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arxiv: 2604.09997 · v1 · submitted 2026-04-11 · 🪐 quant-ph

Quantum Lattice Boltzmann with Denoising Collision Operators

Trong Duong , Matthias M\"oller , Norbert Hosters This is my paper

Pith reviewed 2026-05-10 16:46 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum lattice Boltzmann methoddenoising collision operatororthogonal projectionquantum fluid simulationhydrodynamic recoveryadvection-diffusion
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The pith

A quantum lattice Boltzmann method replaces nonlinear collisions with orthogonal projections onto equilibrium distributions.

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

The paper presents a quantum formulation of the lattice Boltzmann method for simulating fluid dynamics that uses a denoising collision operator based on orthogonal projection. This reformulation avoids the costly repeated quantum tomography and state preparation required in prior approaches. By projecting onto the linearized manifold of equilibrium distributions around a reference state, the method filters non-equilibrium parts while keeping lattice symmetries and approximating the nonlinear terms needed for hydrodynamics. Numerical experiments confirm that advection-diffusion and flow problems are reproduced accurately, with a pathway to coherent multi-timestep simulations using deterministic projectors.

Core claim

The collision dynamics in quantum LBM are reformulated as an orthogonal projection onto the linearized manifold of equilibrium distributions around a reference state. This geometric denoising approach filters non-equilibrium components, preserves lattice symmetries, and approximates nonlinear terms to recover hydrodynamic behavior. A complete pipeline with gate-level realizations for encoding, collision, streaming, boundaries, and measurements is provided, along with deterministic implementation of projectors without postselection.

What carries the argument

Denoising-type collision operator as an orthogonal projection onto the linearized manifold of equilibrium distributions around a reference state.

If this is right

  • Gate-level quantum circuits can implement the full LBM pipeline including collision without tomography overhead.
  • Deterministic projector-based operators enable fully coherent multi-timestep simulations.
  • Macroscopic hydrodynamic behaviors are recovered accurately in advection-diffusion and flow simulations.
  • Performance depends on the choice of reference state for the linearization.

Where Pith is reading between the lines

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

  • The method may generalize to other nonlinear processes in quantum algorithms by similar linear manifold projections.
  • Adaptive selection of the reference state during simulation could further improve accuracy for complex flows.
  • Hybrid quantum-classical implementations might use the projection step to reduce quantum resource demands.

Load-bearing premise

The orthogonal projection onto the linearized equilibrium manifold around a reference state sufficiently approximates the nonlinear collision terms while preserving lattice symmetries and recovering hydrodynamic behavior.

What would settle it

Numerical simulations of standard flow problems that fail to match known hydrodynamic solutions or violate lattice symmetries would indicate the projection does not recover the required behavior.

Figures

Figures reproduced from arXiv: 2604.09997 by Matthias M\"oller, Norbert Hosters, Trong Duong.

Figure 1
Figure 1. Figure 1: Diagram for standard QLBM simulations over t timesteps without boundary treatment. II. LATTICE BOLTZMANN METHOD The Lattice Boltzmann Method (LBM) simulates fluid dynamics by tracking the evolution of particle 2 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Projection onto the local tangent plane (or [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The figure also presents an implementation of the diagonal operator UΣ using controlled phase gates. Meanwhile, U and V † can be implemented using standard gate-decomposition methods [53–55]. How￾ever, since U and V † only act non-identically on the one-hot basis states, it is more efficient and numer￾ically accurate to implement them with Givens ro￾tations. Real-valued Givens rotations are particle￾preser… view at source ↗
Figure 4
Figure 4. Figure 4: The Clements decomposition of U and V † yields a brickwork layout of Givens rotations, with each rotation having a distinct angle. This diagram is shown for q = 5. To sum up, the implementation of UD employs 2(q− rank(D)) controlled phase gates and q(q − 1) Givens rotations. The ancilla must be post-selected in the state |0⟩. This is done by measuring the ancilla and retaining the normalized resulting stat… view at source ↗
Figure 5
Figure 5. Figure 5: General quantum circuit for streaming along [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Quantum circuit for a full LBM step includ [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (a) Concentration profiles C(x, t) obtained with the QLBM at different times. (b) Relative L2 error of the QLBM and CLBM solutions. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: (top) Time evolution of a Gaussian hill initialized with σ0 = 20 in a periodic square domain; the red arrow indicates the advection direction uadv and contours denote 75%, 50%, and 25% of the peak value. (middle) Diffusive decay of the peak density at the hill center. (bottom) Relative L2 error of the QLBM simulations. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: shows the velocity’s magnitude and its error heatmap for the QLBM, as well as the relative error of the macroscopic fields. The velocity field in QLBM exhibits a large discrepancy early in the simulation, but the error decreases gradually. This is because the reference velocity uˆ = 0 is only asymptotically exact; the velocity in early iterations is sinusoidal with mag￾nitude ∼ u0. This suggests that choo… view at source ↗
Figure 11
Figure 11. Figure 11: The speed field ∥u∥ from QLBM and CLBM at different times. Each plot is rotated by 90◦ clockwise for visualization, so the original left bound￾ary becomes the top edge. method, based on one-hot encoding and a denoising￾based collision operator. By formulating the colli￾sion step as an orthogonal projection onto the lo￾cal linearization of the equilibrium manifold, we have demonstrated an alternative path … view at source ↗
read the original abstract

The Lattice Boltzmann method (LBM) is a well-established mesoscopic approach for simulating fluid dynamics by evolving particle distribution functions on discrete lattices. While the LBM is highly parallelizable on classical hardware, its translation to quantum algorithms is impeded by the collision process, which is intrinsically nonlinear and irreversible. Several existing quantum formulations implement this process through repeated quantum tomography and state preparation at every timestep, leading to significant overheads. We introduce a quantum LBM based on a denoising-type collision operator that avoids tomography-based updates. The collision dynamics are reformulated as an orthogonal projection onto the linearized manifold of equilibrium distributions around a reference state. This geometric approach filters non-equilibrium components while preserving lattice symmetries and approximating nonlinear terms needed to recover hydrodynamic behavior. A complete pipeline is presented with efficient gate-level realizations, incorporating encoding of distributions, collision, streaming, boundary conditions, and measurement of physical quantities such as hydrodynamic forces. In addition, we outline an approach for implementing projector-based operators deterministically without postselection, paving the way to fully coherent multi-timestep LBM simulations. Numerical experiments for advection-diffusion and flow problems demonstrate that the method reproduces macroscopic behaviors with high accuracy, with performance depending on the choice of reference state.

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 Lattice Boltzmann Method (LBM) that reformulates the nonlinear collision step as a denoising-type operator implemented via orthogonal projection onto the linearized manifold of equilibrium distributions around a chosen reference state. This construction is intended to avoid repeated quantum tomography and state preparation. The authors describe a full quantum pipeline covering distribution encoding, the projection-based collision, streaming, boundary conditions, and measurement of hydrodynamic quantities, together with a deterministic (postselection-free) realization of the projector. Numerical experiments on advection-diffusion and flow problems are reported to recover macroscopic behavior with high accuracy, with performance depending on the reference-state choice.

Significance. If the linear-projection approximation is shown to recover the correct hydrodynamic limit, the work would offer a concrete route to coherent, tomography-free quantum LBM simulations and could lower the resource overhead that has so far limited quantum fluid-dynamics algorithms. The geometric reformulation, the complete gate-level pipeline, and the initial numerical validation are concrete strengths that would be of interest to the quantum-algorithms and computational-fluid-dynamics communities.

major comments (2)
  1. [Abstract] Abstract: the claim that the orthogonal projection 'approximates nonlinear terms needed to recover hydrodynamic behavior' is load-bearing for the central contribution, yet the abstract supplies neither a Chapman-Enskog-style analysis nor an explicit error bound demonstrating that the linearized projection around a fixed reference recovers the correct viscous and advection terms of the BGK operator when local distributions deviate appreciably from that reference.
  2. [Numerical experiments] Numerical experiments section: the reported high accuracy for advection-diffusion and flow problems is presented without quantitative measures of deviation from the reference state or systematic error analysis; if the linear regime is exceeded, conservation properties or effective relaxation rates may shift, undermining the hydrodynamic-limit claim.
minor comments (2)
  1. [Methods] Clarify in the methods section how the reference distribution is encoded and updated (or held fixed) within the quantum circuit, and state explicitly whether it is chosen globally or locally.
  2. [Figures] Figure captions and axis labels should indicate the quantitative deviation of the simulated distributions from the reference state so that readers can judge the regime of validity of the linear approximation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and the positive evaluation of the geometric reformulation, gate-level pipeline, and initial numerical results. We address each major comment below and will incorporate revisions to strengthen the hydrodynamic-limit claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the orthogonal projection 'approximates nonlinear terms needed to recover hydrodynamic behavior' is load-bearing for the central contribution, yet the abstract supplies neither a Chapman-Enskog-style analysis nor an explicit error bound demonstrating that the linearized projection around a fixed reference recovers the correct viscous and advection terms of the BGK operator when local distributions deviate appreciably from that reference.

    Authors: We agree that the abstract is concise and does not itself contain the supporting analysis. The manuscript body (Section 3) derives the projection operator and argues via lattice symmetries that it preserves the necessary moments for hydrodynamics, but we acknowledge the absence of an explicit Chapman-Enskog expansion or error bound in the abstract. In the revision we will (i) qualify the abstract claim to read 'approximates the nonlinear terms of the BGK operator for moderate deviations from a suitably chosen reference state' and (ii) add a short Chapman-Enskog analysis together with a first-order error bound on the viscous and advection terms in the main text (new subsection 3.4). revision: yes

  2. Referee: [Numerical experiments] Numerical experiments section: the reported high accuracy for advection-diffusion and flow problems is presented without quantitative measures of deviation from the reference state or systematic error analysis; if the linear regime is exceeded, conservation properties or effective relaxation rates may shift, undermining the hydrodynamic-limit claim.

    Authors: We accept this criticism. The current numerical section reports only macroscopic agreement without tracking the deviation from the reference equilibrium or performing a systematic study. In the revised version we will add: (a) time-series plots of the L2 deviation ||f - f_eq(ref)|| for each benchmark, (b) a parameter sweep over reference-state choices with tabulated errors in recovered viscosity and advection speed, and (c) explicit verification that mass and momentum are conserved to machine precision and that the extracted relaxation rates remain consistent with the BGK target inside the reported accuracy window. These additions will delineate the regime where the linear-projection approximation remains valid. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation is an independent geometric construction

full rationale

The paper's central step reformulates the collision operator as an orthogonal projection onto the linearized equilibrium manifold around a reference state. This is introduced as a new geometric approach that filters non-equilibrium components and approximates the needed nonlinear terms. No equations, fitted parameters, or self-citations are shown to reduce the claimed hydrodynamic recovery or quantum implementation to the inputs by definition. The abstract explicitly notes that performance depends on reference choice and that the projection only approximates nonlinear terms, treating this as an approximation rather than an identity. The derivation chain therefore remains self-contained against external benchmarks and does not match any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that a linear projection around a reference state can stand in for the full nonlinear collision operator while still recovering correct hydrodynamics.

axioms (1)
  • domain assumption Collision dynamics can be reformulated as an orthogonal projection onto the linearized manifold of equilibrium distributions around a reference state.
    This is the core geometric reformulation stated in the abstract.

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Reference graph

Works this paper leans on

74 extracted references · 74 canonical work pages · 1 internal anchor

  1. [1]

    J. P. Slotnick, A. Khodadoust, J. Alonso, D. Darmofal, W. Gropp, E. Lurie, and D. J. Mavriplis.CFD Vision 2030 Study: A Path to Revolutionary Computational Aerosciences. Tech. rep. NF1676L-18332. 2014

    work page 2030

  2. [2]

    On the role and challenges of CFD in the aerospace in- dustry

    P. R. Spalart and V. Venkatakrishnan, “On the role and challenges of CFD in the aerospace in- dustry”,The Aeronautical Journal, 120(1223), 2016

    work page 2016

  3. [3]

    CFD Vision 2030 Road Map: Progress and Perspectives

    A. W. Cary, J. Chawner, E. P. Duque, W. Gropp, W. L. Kleb, R. M. Kolonay, E. Nielsen, and B. Smith, “CFD Vision 2030 Road Map: Progress and Perspectives”,AIAA AVIATION 2021 FORUM, 2021

    work page 2030

  4. [4]

    Overview of the Coupled Model Intercompari- son Project Phase 6 (CMIP6) experimental de- sign and organization

    V. Eyring, S. Bony, G. A. Meehl, C. A. Senior, B. Stevens, R. J. Stouffer, and K. E. Taylor, “Overview of the Coupled Model Intercompari- son Project Phase 6 (CMIP6) experimental de- sign and organization”,Geoscientific Model De- velopment, 9(5), 2016

    work page 2016

  5. [5]

    Global Cloud-Resolving Models

    M. Satoh, B. Stevens, F. Judt, M. Khairoutdi- nov, S.-J. Lin, W. M. Putman, and P. Düben, “Global Cloud-Resolving Models”,Current Cli- mate Change Reports, 5(3), 2019

    work page 2019

  6. [6]

    Quarteroni, L

    A. Quarteroni, L. Dede’, A. Manzoni, and C. Vergara,Mathematical Modelling of the Human Cardiovascular System: Data, Numerical Ap- proximation, Clinical Applications, Cambridge University Press, 2019

    work page 2019

  7. [7]

    Computa- tionalmodelingandengineeringinpediatricand congenital heart disease

    A. L. Marsden and J. A. Feinstein, “Computa- tionalmodelingandengineeringinpediatricand congenital heart disease”,Current Opinion in Pediatrics, 27(5), 2015

    work page 2015

  8. [8]

    A computational model of coronary arteries with in-stent restenosis coupling hemodynamics and pharmacokinetics with growth mechanics

    A. Ranno, K. Manjunatha, T. Koritzius, I. Steinbrecher, N. Hosters, M. Nachtsheim, P. Nilcham, N. Schaaps, A. Turoni-Glitz, J. Datz, A. Popp, K. Linka, F. Vogt, and M. Behr, “A computational model of coronary arteries with in-stent restenosis coupling hemodynamics and pharmacokinetics with growth mechanics”,Sci- entific Reports, 15(1), 2025

    work page 2025

  9. [9]

    The future of computational fluid dynamics (CFD) simulation in the chem- ical process industries

    D. F. Fletcher, “The future of computational fluid dynamics (CFD) simulation in the chem- ical process industries”,Chemical Engineering Research and Design, 187, 2022

    work page 2022

  10. [10]

    CFD-based simulation to reduce green- house gas emissions from industrial plants

    Sarjito, M. Elveny, A. T. Jalil, A. Davarpanah, M. Alfakeer, A. A. A. Bahajjaj, and M. Oulads- mane, “CFD-based simulation to reduce green- house gas emissions from industrial plants”,In- ternational Journal of Chemical Reactor Engi- neering, 19(11), 2021

    work page 2021

  11. [11]

    CFD Ap- pliedtoProcessDevelopmentintheOilandGas Industry- A Review

    L. Raynal, F. Augier, F. Bazer-Bachi, Y. Haroun, and C. A. Pereira da Fonte, “CFD Ap- pliedtoProcessDevelopmentintheOilandGas Industry- A Review”,Oil & Gas Science and Technology - Revue d’IFP Energies nouvelles, 71(3), 2016. 16

    work page 2016

  12. [12]

    Special issue on the lattice Boltzmann method

    C. Shu and N. Phan-Thien, “Special issue on the lattice Boltzmann method”,Physics of Fluids, 34(10), 2022

    work page 2022

  13. [13]

    On the lattice Boltzmann method and its application to turbulent, multiphase flows of various fluids includingcryogens:Areview

    K. J. Petersen and J. R. Brinkerhoff, “On the lattice Boltzmann method and its application to turbulent, multiphase flows of various fluids includingcryogens:Areview”,Physics of Fluids, 33(4), 2021

    work page 2021

  14. [14]

    Lattice- BoltzmannMethodforComplexFlows

    C. K. Aidun and J. R. Clausen, “Lattice- BoltzmannMethodforComplexFlows”,Annual Review of Fluid Mechanics, 42(1), 2010

    work page 2010

  15. [15]

    Quantum Algorithm for Linear Systems of Equations

    A. W. Harrow, A. Hassidim, and S. Lloyd, “Quantum Algorithm for Linear Systems of Equations”,Physical Review Letters, 103(15), 2009

    work page 2009

  16. [16]

    Quantum Algorithm for Systems of Linear Equations with Exponentially Improved Depen- dence on Precision

    A. M. Childs, R. Kothari, and R. D. Somma, “Quantum Algorithm for Systems of Linear Equations with Exponentially Improved Depen- dence on Precision”,SIAM Journal on Comput- ing, 46(6), 2017

    work page 2017

  17. [17]

    Variable time amplitude am- plification and quantum algorithms for linear algebra problems

    A. Ambainis, “Variable time amplitude am- plification and quantum algorithms for linear algebra problems”,29th International Sympo- sium on Theoretical Aspects of Computer Sci- ence (STACS 2012),SchlossDagstuhl–Leibniz- Zentrum für Informatik, vol. 14, 2012

    work page 2012

  18. [18]

    Universal Quantum Simulators

    S. Lloyd, “Universal Quantum Simulators”,Sci- ence, 273(5278), 1996

    work page 1996

  19. [19]

    Exponential im- provement in precision for simulating sparse Hamiltonians

    D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, “Exponential im- provement in precision for simulating sparse Hamiltonians”,Proceedings of the forty-sixth annual ACM symposium on Theory of comput- ing, 2014

    work page 2014

  20. [20]

    Simulating Hamil- tonian Dynamics with a Truncated Taylor Se- ries

    D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, “Simulating Hamil- tonian Dynamics with a Truncated Taylor Se- ries”,Physical Review Letters, 114(9), 2015

    work page 2015

  21. [21]

    Optimal Hamil- tonian Simulation by Quantum Signal Process- ing

    G. H. Low and I. L. Chuang, “Optimal Hamil- tonian Simulation by Quantum Signal Process- ing”,Physical Review Letters, 118(1), 2017

    work page 2017

  22. [22]

    Quantum computing for simulation of fluid dynamics

    C. Sanavio and S. Succi, “Quantum computing for simulation of fluid dynamics”,Quantum in- formation science - recent advances and compu- tational science applications, IntechOpen, 2024

    work page 2024

  23. [23]

    Quantum comput- ing of fluid dynamics using the hydrodynamic Schrödinger equation

    Z. Meng and Y. Yang, “Quantum comput- ing of fluid dynamics using the hydrodynamic Schrödinger equation”,Physical Review Re- search, 5(3), 2023

    work page 2023

  24. [24]

    Finding flows of a Navier–Stokes fluid through quantum computing

    F. Gaitan, “Finding flows of a Navier–Stokes fluid through quantum computing”,npj Quan- tum Information, 6(1), 2020

    work page 2020

  25. [25]

    Quantum lattice-gas model for com- putational fluid dynamics

    J. Yepez, “Quantum lattice-gas model for com- putational fluid dynamics”,Physical Review E, 63(4), 2001

    work page 2001

  26. [26]

    From quantum cellular automata to quantum lattice gases

    D. A. Meyer, “From quantum cellular automata to quantum lattice gases”,Journal of Statistical Physics, 85(5-6), 1996

    work page 1996

  27. [27]

    Quantum lattice-gas model for the diffusion equation

    J. Yepez, “Quantum lattice-gas model for the diffusion equation”,International Journal of Modern Physics C, 12(09), 2001

    work page 2001

  28. [28]

    Quantum lattice Boltzmann is a quantum walk

    S. Succi, F. Fillion-Gourdeau, and S. Palpacelli, “Quantum lattice Boltzmann is a quantum walk”,EPJ Quantum Technology, 2(1), 2015

    work page 2015

  29. [29]

    Quantum algo- rithm for the collisionless Boltzmann equation

    B. N. Todorova and R. Steijl, “Quantum algo- rithm for the collisionless Boltzmann equation”, Journal of Computational Physics, 409, 2020

    work page 2020

  30. [30]

    A multiple- circuit approach to quantum resource reduction with application to the quantum lattice Boltz- mann method

    M. Lee, Z. Song, S. Kocherla, A. Adams, A. Alexeev, and S. H. Bryngelson, “A multiple- circuit approach to quantum resource reduction with application to the quantum lattice Boltz- mann method”,Future Generation Computer Systems, 174, 2026

    work page 2026

  31. [31]

    Quantum algorithm for lattice Boltzmann (QALB) simulation of incompressible fluids with a nonlinear collision term

    W. Itani, K. R. Sreenivasan, and S. Succi, “Quantum algorithm for lattice Boltzmann (QALB) simulation of incompressible fluids with a nonlinear collision term”,Physics of Flu- ids, 36(1), 2024

    work page 2024

  32. [32]

    A quantum computing-based latticeBoltzmannmethodwithalinearizednon- equilibrium collision operator and modular cir- cuit for practical flow simulation

    K.-Y. Zeng, X.-D. Niu, A. Khan, D.-C. Li, and H. Yamaguchi, “A quantum computing-based latticeBoltzmannmethodwithalinearizednon- equilibrium collision operator and modular cir- cuit for practical flow simulation”,Physics of Fluids, 37(8), 2025

    work page 2025

  33. [33]

    Decomposition of nonlinear collision operator in quantum Lat- tice Boltzmann algorithm

    E. D. Kumar and S. H. Frankel, “Decomposition of nonlinear collision operator in quantum Lat- tice Boltzmann algorithm”,Europhysics Letters, 148(3), 2024

    work page 2024

  34. [34]

    Quantum algorithm for the advec- tion–diffusion equation simulated with the lat- tice Boltzmann method

    L. Budinski, “Quantum algorithm for the advec- tion–diffusion equation simulated with the lat- tice Boltzmann method”,Quantum Information Processing, 20(2), 2021. 17

    work page 2021

  35. [35]

    Linearized quantum lattice-Boltzmann method for the advection-diffusion equation us- ing dynamic circuits

    D. Wawrzyniak, J. Winter, S. Schmidt, T. In- dinger, C. F. Janßen, U. Schramm, and N. A. Adams, “Linearized quantum lattice-Boltzmann method for the advection-diffusion equation us- ing dynamic circuits”,Computer Physics Com- munications, 317, 2025

    work page 2025

  36. [36]

    Float Lattice Gas Automata: A con- nection between Molecular Dynamics and Lat- tice Boltzmann Method for quantum comput- ers

    A. D. B. Zamora, L. Budinski, V. Lahtinen, and P. Sagaut, “Float Lattice Gas Automata: A con- nection between Molecular Dynamics and Lat- tice Boltzmann Method for quantum comput- ers”,Physical Review E, 112(1), 2025

    work page 2025

  37. [37]

    Fully quantum algorithm for mesoscale fluid simulations with application to partial differ- ential equations

    S. Kocherla, Z. Song, F. E. Chrit, B. Gard, E. F. Dumitrescu, A. Alexeev, and S. H. Bryngelson, “Fully quantum algorithm for mesoscale fluid simulations with application to partial differ- ential equations”,AVS Quantum Science, 6(3), 2024

    work page 2024

  38. [38]

    Quantum lattice Boltzmann method for simu- lating nonlinear fluid dynamics

    B. Wang, Z. Meng, Y. Zhao, and Y. Yang, “Quantum lattice Boltzmann method for simu- lating nonlinear fluid dynamics”,npj Quantum Information, 2025

    work page 2025

  39. [39]

    C. A. Georgescu and M. Möller.Quantum Search in Superposed Quantum Lattice Gas Au- tomata and Lattice Boltzmann Systems. 2025. doi:10.48550/arXiv.2510.14062

    work page doi:10.48550/arxiv.2510.14062 2025

  40. [40]

    Fully Quantum Lattice Gas Automata Building Blocks for Computational Basis State Encodings

    C. A. Georgescu, M. A. Schalkers, and M. Möller, “Fully Quantum Lattice Gas Automata Building Blocks for Computational Basis State Encodings”,Journal of Computational Physics, 549, 2026

    work page 2026

  41. [41]

    Lăcătuş and M

    M. Lăcătuş and M. Möller.Surrogate Quantum Circuit Design for the Lattice Boltzmann Col- lision Operator. 2025.doi:10 . 48550 / arXiv . 2507.12256

    work page arXiv 2025

  42. [42]

    Y. Ji, M. Lacatus, and M. Möller.IGA-LBM: Isogeometric lattice Boltzmann method. 2025. doi:10.48550/arXiv.2509.11427

    work page doi:10.48550/arxiv.2509.11427 2025

  43. [43]

    qlbm – A quantum lattice Boltzmann software framework

    C. A. Georgescu, M. A. Schalkers, and M. Möller, “qlbm – A quantum lattice Boltzmann software framework”,Computer Physics Com- munications, 315, 2025

    work page 2025

  44. [44]

    Utilizing classical programming principles in the Intel Quantum SDK: implementation of quantum lattice Boltz- mannmethod

    T. Shinde, L. Budinski, O. Niemimäki, V. Lahti- nen, H. Liebelt, and R. Li, “Utilizing classical programming principles in the Intel Quantum SDK: implementation of quantum lattice Boltz- mannmethod”,ACM Transactions on Quantum Computing, 6(1), 2025

    work page 2025

  45. [45]

    Simulating Non-Trivial Incompressible Flows With a Quantum Lattice Boltzmann Al- gorithm

    D. Jennings, K. Korzekwa, M. Lostaglio, P. Mannix, R. Ashworth, E. Marsili, and S. Rol- ston, “Simulating Non-Trivial Incompressible Flows With a Quantum Lattice Boltzmann Al- gorithm”,AIAA SCITECH 2026 Forum, 2026

    work page 2026

  46. [46]

    Turro, A

    F. Turro, A. Lignarolo, and D. Dragoni.Practi- cal Application of the Quantum Carleman Lat- tice Boltzmann Method in Industrial CFD Sim- ulations. 2025.doi:10 . 48550 / arXiv . 2504 . 13033

    work page 2025

  47. [47]

    Lattice Boltz- mann–Carleman quantum algorithm and circuit for fluid flows at moderate Reynolds number

    C. Sanavio and S. Succi, “Lattice Boltz- mann–Carleman quantum algorithm and circuit for fluid flows at moderate Reynolds number”, AVS Quantum Science, 6(2), 2024

    work page 2024

  48. [48]

    Quantum lattice Boltzmann method for several time steps: A local Carle- man linearization algorithm

    A. D. B. Zamora, L. Budinski, V. Lahtinen, and P. Sagaut, “Quantum lattice Boltzmann method for several time steps: A local Carle- man linearization algorithm”,Physical Review E, 113(3), 2026

    work page 2026

  49. [49]

    Carleman-lattice-Boltzmannquantum circuit with matrix access oracles

    C. Sanavio, W. A. Simon, A. Ralli, P. Love, and S.Succi,“Carleman-lattice-Boltzmannquantum circuit with matrix access oracles”,Physics of Fluids, 37(3), 2025

    work page 2025

  50. [50]

    Krüger, H

    T. Krüger, H. Kusumaatmaja, A. Kuzmin, O. Shardt, G. Silva, and E. M. Viggen,The Lat- tice Boltzmann Method: Principles and Practice, Springer International Publishing, 2017

    work page 2017

  51. [51]

    TowardlearningLattice Boltzmann collision operators

    A.Corbetta,A.Gabbana,V.Gyrya,D.Livescu, J.Prins,andF.Toschi,“TowardlearningLattice Boltzmann collision operators”,The European Physical Journal E, 46(3), 2023

    work page 2023

  52. [52]

    Quantum state preparation and nonunitary evolution with diagonal operators

    A. W. Schlimgen, K. Head-Marsden, L. M. Sager-Smith, P. Narang, and D. A. Mazziotti, “Quantum state preparation and nonunitary evolution with diagonal operators”,Physical Re- view A, 106(2), 2022

    work page 2022

  53. [53]

    Quantum Circuits for General Multiqubit Gates

    M. Möttönen, J. J. Vartiainen, V. Bergholm, and M. M. Salomaa, “Quantum Circuits for General Multiqubit Gates”,Physical Review Letters, 93(13), 2004

    work page 2004

  54. [54]

    Efficient Decomposition of Quantum Gates

    J. J. Vartiainen, M. Möttönen, and M. M. Sa- lomaa, “Efficient Decomposition of Quantum Gates”,Physical Review Letters, 92(17), 2004

    work page 2004

  55. [55]

    Approaching the theoretical limit in quantum gate decomposi- tion

    P. Rakyta and Z. Zimborás, “Approaching the theoretical limit in quantum gate decomposi- tion”,Quantum, 6, 2022. 18

    work page 2022

  56. [56]

    Univer- sal quantum circuits for quantum chemistry

    J. M. Arrazola, O. D. Matteo, N. Quesada, S. Jahangiri, A. Delgado, and N. Killoran, “Univer- sal quantum circuits for quantum chemistry”, Quantum, 6, 2022

    work page 2022

  57. [57]

    Optimal design for universal multiport inter- ferometers

    W. R. Clements, P. C. Humphreys, B. J. Met- calf, W. S. Kolthammer, and I. A. Walsmley, “Optimal design for universal multiport inter- ferometers”,Optica, 3(12), 2016

    work page 2016

  58. [58]

    Dickerson

    D. Cilluffo.Commentary on the decomposition of universal multiport interferometers: how it works in practice. 2024.doi:10.48550/arXiv. 2412.11955

    work page internal anchor Pith review doi:10.48550/arxiv 2024

  59. [59]

    Efficient and fail-safe quantum algorithm for the transport equation

    M. A. Schalkers and M. Möller, “Efficient and fail-safe quantum algorithm for the transport equation”,Journal of Computational Physics, 502, 2024

    work page 2024

  60. [60]

    Burel and Q

    T. Burel and Q. Gu.An improved boundary condition at a low grid resolution and Reynolds number. 2018.doi:10 . 48550 / arXiv . 1806 . 03623

    work page 2018

  61. [61]

    Boundary conditions for lattice Boltzmann simulations

    D. P. Ziegler, “Boundary conditions for lattice Boltzmann simulations”,Journal of Statistical Physics, 71(5), 1993

    work page 1993

  62. [62]

    Momentum ex- change method for quantum Boltzmann meth- ods

    M. A. Schalkers and M. Möller, “Momentum ex- change method for quantum Boltzmann meth- ods”,Computers & Fluids, 285, 2024

    work page 2024

  63. [63]

    Improved amplitude amplification strategies for the quantum simulation of classi- cal transport problems

    A. A. Zecchi, C. Sanavio, S. Perotto, and S. Succi, “Improved amplitude amplification strategies for the quantum simulation of classi- cal transport problems”,Quantum Science and Technology, 10(3), 2025

    work page 2025

  64. [64]

    Double-Bracket Quantum Algorithms for Quantum Imaginary-Time Evolution

    M. Gluza, J. Son, B. H. Tiang, R. Zander, R. Seidel, Y. Suzuki, Z. Holmes, and N. H. Y. Ng, “Double-Bracket Quantum Algorithms for Quantum Imaginary-Time Evolution”,Physical Review Letters, 136(2), 2026

    work page 2026

  65. [65]

    Double-bracket al- gorithm for quantum signal processing without post-selection

    Y. Suzuki, B. H. Tiang, J. Son, N. H. Y. Ng, Z. Holmes, and M. Gluza, “Double-bracket al- gorithm for quantum signal processing without post-selection”,Quantum, 9, 2025

    work page 2025

  66. [66]

    Double-bracket quantum algorithms for diagonalization

    M. Gluza, “Double-bracket quantum algorithms for diagonalization”,Quantum, 8, 2024

    work page 2024

  67. [67]

    Quantum-circuit design for efficient simula- tions of many-body quantum dynamics

    S. Raeisi, N. Wiebe, and B. C. Sanders, “Quantum-circuit design for efficient simula- tions of many-body quantum dynamics”,New Journal of Physics, 14(10), 2012

    work page 2012

  68. [68]

    Hamiltonian Sim- ulation by Qubitization

    G. H. Low and I. L. Chuang, “Hamiltonian Sim- ulation by Qubitization”,Quantum, 3, 2019

    work page 2019

  69. [69]

    Hamiltonian simu- lation using linear combinations of unitary oper- ations

    A. M. Childs and N. Wiebe, “Hamiltonian simu- lation using linear combinations of unitary oper- ations”,Quantum Information & Computation, 12(11-12), 2012

    work page 2012

  70. [70]

    Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics

    A. Gilyén, Y. Su, G. H. Low, and N. Wiebe, “Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics”,Proceedings of the 51st An- nual ACM SIGACT Symposium on Theory of Computing, 2019

    work page 2019

  71. [71]

    Efficient Quantum Algorithms for Simulating Sparse Hamiltonians

    D. W. Berry, G. Ahokas, R. Cleve, and B. C. Sanders, “Efficient Quantum Algorithms for Simulating Sparse Hamiltonians”,Communica- tions in Mathematical Physics, 270(2), 2007

    work page 2007

  72. [72]

    Quantum principal component analysis

    S. Lloyd, M. Mohseni, and P. Rebentrost, “Quantum principal component analysis”,Na- ture Physics, 10(9), 2014

    work page 2014

  73. [73]

    Quantum approximated cloning-assisted den- sity matrix exponentiation

    P. Rodriguez-Grasa, R. Ibarrondo, J. Gonzalez- Conde, Y. Ban, P. Rebentrost, and M. Sanz, “Quantum approximated cloning-assisted den- sity matrix exponentiation”,Physical Review Research, 7(1), 2025

    work page 2025

  74. [74]

    On the New- ton–Kantorovich Theorem

    P. G. Ciarlet and C. Mardare, “On the New- ton–Kantorovich Theorem”,Analysis and Ap- plications, 10(03), 2012. 19 A. SYMMETRY EQUIVARIANCE LetGbe the symmetry group of the LBM lattice. A lattice symmetryg∈Gis an orthogonal transformation Rg ∈O(d)in physical space consisting of rotations and reflections that preserve the lattice structure. It acts on the d...

    work page 2012