and Y. Yang, Quantum computing of fluid dy- namics using the hydrodynamic Schr¨ odinger equation, Phys. Rev. Res.5, 033182 (2023)

    2023

  53. [53]

    Ingelmann, S

    J. Ingelmann, S. S. Bharadwaj, P. Pfeffer, K. R. Sreeni- vasan, and J. Schumacher, Two quantum algorithms for solving the one-dimensional advection–diffusion equa- tion, Comput. Fluids281, 106369 (2024)

    2024

  54. [54]

    Ahmed, F

    O. Ahmed, F. Tennie, and L. Magri, Prediction of chaotic dynamics and extreme events: A recurrence-free quan- tum reservoir computing approach, Phys. Rev. Res.6, 043082 (2024)

    2024

  55. [55]

    A. J. Pool, A. D. Somoza, C. Mc Keever, M. Lubasch, and B. Horstmann, Nonlinear dynamics as a ground-state so- lution on quantum computers, Phys. Rev. Res.6, 033257 (2024)

    2024

  56. [56]

    Brearley and S

    P. Brearley and S. Laizet, Quantum algorithm for solv- ing the advection equation using Hamiltonian simulation, Phys. Rev. A110, 012430 (2024)

    2024

  57. [57]

    Z. Meng, J. Zhong, S. Xu et al., Simulating unsteady flows on a superconducting quantum processor, Com- mun. Phys.7, 349 (2024)

    2024

  58. [58]

    K¨ ocher, H

    N. K¨ ocher, H. Rose, S. S. Bharadwaj, J. Schumacher, and S. Schumacher, Numerical solution of nonlin- ear Schr¨ odinger equation by a hybrid pseudospectral- variational quantum algorithm, Sci. Rep.15, 23478 (2025). 22

    2025

  59. [59]

    S. S. Bharadwaj and K. R. Sreenivasan, Compact quan- tum algorithms for time-dependent differential equations, Phys. Rev. Res.7, 023262 (2025)

    2025

  60. [60]

    P. Over, S. Bengoechea, T. Rung, F. C. L. Scandurra, E. de Villiers, and D. Jaksch, Boundary treatment for variational quantum simulations of partial differential equations on quantum computers, Comput. Fluids288, 106508 (2025)

    2025

  61. [61]

    Pfeffer, P

    P. Pfeffer, P. Brearley, S. Laizet, and J. Schumacher, Spectral quantum algorithm for passive scalar transport in shear flows, Sci. Rep.15, 41172 (2025)

    2025

  62. [62]

    S. S. Bharadwaj and K. R. Sreenivasan, Quantum com- putation of fluid dynamics, Indian Academy of Sciences Conference Series3, 77 (2020)

    2020

  63. [63]

    Succi, W

    S. Succi, W. Itani, K. R. Sreenivasan, and R. Steijl, Quantum computing for fluids: Where do we stand?, Eu- rophys. Lett.144, 10001 (2023)

    2023

  64. [64]

    Tennie, S

    F. Tennie, S. Laizet, S. Lloyd, and L. Magri, Quantum computing for nonlinear differential equations and tur- bulence, Nat. Rev. Phys.7, 1 (2025)

    2025

  65. [65]

    Shen and P

    P. Shen and P. K. Yeung, Fluid particle dispersion in homogeneous turbulent shear flow, Phys. Fluids9, 3472 (1997)

    1997

  66. [66]

    S. B. Pope, Stochastic Lagrangian models of velocity in homogeneous turbulent shear flow, Phys. Fluids14, 1696 (2002)

    2002

  67. [67]

    Schumacher and B

    J. Schumacher and B. Eckhardt, Derivative moments in stationary homogeneous shear turbulence, J. Fluid Mech. 441, 109 (2001)

    2001

  68. [68]

    Schumacher, Relation between shear parameter and Reynolds number in statistically stationary turbulent shear flows, Phys

    J. Schumacher, Relation between shear parameter and Reynolds number in statistically stationary turbulent shear flows, Phys. Fluids16, 3094 (2004)

    2004

  69. [69]

    S. B. Pope,Turbulent Flows(Cambridge University Press, Cambridge, UK, 2000)

    2000

  70. [70]

    J. I. Polanco, I. Vinkovic, N. Stelzenmuller, N. Mordant, and M. Bourgoin, Relative dispersion of particle pairs in turbulent channel flow, Int. J. Heat Fluid Flow71, 231 (2018)

    2018

  71. [71]

    Y. Li, E. Perlman, M. Wan, Y. Yang, C. Meneveau, R. Burns, S. Chen, A. Szalay, and G. Eyink, A public turbulence database cluster and applications to study La- grangian evolution of velocity increments in turbulence, J. Turb.9, N31 (2008)

    2008

  72. [72]

    Schumacher and B

    J. Schumacher and B. Eckhardt, On statistically station- ary homogeneous shear turbulence, Europhys. Lett.52, 627 (2000)

    2000

  73. [73]

    Gualtieri, C

    P. Gualtieri, C. M. Casciola, R. Benzi, G. Amati, and R. Piva, Scaling laws and intermittency in homogeneous shear flow, Phys. Fluids14, 583 (2002)

    2002

  74. [74]

    S. C. C. Bailey and A. J. Smits, Experimental investi- gation of the structure of large- and very-large-scale mo- tions in turbulent pipe flow, J. Fluid Mech.651, 339 (2010)

    2010

  75. [75]

    A. J. Smits and I. Marusic, Wall-bounded turbulence, Phys. Today66, 25 (2013)

    2013

  76. [76]

    D. J. Higham and P. E. Kloeden,An Introduction to the Numerical Simulation of Stochastic Differential Equa- tions(SIAM, Philadelphia, 2021)

    2021

  77. [77]

    A. M. Obukhov, Distribution of energy in the spectrum of turbulent flow, Izv. Akad. Nauk Ser. Geogr. i Geophys. 4-5(1941)

    1941

  78. [78]

    Boffetta and I

    G. Boffetta and I. M. Sokolov, Statistics of two-particle dispersion in two-dimensional turbulence, Phys. Fluids 14, 3224 (2002)

    2002

  79. [79]

    Birnir and L

    B. Birnir and L. Angheluta, Scaling of Lagrangian struc- ture functions, Phys. Rev. Res.7, 023225 (2025)

    2025

  80. [80]

    Schumacher, Lagrangian dispersion and heat transport in convective turbulence, Phys

    J. Schumacher, Lagrangian dispersion and heat transport in convective turbulence, Phys. Rev. Lett.100, 134502 (2008)

    2008

Showing first 80 references.