pith. sign in

arxiv: 2208.11780 · v1 · submitted 2022-08-24 · 🪐 quant-ph · physics.ao-ph· physics.comp-ph· physics.flu-dyn

Variational Quantum Solutions to the Advection-Diffusion Equation for Applications in Fluid Dynamics

Reuben Demirdjian , Daniel Gunlycke , Carolyn A. Reynolds , James D. Doyle , Sergio Tafur This is my paper

Pith reviewed 2026-05-24 11:29 UTC · model grok-4.3

classification 🪐 quant-ph physics.ao-phphysics.comp-phphysics.flu-dyn
keywords advection-diffusion equationvariational quantum algorithmsfluid dynamicsquantum computingnumerical weather predictionhybrid quantum-classical methodsIBM quantum computers
0
0 comments X

The pith

A hybrid quantum-classical variational method solves the advection-diffusion equation on noisy quantum computers with logarithmic scaling in vector space dimension.

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 approach to solve the advection-diffusion equation for fluid dynamics applications such as numerical weather prediction. This method combines variational algorithms to represent and optimize solutions to the linear operator of the discretized system. It achieves scaling logarithmic in the dimension of the vector space and quadratic in the number of nonzero terms within the linear combination of unitary operators. The authors demonstrate the approach on a small system using IBM quantum computers and report that reliable solutions are obtained even on currently available noisy hardware. This opens a path for quantum methods to address power and computational constraints that limit classical simulations.

Core claim

The hybrid quantum-classical method scales logarithmically with the dimension of the vector space and quadratically with the number of nonzero terms in the linear combination of unitary operators that specifies the linear operator describing the system, allowing reliable solutions of the advection-diffusion equation on noisy quantum computers as shown in a demonstration on IBM devices.

What carries the argument

The hybrid quantum-classical variational algorithm that decomposes the discretized advection-diffusion linear operator into a linear combination of unitary operators and optimizes the solution variationally.

Load-bearing premise

The discretized advection-diffusion linear operator admits an efficient decomposition into a linear combination of unitary operators whose number remains manageable, and variational optimization on noisy hardware converges to a sufficiently accurate solution.

What would settle it

Running the variational procedure on the demonstrated small system and finding that the quantum solution deviates substantially from the known classical solution, or that the number of unitary terms grows too rapidly with system size, would show the claimed reliability and scaling do not hold.

Figures

Figures reproduced from arXiv: 2208.11780 by Carolyn A. Reynolds, Daniel Gunlycke, James D. Doyle, Reuben Demirdjian, Sergio Tafur.

Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
read the original abstract

Constraints in power consumption and computational power limit the skill of operational numerical weather prediction by classical computing methods. Quantum computing could potentially address both of these challenges. Herein, we present one method to perform fluid dynamics calculations that takes advantage of quantum computing. This hybrid quantum-classical method, which combines several algorithms, scales logarithmically with the dimension of the vector space and quadratically with the number of nonzero terms in the linear combination of unitary operators that specifies the linear operator describing the system of interest. As a demonstration, we apply our method to solve the advection-diffusion equation for a small system using IBM quantum computers. We find that reliable solutions of the equation can be obtained on even the noisy quantum computers available today. This and other methods that exploit quantum computers could replace some of our traditional methods in numerical weather prediction as quantum hardware continues to improve.

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

3 major / 2 minor

Summary. The paper presents a hybrid quantum-classical variational method for solving the advection-diffusion equation, claiming it scales logarithmically with the dimension of the vector space and quadratically with the number of nonzero terms in the linear combination of unitary operators (LCU) decomposition of the linear operator. As a demonstration, the method is applied to a small system on IBM quantum computers, with the assertion that reliable solutions can be obtained on current noisy hardware for applications in fluid dynamics and numerical weather prediction.

Significance. If the scaling properties and reliable convergence on noisy hardware hold beyond the small-system case, the approach could provide a pathway for quantum-enhanced fluid dynamics simulations. The manuscript does not include machine-checked proofs, reproducible code, or falsifiable predictions that would strengthen this assessment.

major comments (3)
  1. [Abstract] Abstract: The central claim that 'reliable solutions of the equation can be obtained on even the noisy quantum computers available today' lacks supporting quantitative error metrics, classical baseline comparisons, details on noise mitigation, or verification that obtained solutions satisfy the original PDE; the small-system demo alone does not substantiate this for the claimed applications.
  2. [Abstract] Abstract and scaling discussion: The asserted logarithmic scaling with vector-space dimension and quadratic scaling with LCU term count are presented as properties of the algorithm, but no explicit construction or bound is given showing that the finite-difference discretization of the advection-diffusion operator admits an LCU decomposition whose term count remains poly(log N); without this, the quadratic prefactor undermines the claimed advantage for larger grids.
  3. [Results/Demonstration] Demonstration/results: The variational optimization on noisy hardware is reported to converge for the small system, but no analysis addresses whether convergence occurs without post-selection or error-mitigation steps that would not scale, leaving the weakest assumption (efficient LCU and reliable noisy convergence) unverified beyond the demo.
minor comments (2)
  1. [Methods] Notation for the LCU decomposition and variational ansatz should be defined more explicitly with equation numbers for clarity.
  2. [Figures] Figure captions could include quantitative error values or grid sizes to aid interpretation.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive comments, which help clarify and strengthen the presentation of our work. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that 'reliable solutions of the equation can be obtained on even the noisy quantum computers available today' lacks supporting quantitative error metrics, classical baseline comparisons, details on noise mitigation, or verification that obtained solutions satisfy the original PDE; the small-system demo alone does not substantiate this for the claimed applications.

    Authors: We agree that the abstract claim would be strengthened by additional quantitative support. In the revised manuscript we will add error metrics from the hardware executions, direct comparisons to classical finite-difference solutions for the same small system, explicit verification that the obtained quantum solutions satisfy the discretized PDE within expected truncation error, and a brief description of the standard readout-error mitigation used on the IBM hardware. These elements are already present in the full results but will be highlighted to better substantiate the claim for the demonstrated cases. revision: yes

  2. Referee: [Abstract] Abstract and scaling discussion: The asserted logarithmic scaling with vector-space dimension and quadratic scaling with LCU term count are presented as properties of the algorithm, but no explicit construction or bound is given showing that the finite-difference discretization of the advection-diffusion operator admits an LCU decomposition whose term count remains poly(log N); without this, the quadratic prefactor undermines the claimed advantage for larger grids.

    Authors: The logarithmic dependence on vector-space dimension follows directly from the qubit encoding of the grid (n = log2 N qubits). The manuscript states the quadratic dependence on the number of LCU terms as a general property of the algorithm but does not supply an explicit LCU decomposition or term-count bound for the finite-difference advection-diffusion operator. We will add this construction in the revision, demonstrating that the local finite-difference stencil on a periodic grid yields an LCU with O(log N) terms, preserving the overall scaling advantage. revision: yes

  3. Referee: [Results/Demonstration] Demonstration/results: The variational optimization on noisy hardware is reported to converge for the small system, but no analysis addresses whether convergence occurs without post-selection or error-mitigation steps that would not scale, leaving the weakest assumption (efficient LCU and reliable noisy convergence) unverified beyond the demo.

    Authors: The reported convergence used only the hardware-native error mitigation available at the time; no non-scalable post-selection was applied. In the revised manuscript we will add a short analysis clarifying the mitigation steps employed and confirming that convergence was obtained without additional post-selection. We note that the demonstration remains limited to small systems and that reliable convergence on larger noisy devices is an assumption requiring future verification, consistent with the proof-of-concept nature of the work. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation remains self-contained

full rationale

The abstract and provided text present the hybrid quantum-classical method's scaling (logarithmic in vector dimension, quadratic in unitary terms) as an intrinsic property of the algorithm construction rather than a result derived from or fitted to the small-system demonstration. The empirical finding that reliable solutions are obtained on noisy IBM hardware is reported as an observation from the demo, not as a self-referential definition or a prediction forced by prior fits. No load-bearing self-citations, uniqueness theorems imported from the authors' prior work, or ansatzes smuggled via citation are visible in the text. The central claims do not reduce by construction to the inputs; the derivation chain is independent of the reported results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; the method implicitly relies on standard quantum computing assumptions about efficient operator decomposition and variational convergence, with no free parameters, invented entities, or non-standard axioms explicitly stated.

axioms (1)
  • domain assumption The discretized advection-diffusion operator can be expressed as a linear combination of unitary operators whose count permits quadratic scaling.
    Invoked to support the claimed complexity; location is the scaling statement in the abstract.

pith-pipeline@v0.9.0 · 5696 in / 1284 out tokens · 34155 ms · 2026-05-24T11:29:21.965958+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages · 1 internal anchor

  1. [1]

    Report of the National Science Foundation Blue Ribbon Panel on Simulation-Based Engineering Science,

    J. T. Oden, T. Belytschko, J. Fish, T. J.R. Hughes, C. Johnshon, D. Keyes, A. Laub, L. Petzold, D. Srolovitz, S. Yip and J. Bass, "Report of the National Science Foundation Blue Ribbon Panel on Simulation-Based Engineering Science," 2006

    work page 2006

  2. [2]

    The quiet revolution of numerical weather prediction,

    P. Bauer, A. Thorpe and G. Brunet, "The quiet revolution of numerical weather prediction," Nature, vol. 525, pp. 47-55, 2015

    work page 2015

  3. [3]

    Earth System Modeling Must Become More Energy Efficient,

    R. Loft, "Earth System Modeling Must Become More Energy Efficient," Eos, 28 July 2020

    work page 2020

  4. [4]

    Quantum technologies for climate change: Preliminary assessment,

    C. Berger, A. Di Paolo, T. Forrest, S. Hadfield, N. Sawaya, M. Stęchły and K. Thibault, "Quantum technologies for climate change: Preliminary assessment," arXiv preprint arXiv:2107.05362, 2021

    work page arXiv 2021

  5. [5]

    Quantum computing 40 years later,

    J. Preskill, "Quantum computing 40 years later," arXiv preprint arXiv:2106.10522, 2021. DISTRIBUTION A. Approved for public release: distribution unlimited

    work page arXiv 2021

  6. [6]

    Moore's law: past, present and future,

    R. R. Schaller, "Moore's law: past, present and future," IEEE spectrum, vol. 34, p. 52–59, 1997

    work page 1997

  7. [7]

    Moore’s Law revisited through Intel chip density,

    D. Burg and J. H. Ausubel, "Moore’s Law revisited through Intel chip density," PloS one, vol. 16, p. e0256245, 2021

    work page 2021

  8. [8]

    Can deep learning beat numerical weather prediction?,

    M. G. Schultz, C. Betancourt, B. Gong, F. Kleinert, M. Langguth, L. H. Leufen, A. Mozaffari and S. Stadtler, "Can deep learning beat numerical weather prediction?," Philosophical Transactions of the Royal Society A, vol. 379, p. 20200097, 2021

    work page 2021

  9. [9]

    Can machines learn to predict weather? Using deep learning to predict gridded 500-hPa geopotential height from historical weather data,

    J. A. Weyn, D. R. Durran and R. Caruana, "Can machines learn to predict weather? Using deep learning to predict gridded 500-hPa geopotential height from historical weather data," Journal of Advances in Modeling Earth Systems, vol. 11, p. 2680–2693, 2019

    work page 2019

  10. [10]

    Improving data-driven global weather prediction using deep convolutional neural networks on a cubed sphere,

    J. A. Weyn, D. R. Durran and R. Caruana, "Improving data-driven global weather prediction using deep convolutional neural networks on a cubed sphere," Journal of Advances in Modeling Earth Systems, vol. 12, p. e2020MS002109, 2020

    work page 2020

  11. [11]

    Clay Math Institute,

    "Clay Math Institute," [Online]. Available: http://www.claymath.org/millennium- problems/navier%E2%80%93stokes-equation

    work page

  12. [12]

    Finding flows of a Navier–Stokes fluid through quantum computing,

    F. Gaitan, "Finding flows of a Navier–Stokes fluid through quantum computing," Quantum Information Processing , vol. 6, no. 1, 2020

    work page 2020

  13. [13]

    Solving Burgers' equation with quantum computing,

    F. Oz, R. K. S. S. Vuppala, K. Kara and F. Gaitan, "Solving Burgers' equation with quantum computing," Quantum Information Processing, vol. 21, no. 1, 2022

    work page 2022

  14. [14]

    Solving partial differential equations in quantum computers,

    P. Garcı́a-Molina, J. Rodrı́guez-Mediavilla and J. J. Garcı́a-Ripoll, "Solving partial differential equations in quantum computers," arXiv preprint arXiv:2104.02668, 2021

    work page arXiv 2021

  15. [15]

    Linear embedding of nonlinear dynamical systems and prospects for efficient quantum algorithms,

    A. Engel, G. Smith and S. E. Parker, "Linear embedding of nonlinear dynamical systems and prospects for efficient quantum algorithms," Physics of Plasmas, vol. 28, p. 062305, 2021

    work page 2021

  16. [16]

    Fluid Dynamicists Need to Add Quantum Mechanics into their Toolbox,

    W. Itani, "Fluid Dynamicists Need to Add Quantum Mechanics into their Toolbox," 2021

    work page 2021

  17. [17]

    High-order quantum algorithm for solving linear differential equations,

    D. W. Berry, "High-order quantum algorithm for solving linear differential equations," Journal of Physics A: Mathematical and Theoretical, vol. 47, p. 105301, 2014

    work page 2014

  18. [18]

    Quantum algorithm for linear differential equations with exponentially improved dependence on precision,

    D. W. Berry, A. M. Childs, A. Ostrander and G. Wang, "Quantum algorithm for linear differential equations with exponentially improved dependence on precision," Communications in Mathematical Physics, vol. 356, p. 1057–1081, 2017

    work page 2017

  19. [19]

    Quantum algorithm for solving linear differential equations: Theory and experiment,

    T. Xin, S. Wei, J. Cui, J. Xiao, I. Arrazola, L. Lamata, X. Kong, D. Lu, E. Solano and G. Long, "Quantum algorithm for solving linear differential equations: Theory and experiment," Physical Review A, vol. 101, p. 032307, 2020

    work page 2020

  20. [20]

    Efficient quantum algorithm for dissipative nonlinear differential equations,

    J.-P. Liu, H. Ø. Kolden, H. K. Krovi, N. F. Loureiro, K. Trivisa and A. M. Childs, "Efficient quantum algorithm for dissipative nonlinear differential equations," Proceedings of the National Academy of Sciences, vol. 118, 2021

    work page 2021

  21. [21]

    Practical Quantum Computing: solving the wave equation using a quantum approach,

    A. Suau, G. Staffelbach and H. Calandra, "Practical Quantum Computing: solving the wave equation using a quantum approach," ACM Transactions on Quantum Computing, vol. 2, p. 1–35, 2021. DISTRIBUTION A. Approved for public release: distribution unlimited

    work page 2021

  22. [22]

    Quantum algorithm for nonhomogeneous linear partial differential equations,

    J. M. Arrazola, T. Kalajdzievski, C. Weedbrook and S. Lloyd, "Quantum algorithm for nonhomogeneous linear partial differential equations," Physical Review A, vol. 100, p. 032306, 2019

    work page 2019

  23. [23]

    A quantum algorithm to solve nonlinear differential equations

    S. K. Leyton and T. J. Osborne, "A quantum algorithm to solve nonlinear differential equations," arXiv preprint arXiv:0812.4423, 2008

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  24. [24]

    Quantum spectral methods for differential equations,

    A. M. Childs and J.-P. Liu, "Quantum spectral methods for differential equations," Communications in Mathematical Physics, vol. 375, p. 1427–1457, 2020

    work page 2020

  25. [25]

    High-precision quantum algorithms for partial differential equations,

    A. M. Childs, J.-P. Liu and A. Ostrander, "High-precision quantum algorithms for partial differential equations," arXiv preprint arXiv:2002.07868, 2020

    work page arXiv 2002

  26. [26]

    Quantum algorithm for simulating the wave equation,

    P. C. S. Costa, S. Jordan and A. Ostrander, "Quantum algorithm for simulating the wave equation," Physical Review A, vol. 99, p. 012323, 2019

    work page 2019

  27. [27]

    Quantum algorithm for nonlinear differential equations,

    S. Lloyd, G. De Palma, C. Gokler, B. Kiani, Z.-W. Liu, M. Marvian, F. Tennie and T. Palmer, "Quantum algorithm for nonlinear differential equations," arXiv preprint arXiv:2011.06571, 2020

    work page arXiv 2011

  28. [28]

    Solving nonlinear differential equations with differentiable quantum circuits,

    O. Kyriienko, A. E. Paine and V. E. Elfving, "Solving nonlinear differential equations with differentiable quantum circuits," Physical Review A, vol. 103, p. 052416, 2021

    work page 2021

  29. [29]

    A variational eigenvalue solver on a photonic quantum processor,

    A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik and J. L. O’brien, "A variational eigenvalue solver on a photonic quantum processor," Nature communications, vol. 5, p. 1–7, 2014

    work page 2014

  30. [30]

    IBM’s roadmap for building an open quantum software ecosystem,

    J. Gambetta, I. Faro and K. Wehden, "IBM’s roadmap for building an open quantum software ecosystem," IBM, 4 February 2021

    work page 2021

  31. [31]

    Efficient algorithm for generating Pauli coordinates for an arbitrary linear operator,

    D. Gunlycke, M. C. Palenik, A. R. Emmert and S. A. Fischer, "Efficient algorithm for generating Pauli coordinates for an arbitrary linear operator," arXiv preprint arXiv:2011.08942, 2020

    work page arXiv 2011

  32. [32]

    Variational quantum linear solver,

    C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio and P. J. Coles, "Variational quantum linear solver," arXiv preprint arXiv:1909.05820, 2019

    work page arXiv 1909

  33. [33]

    Pennylane,

    Andrea, "Pennylane," 20 January 2021. [Online]. Available: https://pennylane.ai/qml/demos/tutorial_vqls.html

    work page 2021

  34. [34]

    IBM Quantum,

    "IBM Quantum," 2021. [Online]. Available: https://quantum-computing.ibm.com/

    work page 2021

  35. [35]

    Cost function dependent barren plateaus in shallow parametrized quantum circuits,

    M. Cerezo, A. Sone, T. Volkoff, L. Cincio and P. J. Coles, "Cost function dependent barren plateaus in shallow parametrized quantum circuits," Nature communications, vol. 12, p. 1–12, 2021

    work page 2021

  36. [36]

    Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms,

    S. Sim, P. D. Johnson and A. Aspuru-Guzik, "Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms," Advanced Quantum Technologies, vol. 2, p. 1900070, 2019

    work page 2019

  37. [37]

    An overview of the simultaneous perturbation method for efficient optimization,

    J. C. Spall, "An overview of the simultaneous perturbation method for efficient optimization," Johns Hopkins apl technical digest, vol. 19, p. 482–492, 1998

    work page 1998

  38. [38]

    Implementation of the simultaneous perturbation algorithm for stochastic optimization,

    J. C. Spall, "Implementation of the simultaneous perturbation algorithm for stochastic optimization," IEEE Transactions on aerospace and electronic systems, vol. 34, p. 817–823, 1998. DISTRIBUTION A. Approved for public release: distribution unlimited

    work page 1998

  39. [39]

    Validating quantum computers using randomized model circuits,

    A. W. Cross, L. S. Bishop, S. Sheldon, P. D. Nation and J. M. Gambetta, "Validating quantum computers using randomized model circuits," Physical Review A, vol. 100, p. 032328, 2019

    work page 2019

  40. [40]

    On quantum supremacy,

    E. Pednault, J. Gunnels, D. Maslov and J. Gambetta, "On quantum supremacy," IBM Research Blog, vol. 21, 2019

    work page 2019

  41. [41]

    Establishing Moore's law,

    E. Mollick, "Establishing Moore's law," IEEE Annals of the History of Computing, vol. 28, p. 62–75, 2006

    work page 2006

  42. [42]

    The quiet revolution of numerical weather prediction,

    P. Bauer, A. Thorpe and G. Brunet, "The quiet revolution of numerical weather prediction," Nature, vol. 525, p. 47–55, 2015

    work page 2015

  43. [43]

    Quantum supremacy using a programmable superconducting processor,

    F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buell and others, "Quantum supremacy using a programmable superconducting processor," Nature, vol. 574, p. 505–510, 2019

    work page 2019

  44. [44]

    Quantum algorithm for systems of linear equations with exponentially improved dependence on precision,

    A. M. Childs, R. Kothari and R. D. Somma, "Quantum algorithm for systems of linear equations with exponentially improved dependence on precision," SIAM Journal on Computing, vol. 46, p. 1920–1950, 2017

    work page 1920