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arxiv: 2510.14099 · v2 · submitted 2025-10-15 · 🪐 quant-ph

A review of quantum machine learning and quantum-inspired applied methods to computational fluid dynamics

Cesar A. Amaral , Vin\'icius L. Oliveira , Juan P. L. C. Salazar , Eduardo I. Duzzioni This is my paper

Pith reviewed 2026-05-18 06:39 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum machine learningtensor networkscomputational fluid dynamicsvariational quantum algorithmsquantum neural networkshybrid quantum-classical methodsNISQ erapartial differential equations
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The pith

Quantum-inspired tensor networks already cut memory and runtime for fluid simulations by orders of magnitude, while true quantum CFD awaits better hardware.

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

This review surveys quantum computing, quantum machine learning, and tensor network techniques applied to computational fluid dynamics. It shows that variational quantum algorithms can act as hybrid solvers for the governing partial differential equations, with quantum nonlinear processing units handling the nonlinear terms. Quantum neural networks and physics-informed versions demonstrate gains in parameter count and accuracy on selected benchmarks. Tensor networks, drawn from quantum many-body methods, compress high-dimensional flow data to achieve large savings in memory and computation time. The authors conclude that these quantum-inspired methods already deliver practical value for CFD, whereas full quantum approaches stay out of reach on current noisy intermediate-scale devices.

Core claim

Quantum CFD remains out of reach in the NISQ era, but quantum-inspired tensor networks already show practical benefits, with hybrid approaches offering the most promising near-term strategy.

What carries the argument

Tensor networks and variational quantum algorithms adapted as solvers for PDEs in CFD, with quantum nonlinear processing units to manage nonlinearities in the fluid equations.

If this is right

  • Hybrid quantum-classical solvers can incorporate nonlinear fluid terms through quantum processing units without requiring fully quantum hardware.
  • Quantum neural networks solve selected CFD problems with fewer trainable parameters while preserving or improving accuracy.
  • Tensor network compression scales to high-dimensional flow fields that overwhelm conventional discretizations.
  • These methods may extend the range of simulatable fluid regimes without proportional growth in computational resources.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If tensor networks retain their reported efficiency on fully developed turbulence, they could become standard tools in industrial CFD codes.
  • Direct integration of the reviewed quantum machine learning layers into existing classical CFD packages would create new hybrid workflows.
  • Progress on quantum hardware could move the variational algorithms from benchmarks to real-time flow control applications.

Load-bearing premise

The memory and runtime reductions reported in the surveyed studies hold for general high-dimensional or turbulent CFD problems rather than only for limited benchmarks.

What would settle it

A head-to-head comparison of a tensor-network CFD solver against a standard finite-volume or finite-element code on a three-dimensional turbulent flow at high Reynolds number, verifying whether solution accuracy is maintained at substantially lower memory and runtime cost.

read the original abstract

Computational Fluid Dynamics (CFD) is central to science and engineering, but faces severe scalability challenges, especially in high-dimensional, multiscale, and turbulent regimes. Traditional numerical methods often become prohibitively expensive under these conditions. Quantum computing and quantum-inspired methods have been investigated as promising alternatives. This review surveys advances at the intersection of quantum computing, quantum algorithms, machine learning, and tensor network techniques for CFD. We discuss the use of Variational Quantum Algorithms as hybrid quantum-classical solvers for PDEs, emphasizing their ability to incorporate nonlinearities through Quantum Nonlinear Processing Units. We further review Quantum Neural Networks and Quantum Physics-Informed Neural Networks, which extend classical machine learning frameworks to quantum hardware and have shown advantages in parameter efficiency and solution accuracy for certain CFD benchmarks. Beyond quantum computing, we examine tensor network methods, originally developed for quantum many-body systems and now adapted to CFD as efficient high-dimensional compression and solver tools. Reported studies include several orders of magnitude reductions in memory and runtime while preserving accuracy. Together, these approaches highlight quantum and quantum-inspired strategies that may enable more efficient CFD solvers. This review closes with perspectives: quantum CFD remains out of reach in the NISQ era, but quantum-inspired tensor networks already show practical benefits, with hybrid approaches offering the most promising near-term strategy.

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. This review surveys quantum computing, quantum machine learning, and quantum-inspired tensor network methods applied to computational fluid dynamics (CFD). It covers variational quantum algorithms for PDEs (including quantum nonlinear processing units), quantum neural networks and physics-informed variants, and tensor network techniques adapted from quantum many-body physics for high-dimensional compression and solving. The manuscript highlights scalability challenges in high-dimensional, multiscale, and turbulent CFD regimes and concludes that quantum CFD remains out of reach on NISQ hardware while quantum-inspired tensor networks already deliver practical benefits and hybrid approaches are the most promising near-term path.

Significance. If the surveyed studies are representative, the review offers a timely synthesis of an emerging interdisciplinary area that could guide researchers toward efficient alternatives to classical CFD solvers. Credit is due for compiling literature on memory and runtime reductions via tensor networks and for framing hybrid quantum-classical strategies as a concrete near-term option. The absence of original derivations or new benchmarks means significance rests on the accuracy and balance of the literature summary rather than novel claims.

major comments (2)
  1. [Abstract / Perspectives] Abstract and perspectives section: the claim that 'quantum-inspired tensor networks already show practical benefits' and constitute 'the most promising near-term strategy' is load-bearing for the manuscript's closing recommendation. The review must explicitly state the dimensionality, flow regime (laminar vs. turbulent), and problem size of each cited tensor-network CFD study; without this, it is unclear whether the reported orders-of-magnitude memory/runtime gains extend to the high-dimensional, multiscale, or turbulent regimes emphasized as the motivating challenge in the introduction.
  2. [Tensor network methods for CFD] Section surveying tensor-network CFD applications: the manuscript should include a table or dedicated paragraph that maps each referenced work to its benchmark characteristics (e.g., Reynolds number, grid size, dimensionality). If most examples remain confined to low-dimensional laminar or simplified test cases, the generalization to 'practical benefits' for general CFD problems requires qualification or additional caveats.
minor comments (2)
  1. Ensure consistent notation for quantum nonlinear processing units and quantum physics-informed neural networks across sections; a short glossary or acronym table would improve readability.
  2. Add a brief discussion of the hardware requirements or classical simulation overhead for the tensor-network methods to help readers assess near-term practicality.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the detailed suggestions that will improve the precision and balance of our review. We address the two major comments below and will revise the manuscript to incorporate the requested clarifications on the tensor-network studies.

read point-by-point responses

  1. Referee: [Abstract / Perspectives] Abstract and perspectives section: the claim that 'quantum-inspired tensor networks already show practical benefits' and constitute 'the most promising near-term strategy' is load-bearing for the manuscript's closing recommendation. The review must explicitly state the dimensionality, flow regime (laminar vs. turbulent), and problem size of each cited tensor-network CFD study; without this, it is unclear whether the reported orders-of-magnitude memory/runtime gains extend to the high-dimensional, multiscale, or turbulent regimes emphasized as the motivating challenge in the introduction.

    Authors: We agree that explicit context on the cited studies is necessary to support the claims. In the revised manuscript we will expand the perspectives section with a concise summary of dimensionality, flow regime, and problem size for each key tensor-network CFD reference, together with a brief discussion of how far the reported gains have been demonstrated beyond low-dimensional laminar cases. revision: yes

  2. Referee: [Tensor network methods for CFD] Section surveying tensor-network CFD applications: the manuscript should include a table or dedicated paragraph that maps each referenced work to its benchmark characteristics (e.g., Reynolds number, grid size, dimensionality). If most examples remain confined to low-dimensional laminar or simplified test cases, the generalization to 'practical benefits' for general CFD problems requires qualification or additional caveats.

    Authors: We accept this recommendation. The revised version will contain a new table in the tensor-network section that lists, for every referenced work, the reported Reynolds number, grid size, dimensionality, and flow regime. We will also add a short paragraph noting the current predominance of simplified test cases and the consequent need for caution when extrapolating the observed memory and runtime reductions to fully turbulent, high-dimensional CFD. revision: yes

Circularity Check

0 steps flagged

No circularity: survey paper with no original derivations or self-referential predictions

full rationale

This is a review article that surveys external literature on quantum algorithms, quantum machine learning, and tensor networks applied to CFD. It contains no original equations, derivations, fitted parameters, or predictions generated by the authors themselves. All reported performance claims (e.g., orders-of-magnitude memory/runtime reductions) are explicitly attributed to the cited studies rather than derived or fitted within the paper. The central perspective—that quantum CFD is out of reach on NISQ hardware while tensor networks offer near-term benefits—is framed as a synthesis of existing work, not a self-contained derivation that reduces to the authors' own inputs. No self-citation chains, ansatzes, or uniqueness theorems are invoked to support load-bearing claims. The paper is therefore self-contained against external benchmarks and carries no circularity burden.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a review paper; the authors introduce no new free parameters, axioms, or invented entities. The content rests on standard background assumptions from quantum computing and CFD that are external to the manuscript.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Encoding strategies for quantum enhanced fluid simulations: opportunities and challenges

    quant-ph 2026-04 unverdicted novelty 3.0

    Encoding strategies for quantum fluid simulations trade off compactness against practicality in state preparation, measurement, boundary conditions, and nonlinear operations, with no single approach being universally optimal.

Reference graph

Works this paper leans on

119 extracted references · 119 canonical work pages · cited by 1 Pith paper · 12 internal anchors

  1. [1]

    Tellus7(1), 22–26 (1955) https://doi.org/10.3402/tellusa.v7i1.8796

    Charney, J.G.: The use of the primitive equations of motion in numerical prediction. Tellus7(1), 22–26 (1955) https://doi.org/10.3402/tellusa.v7i1.8796

    work page doi:10.3402/tellusa.v7i1.8796 1955

  2. [2]

    NASA Reference Publication, Washington, DC (1976)

    Lomax, H., Pulliam, T.H., Zingg, D.W.: Fundamentals of Computational Fluid Dynamics. NASA Reference Publication, Washington, DC (1976)

    work page 1976

  3. [3]

    In: Bhattacharyya, S

    Bhattacharyya, S., Abraham, J.P., Cheng, L., Gorman, J.: Introductory chap- ter: A brief history of and introduction to computational fluid dynamics. In: Bhattacharyya, S. (ed.) Applications of Computational Fluid Dynamics Simu- lation and Modeling. IntechOpen, London (2021). Chap. 1. https://doi.org/10. 5772/intechopen.97235 .https://doi.org/10.5772/inte...

    work page doi:10.5772/intechopen.97235 2021

  4. [4]

    Springer, Berlin, Heidelberg (2012)

    Ferziger, J.H., Peri´ c, M.: Computational Methods for Fluid Dynamics. Springer, Berlin, Heidelberg (2012)

    work page 2012

  5. [5]

    Bioengi- neering11(11), 1168 (2024) https://doi.org/10.3390/bioengineering11111168

    Taebi, A.: Computational fluid dynamics in medicine and biology. Bioengi- neering11(11), 1168 (2024) https://doi.org/10.3390/bioengineering11111168 . Editorial, Special Issue on CFD in medicine and biology

    work page doi:10.3390/bioengineering11111168 2024

  6. [6]

    McGraw-Hill, New York (1995)

    Anderson, J.D.: Computational Fluid Dynamics: The Basics with Applications. McGraw-Hill, New York (1995)

    work page 1995

  7. [7]

    Cam- bridge University Press, Cambridge, UK (2002)

    LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cam- bridge University Press, Cambridge, UK (2002). https://doi.org/10.1017/ CBO9780511791253

    work page 2002

  8. [8]

    Stone, J.M., Norman, M.L.: ZEUS-2D: A Radiation Magnetohydrodynamics Code for Astrophysical Flows in Two Space Dimensions. I. The Hydrody- namic Algorithms and Tests. Astrophysical Journal Supplement80, 753 (1992) https://doi.org/10.1086/191680

    work page doi:10.1086/191680 1992

  9. [9]

    The Astro- physical Journal988(1), 56 (2025) https://doi.org/10.3847/1538-4357/addbe2 arXiv:2505.09231 [astro-ph.SR]

    Nishio, E., Tomida, K., Kudoh, Y., Kimura, S.S.: Formation and Early Evolution of Protoplanetary Disks under Nonuniform Cosmic-Ray Ionization. The Astro- physical Journal988(1), 56 (2025) https://doi.org/10.3847/1538-4357/addbe2 arXiv:2505.09231 [astro-ph.SR]

    work page doi:10.3847/1538-4357/addbe2 2025

  10. [10]

    Cambridge University Press, Cambridge, UK (1967)

    Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge, UK (1967)

    work page 1967

  11. [11]

    Pope, S.B.: Turbulent Flows, p. 771. Cambridge University Press, Cambridge; New York (2000)

    work page 2000

  12. [12]

    Elsevier, Oxford, UK (2007)

    Hirsch, C.: Numerical Computation of Internal and External Flows, 2nd edn. Elsevier, Oxford, UK (2007)

    work page 2007

  13. [13]

    Annual Review of Fluid Mechanics41, 37 181–202 (2009) https://doi.org/10.1146/annurev.fluid.010908.165130

    Spalart, P.R.: Detached-eddy simulation. Annual Review of Fluid Mechanics41, 37 181–202 (2009) https://doi.org/10.1146/annurev.fluid.010908.165130

    work page doi:10.1146/annurev.fluid.010908.165130 2009

  14. [14]

    Nature Computational Science2(1), 30–37 (2022) https://doi.org/ 10.1038/s43588-021-00181-1

    Gourianov, N., Lubasch, M., Dolgov, S., Berg, Q.Y.v.d., Babaee, H., Givi, P., Kiffner, M., Jaksch, D.: A Quantum Inspired Approach to Exploit Turbulence Structures. Nature Computational Science2(1), 30–37 (2022) https://doi.org/ 10.1038/s43588-021-00181-1 . arXiv:2106.05782 [physics]. Accessed 2025-07-30

    work page doi:10.1038/s43588-021-00181-1 2022

  15. [15]

    Physical Review A93(3), 032324 (2016)

    Montanaro, A., Pallister, S.: Quantum algorithms and the finite element method. Physical Review A93(3), 032324 (2016)

    work page 2016

  16. [16]

    Physical Review A104(3), 032426 (2021)

    Pollachini, G.G., Salazar, J.P., G´ oes, C.B., Maciel, T.O., Duzzioni, E.I.: Hybrid classical-quantum approach to solve the heat equation using quantum annealers. Physical Review A104(3), 032426 (2021)

    work page 2021

  17. [17]

    Physical review letters103(15), 150502 (2009)

    Harrow, A.W., Hassidim, A., Lloyd, S.: Quantum algorithm for linear systems of equations. Physical review letters103(15), 150502 (2009)

    work page 2009

  18. [18]

    Physical review letters110(25), 250504 (2013)

    Clader, B.D., Jacobs, B.C., Sprouse, C.R.: Preconditioned quantum linear system algorithm. Physical review letters110(25), 250504 (2013)

    work page 2013

  19. [19]

    New Journal of Physics15(1), 013021 (2013)

    Cao, Y., Papageorgiou, A., Petras, I., Traub, J., Kais, S.: Quantum algorithm and circuit design solving the poisson equation. New Journal of Physics15(1), 013021 (2013)

    work page 2013

  20. [20]

    Physical Review A99(1), 012323 (2019)

    Costa, P.C., Jordan, S., Ostrander, A.: Quantum algorithm for simulating the wave equation. Physical Review A99(1), 012323 (2019)

    work page 2019

  21. [21]

    classical algorithms for solving the heat equation

    Linden, N., Montanaro, A., Shao, C.: Quantum vs. classical algorithms for solving the heat equation. Communications in Mathematical Physics395(2), 601–641 (2022)

    work page 2022

  22. [22]

    Quantum Information Processing16(3), 1–65 (2017)

    Scherer, A., Valiron, B., Mau, S.-C., Alexander, S., Berg, E., Chapuran, T.E.: Concrete resource analysis of the quantum linear-system algorithm used to compute the electromagnetic scattering cross section of a 2d target. Quantum Information Processing16(3), 1–65 (2017)

    work page 2017

  23. [24]

    https://arxiv.org/abs/2409.03241

    Bosco, F.S.D., S, D., Lineswala, R., Chopra, A.: Demonstration of Scalability and Accuracy of Variational Quantum Linear Solver for Computational Fluid Dynamics (2024). https://arxiv.org/abs/2409.03241

    work page arXiv 2024

  24. [25]

    Variational Quantum Algorithms,

    Cerezo, M., Arrasmith, A., Babbush, R., Benjamin, S.C., Endo, S., Fujii, K., McClean, J.R., Mitarai, K., Yuan, X., Cincio, L., Coles, P.J.: Variational quan- tum algorithms. Nature Reviews Physics3(9), 625–644 (2021) https://doi.org/ 38 10.1038/s42254-021-00348-9 . Publisher: Nature Publishing Group. Accessed 2025-07-30

    work page doi:10.1038/s42254-021-00348-9 2021

  25. [26]

    Schuld, A

    Schuld, M., Bocharov, A., Svore, K., Wiebe, N.: Circuit-centric quantum classifiers. Physical Review A101(3), 032308 (2020) https://doi.org/10.1103/ PhysRevA.101.032308 . arXiv:1804.00633 [quant-ph]. Accessed 2025-08-31

    work page arXiv 2020

  26. [27]

    [Online]

    Biamonte, J., Wittek, P., Pancotti, N., Rebentrost, P., Wiebe, N., Lloyd, S.: Quantum machine learning. Nature549(7671), 195–202 (2017) https://doi.org/ 10.1038/nature23474 . Publisher: Nature Publishing Group. Accessed 2025-09- 04

    work page doi:10.1038/nature23474 2017

  27. [28]

    Du, Y., Wang, X., Guo, N., Yu, Z., Qian, Y., Zhang, K., Hsieh, M.-H., Reben- trost, P., Tao, D.: Quantum Machine Learning: A Hands-on Tutorial for Machine Learning Practitioners and Researchers. arXiv. arXiv:2502.01146 [quant-ph] (2025). https://doi.org/10.48550/arXiv.2502.01146 . http://arxiv.org/abs/2502. 01146 Accessed 2025-07-25

    work page doi:10.48550/arxiv.2502.01146 2025

  28. [29]

    https://doi.org/10.1017/9781009023405

    Roberts, D.A., Yaida, S., Hanin, B.: The Principles of Deep Learning The- ory, (2022). https://doi.org/10.1017/9781009023405 . arXiv:2106.10165 [cs]. http://arxiv.org/abs/2106.10165Accessed 2025-07-24

    work page doi:10.1017/9781009023405 2022

  29. [30]

    Europhysics Letters134(1), 10002 (2021) https://doi.org/10.1209/0295-5075/134/10002

    Mangini, S., Tacchino, F., Gerace, D., Bajoni, D., Macchiavello, C.: Quantum computing models for artificial neural networks. Europhysics Letters134(1), 10002 (2021) https://doi.org/10.1209/0295-5075/134/10002 . arXiv:2102.03879 [quant-ph]. Accessed 2025-08-29

    work page doi:10.1209/0295-5075/134/10002 2021

  30. [31]

    Wang, H., Cao, Y., Huang, Z., Liu, Y., Hu, P., Luo, X., Song, Z., Zhao, W., Liu, J., Sun, J., Zhang, S., Wei, L., Wang, Y., Wu, T., Ma, Z.-M., Sun, Y.: Recent Advances on Machine Learning for Computational Fluid Dynamics: A Survey. arXiv. arXiv:2408.12171 [cs] (2024). https://doi.org/10.48550/arXiv.2408.12171 . http://arxiv.org/abs/2408.12171 Accessed 2025-09-07

    work page doi:10.48550/arxiv.2408.12171 2024

  31. [32]

    Fourier Neural Operator for Parametric Partial Differential Equations

    Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A., Anandkumar, A.: Fourier Neural Operator for Parametric Partial Differen- tial Equations. arXiv:2010.08895 [cs, math] (2021). arXiv: 2010.08895. Accessed 2021-09-27

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  32. [33]

    Proceedings of the National Academy of Sciences118(21), 2101784118 (2021) https://doi.org/10

    Kochkov, D., Smith, J.A., Alieva, A., Wang, Q., Brenner, M.P., Hoyer, S.: Machine learning accelerated computational fluid dynamics. Proceedings of the National Academy of Sciences118(21), 2101784118 (2021) https://doi.org/10. 1073/pnas.2101784118 . arXiv:2102.01010 [physics]. Accessed 2025-09-07

    work page arXiv 2021

  33. [34]

    Steady States and Well-balanced Schemes for Shallow Water Moment Equations with Topography

    Jin, X., Cai, S., Li, H., Karniadakis, G.E.: NSFnets (Navier-Stokes Flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics426, 109951 (2021) https://doi. 39 org/10.1016/j.jcp.2020.109951 . arXiv:2003.06496 [physics]. Accessed 2025-09- 07

    work page doi:10.1016/j.jcp.2020.109951 2021

  34. [35]

    Raissi, P

    Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics378, 686–707 (2019) https://doi.org/10.1016/j.jcp.2018.10.045 . Accessed 2025-07-30

    work page doi:10.1016/j.jcp.2018.10.045 2019

  35. [36]

    Entropy26(8), 649 (2024) https://doi.org/10.3390/e26080649

    Trahan, C., Loveland, M., Dent, S.: Quantum Physics-Informed Neural Net- works. Entropy26(8), 649 (2024) https://doi.org/10.3390/e26080649 . Pub- lisher: Multidisciplinary Digital Publishing Institute. Accessed 2025-08-28

    work page doi:10.3390/e26080649 2024

  36. [37]

    Farea, A., Khan, S., Celebi, M.S.: QCPINN: Quantum Classical Physics- Informed Neural Networks for Solving PDEs. arXiv. arXiv:2503.16678 [quant-ph] version: 1 (2025). https://doi.org/10.48550/arXiv.2503.16678 . http://arxiv.org/ abs/2503.16678 Accessed 2025-08-28

    work page doi:10.48550/arxiv.2503.16678 2025

  37. [38]

    Nature Reviews Physics7(4), 174–189 (2025)

    Larocca, M., Thanasilp, S., Wang, S., Sharma, K., Biamonte, J., Coles, P.J., Cincio, L., McClean, J.R., Holmes, Z., Cerezo, M.: Barren plateaus in variational quantum computing. Nature Reviews Physics7(4), 174–189 (2025)

    work page 2025

  38. [39]

    Quantum Information Processing 22(12), 435 (2023) https://doi.org/10.1007/s11128-023-04188-7

    Qi, H., Wang, L., Zhu, H., Gani, A., Gong, C.: The barren plateaus of quantum neural networks: review, taxonomy and trends. Quantum Information Processing 22(12), 435 (2023) https://doi.org/10.1007/s11128-023-04188-7

    work page doi:10.1007/s11128-023-04188-7 2023

  39. [40]

    Physics of Fluids37(5) (2025) https://doi.org/10.1063/5.0268240

    Au-Yeung, R., Kendon, V.M., Lind, S.J.: Quantum smoothed particle hydro- dynamics algorithm inspired by quantum walks. Physics of Fluids37(5) (2025) https://doi.org/10.1063/5.0268240

    work page doi:10.1063/5.0268240 2025

  40. [41]

    Annals of Physics326(1), 96–192 (2011) https://doi.org/10

    Schollw¨ ock, U.: The density-matrix renormalization group in the age of matrix product states. Annals of Physics326(1), 96–192 (2011) https://doi.org/10. 1016/j.aop.2010.09.012 . January 2011 Special Issue

    work page 2011

  41. [42]

    Gourianov, P

    Gourianov, N., Givi, P., Jaksch, D., Pope, S.B.: Tensor networks enable the calculation of turbulence probability distributions. Science Advances11(5), 5990 (2025) https://doi.org/10.1126/sciadv.ads5990 https://www.science.org/doi/pdf/10.1126/sciadv.ads5990

    work page doi:10.1126/sciadv.ads5990 2025

  42. [43]

    Kiffner, M., Jaksch, D.: Tensor network reduced order models for wall- bounded flows. Phys. Rev. Fluids8, 124101 (2023) https://doi.org/10.1103/ PhysRevFluids.8.124101

    work page 2023

  43. [44]

    Pisoni, R

    Pisoni, S., Peddinti, R.D., Tiunov, E., Guzman, S.E., Aolita, L.: Compression, simulation, and synthesis of turbulent flows with tensor trains (2025). https: //arxiv.org/abs/2506.05477

    work page arXiv 2025

  44. [45]

    Journal of Computational Physics544, 114439 (2026) https://doi.org/10.1016/j.jcp.2025

    Danis, M.E., Truong, D., Boureima, I., Korobkin, O., Rasmussen, K.Ø., 40 Alexandrov, B.S.: Tensor-train weno scheme for compressible flows. Journal of Computational Physics529, 113891 (2025) https://doi.org/10.1016/j.jcp.2025. 113891

    work page doi:10.1016/j.jcp.2025 2025

  45. [46]

    Kornev, S

    Kornev, E., Dolgov, S., Pinto, K., Pflitsch, M., Perelshtein, M., Melnikov, A.: Numerical solution of the incompressible Navier-Stokes equations for chemical mixers via quantum-inspired Tensor Train Finite Element Method (2023). https: //arxiv.org/abs/2305.10784

    work page arXiv 2023

  46. [47]

    H¨ olscher, L., Rao, P., M¨ uller, L., Klepsch, J., Luckow, A., Stollenwerk, T., Wil- helm, F.K.: Quantum-inspired fluid simulation of two-dimensional turbulence with gpu acceleration. Phys. Rev. Res.7, 013112 (2025) https://doi.org/10. 1103/PhysRevResearch.7.013112

    work page 2025

  47. [48]

    https://arxiv.org/abs/2408.03483

    Danis, M.E., Truong, D.P., DeSantis, D., Petersen, M., Rasmussen, K.O., Alexandrov, B.S.: High-order Tensor-Train Finite Volume Method for Shallow Water Equations (2024). https://arxiv.org/abs/2408.03483

    work page arXiv 2024

  48. [49]

    https://arxiv.org/abs/2502.04868

    Dubois, J., Herty, M., M¨ uller, S.: High-dimensional stochastic finite volumes using the tensor train format (2025). https://arxiv.org/abs/2502.04868

    work page arXiv 2025

  49. [50]

    Journal of Computational Physics538, 114192 (2025) https://doi.org/10.1016/j.jcp.2025.114192

    Walton, S., Tokareva, S., Manzini, G.: The tensor-train stochastic finite volume method for uncertainty quantification. Journal of Computational Physics538, 114192 (2025) https://doi.org/10.1016/j.jcp.2025.114192

    work page doi:10.1016/j.jcp.2025.114192 2025

  50. [51]

    Communi- cations Physics7(1), 135 (2024) https://doi.org/10.1038/s42005-024-01623-8

    Peddinti, R.D., Pisoni, S., Marini, A., Lott, P., Argentieri, H., Tiunov, E., Aolita, L.: Quantum-inspired framework for computational fluid dynamics. Communi- cations Physics7(1), 135 (2024) https://doi.org/10.1038/s42005-024-01623-8

    work page doi:10.1038/s42005-024-01623-8 2024

  51. [52]

    Quantum-Inspired Tensor-Network Fractional-Step Method for Incompressible Flow in Curvilinear Coordinates

    H¨ ulst, N.-L., Siegl, P., Over, P., Bengoechea, S., Hashizume, T., Cecile, M.G., Rung, T., Jaksch, D.: Quantum-Inspired Tensor-Network Fractional- Step Method for Incompressible Flow in Curvilinear Coordinates (2025). https: //arxiv.org/abs/2507.05222

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  52. [53]

    Lubasch, J

    Lubasch, M., Joo, J., Moinier, P., Kiffner, M., Jaksch, D.: Variational quantum algorithms for nonlinear problems. Physical Review A101(1), 010301 (2020) https://doi.org/10.1103/PhysRevA.101.010301 . Publisher: American Physical Society. Accessed 2025-07-25

    work page doi:10.1103/physreva.101.010301 2020

  53. [54]

    Mangini, S.: Variational quantum algorithms for machine learning: theory and applications. arXiv. arXiv:2306.09984 [quant-ph] (2023). https://doi.org/10. 48550/arXiv.2306.09984 . http://arxiv.org/abs/2306.09984 Accessed 2025-07- 24

    work page arXiv 2023

  54. [55]

    Streif, M., Leib, M.: Comparison of QAOA with Quantum and Simu- lated Annealing. arXiv. arXiv:1901.01903 [quant-ph] (2019). https://doi.org/10. 41 48550/arXiv.1901.01903 . http://arxiv.org/abs/1901.01903 Accessed 2025-09- 30

    work page internal anchor Pith review Pith/arXiv arXiv 1901

  55. [56]

    Verdon, G., Broughton, M., Biamonte, J.: A quantum algorithm to train neural networks using low-depth circuits. arXiv. arXiv:1712.05304 [quant-ph] (2019). https://doi.org/10.48550/arXiv.1712.05304 . http://arxiv.org/abs/1712.05304 Accessed 2025-09-30

    work page doi:10.48550/arxiv.1712.05304 2019

  56. [57]

    Iannelli, G., Jansen, K.: Noisy Bayesian optimization for variational quan- tum eigensolvers. arXiv. arXiv:2112.00426 [quant-ph] (2021). https://doi.org/ 10.48550/arXiv.2112.00426 . http://arxiv.org/abs/2112.00426 Accessed 2025- 09-30

    work page doi:10.48550/arxiv.2112.00426 2021

  57. [58]

    A variational eigenvalue solver on a quantum processor

    Peruzzo, A., McClean, J., Shadbolt, P., Yung, M.-H., Zhou, X.-Q., Love, P.J., Aspuru-Guzik, A., O’Brien, J.L.: A variational eigenvalue solver on a quantum processor. Nature Communications5(1), 4213 (2014) https://doi.org/10.1038/ ncomms5213 . arXiv:1304.3061 [quant-ph]. Accessed 2025-09-30

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  58. [59]

    Quantum4, 248 (2020) https://doi.org/10.22331/q-2020-03-26-248

    Cerezo, M., Poremba, A., Cincio, L., Coles, P.J.: Variational Quantum Fidelity Estimation. Quantum4, 248 (2020) https://doi.org/10.22331/q-2020-03-26-248 . arXiv:1906.09253 [quant-ph]. Accessed 2025-09-30

    work page doi:10.22331/q-2020-03-26-248 2020

  59. [60]

    Quantum autoencoders for efficient compression of quantum data

    Romero, J., Olson, J.P., Aspuru-Guzik, A.: Quantum autoencoders for effi- cient compression of quantum data. Quantum Science and Technology2(4), 045001 (2017) https://doi.org/10.1088/2058-9565/aa8072 . arXiv:1612.02806 [quant-ph]. Accessed 2025-09-30

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/2058-9565/aa8072 2017

  60. [61]

    Quantum generative adversarial networks

    Dallaire-Demers, P.-L., Killoran, N.: Quantum generative adversarial networks. Physical Review A98(1), 012324 (2018) https://doi.org/10.1103/PhysRevA.98. 012324 . arXiv:1804.08641 [quant-ph]. Accessed 2025-09-30

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physreva.98 2018

  61. [62]

    Physical Review Research3(3), 033251 (2021) https://doi.org/10.1103/ PhysRevResearch.3.033251

    Tan, K.C., Volkoff, T.: Variational quantum algorithms to estimate rank, quantum entropies, fidelity and Fisher information via purity minimiza- tion. Physical Review Research3(3), 033251 (2021) https://doi.org/10.1103/ PhysRevResearch.3.033251 . arXiv:2103.15956 [quant-ph]. Accessed 2025-09-30

    work page arXiv 2021

  62. [63]

    Physical Review A103(1), 012405 (2021) https://doi.org/10.1103/PhysRevA.103.012405

    Mari, A., Bromley, T.R., Killoran, N.: Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A103(1), 012405 (2021) https://doi.org/10.1103/PhysRevA.103.012405 . Publisher: American Physical Society. Accessed 2025-09-03

    work page doi:10.1103/physreva.103.012405 2021

  63. [64]

    Guerreschi, G.G., Smelyanskiy, M.: Practical optimization for hybrid quantum- classical algorithms. arXiv. arXiv:1701.01450 [quant-ph] (2017). https://doi.org/ 10.48550/arXiv.1701.01450 . http://arxiv.org/abs/1701.01450 Accessed 2025- 08-04

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1701.01450 2017

  64. [65]

    Effect of data encoding on the expressive power of variational quantum-machine-learning models

    Schuld, M., Sweke, R., Meyer, J.J.: The effect of data encoding on the 42 expressive power of variational quantum machine learning models. Physi- cal Review A103(3) (2021) https://doi.org/10.1103/PhysRevA.103.032430 . arXiv:2008.08605 [quant-ph]. Accessed 2025-07-24

    work page doi:10.1103/physreva.103.032430 2021

  65. [66]

    Quantum4, 314 (2020) https://doi.org/10.22331/ q-2020-08-31-314

    Sweke, R., Wilde, F., Meyer, J., Schuld, M., Faehrmann, P.K., Meynard- Piganeau, B., Eisert, J.: Stochastic gradient descent for hybrid quantum- classical optimization. Quantum4, 314 (2020) https://doi.org/10.22331/ q-2020-08-31-314 . arXiv:1910.01155 [quant-ph]. Accessed 2025-07-30

    work page arXiv 2020

  66. [67]

    Crooks, G.E.: Gradients of parameterized quantum gates using the parameter- shift rule and gate decomposition. arXiv. arXiv:1905.13311 [quant-ph] (2019). https://doi.org/10.48550/arXiv.1905.13311 . http://arxiv.org/abs/1905.13311 Accessed 2025-07-30

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1905.13311 1905

  67. [68]

    AIAA Journal61(5), 1885–1894 (2023) https: //doi.org/10.2514/1.J062426

    Jaksch, D., Givi, P., Daley, A.J., Rung, T.: Variational Quantum Algorithms for Computational Fluid Dynamics. AIAA Journal61(5), 1885–1894 (2023) https: //doi.org/10.2514/1.J062426 . Publisher: American Institute of Aeronautics and Astronautics. Accessed 2025-07-25

    work page doi:10.2514/1.j062426 2023

  68. [69]

    Kaltenbaek, M

    Rath, M., Date, H.: Quantum data encoding: a comparative analysis of classical- to-quantum mapping techniques and their impact on machine learning accuracy. EPJ Quantum Technology11(1), 1–22 (2024) https://doi.org/10.1140/epjqt/ s40507-024-00285-3 . Number: 1 Publisher: SpringerOpen. Accessed 2025-07-24

    work page doi:10.1140/epjqt/ 2024

  69. [70]

    Multigrid Renormalization

    Lubasch, M., Moinier, P., Jaksch, D.: Multigrid Renormalization. Journal of Computational Physics372, 587–602 (2018) https://doi.org/10.1016/j.jcp.2018. 06.065 . arXiv:1802.07259 [physics]. Accessed 2025-07-25

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.jcp.2018 2018

  70. [71]

    Advances in Applied Mechanics, vol

    Burgers, J.M.: A Mathematical Model Illustrating the Theory of Turbulence. Advances in Applied Mechanics, vol. 1, pp. 171–199. Elsevier, ??? (1948). https: //doi.org/10.1016/S0065-2156(08)70100-5

    work page doi:10.1016/s0065-2156(08)70100-5 1948

  71. [72]

    Vedral, V., Barenco, A., Ekert, A.: Quantum networks for elementary arith- metic operations. Phys. Rev. A54, 147–153 (1996) https://doi.org/10.1103/ PhysRevA.54.147

    work page 1996

  72. [73]

    https://quantumalgorithms.org/ chapter-intro.html#hadamard-test Accessed 2025-08-11

    Luongo, A.: Chapter 2 Quantum Computing and Quantum Algorithms |Quantum Algorithms For data analysis. https://quantumalgorithms.org/ chapter-intro.html#hadamard-test Accessed 2025-08-11

    work page 2025

  73. [74]

    Multilayer feedforward ne tworks are universal approximators

    Hornik, K., Stinchcombe, M., White, H.: Multilayer feedforward networks are universal approximators. Neural Networks2(5), 359–366 (1989) https://doi.org/ 10.1016/0893-6080(89)90020-8 . Accessed 2025-07-30

    work page doi:10.1016/0893-6080(89)90020-8 1989

  74. [75]

    Gazoulis, D., Gkanis, I., Makridakis, C.G.: On the Stability and Convergence of Physics Informed Neural Networks. arXiv. arXiv:2308.05423 [math] (2025). https://doi.org/10.48550/arXiv.2308.05423 . http://arxiv.org/abs/2308.05423 43 Accessed 2025-08-29

    work page doi:10.48550/arxiv.2308.05423 2025

  75. [76]

    Raissi, M., Yazdani, A., Karniadakis, G.E.: Hidden Fluid Mechanics: A Navier- Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data. arXiv. arXiv:1808.04327 [cs] (2018). https://doi.org/10.48550/arXiv.1808. 04327 . http://arxiv.org/abs/1808.04327 Accessed 2025-08-29

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1808 2018

  76. [77]

    Physics of Fluids34(7), 075117 (2022) https://doi.org/10.1063/5.0095270

    Eivazi, H., Tahani, M., Schlatter, P., Vinuesa, R.: Physics-informed neural net- works for solving Reynolds-averaged Navier$\unicode{x2013}$Stokes equations. Physics of Fluids34(7), 075117 (2022) https://doi.org/10.1063/5.0095270 . arXiv:2107.10711 [physics]. Accessed 2025-08-29

    work page doi:10.1063/5.0095270 2022

  77. [78]

    Kobrin, Z

    Goto, T., Tran, Q.H., Nakajima, K.: Universal Approximation Property of Quan- tum Machine Learning Models in Quantum-Enhanced Feature Spaces. Physical Review Letters127(9), 090506 (2021) https://doi.org/10.1103/PhysRevLett. 127.090506 . arXiv:2009.00298 [quant-ph]. Accessed 2025-08-30

    work page doi:10.1103/physrevlett 2021

  78. [79]

    Data re- uploading for a universal quantum classifier,

    P´ erez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re- uploading for a universal quantum classifier. Quantum4, 226 (2020) https: //doi.org/10.22331/q-2020-02-06-226 . arXiv:1907.02085 [quant-ph]. Accessed 2025-08-30

    work page doi:10.22331/q-2020-02-06-226 2020

  79. [80]

    Mathematics12(21), 3318 (2024) https:// doi.org/10.3390/math12213318

    Ranga, D., Rana, A., Prajapat, S., Kumar, P., Kumar, K., Vasilakos, A.V.: Quantum Machine Learning: Exploring the Role of Data Encoding Techniques, Challenges, and Future Directions. Mathematics12(21), 3318 (2024) https:// doi.org/10.3390/math12213318 . Publisher: Multidisciplinary Digital Publishing Institute. Accessed 2025-08-29

    work page doi:10.3390/math12213318 2024

  80. [81]

    Evaluating analytic gradients on quantum hardware

    Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating ana- lytic gradients on quantum hardware. Physical Review A99(3) (2019) https:// doi.org/10.1103/PhysRevA.99.032331 . arXiv:1811.11184 [quant-ph]. Accessed 2025-07-25

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physreva.99.032331 2019

Showing first 80 references.