pith. sign in

arxiv: 2607.00281 · v1 · pith:KSTKTUJ4new · submitted 2026-07-01 · 🧮 math.NA · cs.NA

Quantum Simulation of Stokes Flow via Schr\"odingerisation and Artificial Compressibility

Shi Jin , Jiaqi Tang , Qilong Zhai , Lei Zhang This is my paper

Pith reviewed 2026-07-02 08:29 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords quantum algorithmStokes equationsSchrödingerisationartificial compressibilityquantum simulationincompressible flowhigh-dimensional problemscomputational complexity
0
0 comments X

The pith

A quantum algorithm simulates incompressible Stokes flow with exponential speedup in problem dimensionality.

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

The paper develops a quantum algorithm for the Stokes equations by regularizing them with artificial compressibility and then applying the Schrödingerisation technique to produce an explicit quantum circuit. This addresses the high cost of solving the associated saddle-point problem, which grows rapidly with dimensionality in classical approaches. Rigorous analysis establishes quantum computational advantage, specifically an exponential speedup in dimensionality, and Qiskit simulations confirm the method works and scales. Readers would care because accurate simulations of low-Reynolds-number flows in microfluidics could become practical in regimes where classical computation is infeasible.

Core claim

The artificial compressibility formulation unifies the Stokes system so that the Schrödingerisation procedure maps it directly to a quantum circuit; complexity analysis then shows this yields quantum advantage with exponential speedup in high-dimensional settings, while numerical tests on Qiskit corroborate validity and scalability.

What carries the argument

Schrödingerisation applied to the artificial-compressibility-regularized Stokes equations, which encodes the regularized system into an explicit quantum circuit.

If this is right

  • High-dimensional Stokes flow problems that are intractable on classical computers become accessible.
  • The unified artificial compressibility framework allows the entire system to be handled by a single quantum procedure.
  • The approach scales as shown by both complexity bounds and Qiskit experiments.
  • Similar quantum mappings could apply to other saddle-point problems arising in fluid dynamics.

Where Pith is reading between the lines

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

  • If the regularization error can be controlled independently of dimension, the method might extend to time-dependent or nonlinear flow problems.
  • The explicit circuit construction could serve as a template for quantum simulation of other linear PDE systems that reduce to saddle-point forms.
  • Practical deployment would require checking whether the predicted speedup survives noise and limited qubit counts on near-term hardware.

Load-bearing premise

The artificial compressibility regularization produces a system whose solution remains sufficiently close to the true incompressible Stokes solution for the quantum simulation to be useful.

What would settle it

Numerical evidence that the regularized solution fails to approach the true incompressible Stokes solution as the artificial compressibility parameter tends to zero, or that the quantum circuit complexity does not exhibit the claimed exponential scaling with dimension.

Figures

Figures reproduced from arXiv: 2607.00281 by Jiaqi Tang, Lei Zhang, Qilong Zhai, Shi Jin.

Figure 1
Figure 1. Figure 1: 3.5. Success probability estimation. The state |vˆiD(T)⟩, i = 1, · · · , d, generated by applying VStokes(τ ) r = T /τ times to |vˆi(0)⟩, is transformed via an inverse quantum Fourier transform into |viD(T)⟩. A projective measurement Mk = I ⊗4nxd ⊗ |k⟩ ⟨k| (qk > 0) then selects the |k⟩ component. From the relation vi = e −qu˜i (q > 0), it follows that: ⟨viD(T)| Mk |viD(T)⟩ ≈ e −2qk ∥u˜i(T)∥ 2 ∥q∥ 2∥u˜i(0)∥… view at source ↗
Figure 1
Figure 1. Figure 1: Quantum circuit. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Result of the collocated grid. demonstrate the full quantum algorithm for the coupled Stokes system on staggered grids, consider (3.1) with the analytic solution u =  e −t sin ( 2πx L ) cos ( 2πx L ) −e −t cos ( 2πx L ) sin ( 2πx L )  , p = e −t L 2π cos (2πx L ) sin (2πx L ), L = 2nx . where a = 1, ε = 0.1. We set nx = 4, np = 8, R = 10, T = 0.4, dt = 0.08 for the Schrödingerisation method. The numerica… view at source ↗
Figure 3
Figure 3. Figure 3: Result of Stokes flow. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: nx = 4, nq = 7, dt = 0.04, Nt = 3, ε = 0.2. nq. Keeping all other parameters fixed, we compute solutions for nq = 6, 8. The results are shown in [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: and [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: nx = 4, nq = 8, dt = 0.04, Nt = 3, ε = 0.2 [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: nx = 3, nq = 7, dt = 0.04, Nt = 3, ε = 0.2 [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: nx = 5, nq = 7, dt = 0.04, Nt = 3, ε = 0.2. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: nx = 4, nq = 7, dt = 0.02, Nt = 6, ε = 0.2 [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: nx = 4, nq = 7, dt = 0.04, Nt = 3, ε = 0.1. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
read the original abstract

Simulating incompressible Stokes flow is essential for studies in microfluidics and low-Reynolds number hydrodynamics. However, the computational cost of resolving the associated saddle-point problem grows prohibitively with the dimensionality of the problem. In this work, we present a quantum algorithm based on the Schr\"odingerisation technique for the Stokes equations, incorporating an artificial compressibility regularization. The core of our approach is the design of an explicit quantum circuit that encodes the resulting regularized system. The artificial compressibility formulation provides a unified framework for the system, which is then efficiently mapped to a quantum circuit via the Schr\"odingerisation procedure. A rigorous complexity analysis demonstrates the quantum computational advantage of our algorithms in high-dimensional settings, notably an exponential speedup in problem dimensionality. The validity and scalability of the proposed method are corroborated by numerical simulations performed on Qiskit.

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 paper presents a quantum algorithm for simulating incompressible Stokes flow that combines artificial compressibility regularization with the Schrödingerisation technique to produce an explicit quantum circuit. It asserts that a rigorous complexity analysis establishes an exponential speedup with respect to problem dimensionality and that the approach is validated by Qiskit simulations.

Significance. If the claimed complexity bounds hold and the regularization error can be controlled to a level that preserves the essential features of the incompressible solution, the work would constitute a concrete route to quantum advantage for high-dimensional saddle-point problems arising in low-Reynolds-number hydrodynamics.

major comments (2)
  1. [Abstract, §4] Abstract and §4 (complexity analysis): the manuscript states that a rigorous complexity analysis is provided, yet no explicit error bounds relating the artificial-compressibility parameter to the deviation from the true incompressible Stokes solution appear in the abstract or are referenced in the complexity derivation; without these bounds the claimed usefulness of the quantum simulation for the original problem cannot be assessed.
  2. [§3] §3 (regularization and circuit construction): the mapping via Schrödingerisation assumes that the regularized system remains sufficiently close to the incompressible limit, but the manuscript does not quantify how the free parameter in the artificial-compressibility term propagates into the final circuit depth or solution error, which is load-bearing for the exponential-speedup claim.
minor comments (2)
  1. [Abstract] The abstract refers to “numerical simulations performed on Qiskit” without specifying the problem sizes, circuit depths, or error metrics used; these details should be added for reproducibility.
  2. Notation for the artificial-compressibility parameter should be introduced once and used consistently throughout the text and equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. The comments identify important gaps in the presentation of error bounds and parameter dependence that we will address in revision. We respond point by point below.

read point-by-point responses

  1. Referee: [Abstract, §4] Abstract and §4 (complexity analysis): the manuscript states that a rigorous complexity analysis is provided, yet no explicit error bounds relating the artificial-compressibility parameter to the deviation from the true incompressible Stokes solution appear in the abstract or are referenced in the complexity derivation; without these bounds the claimed usefulness of the quantum simulation for the original problem cannot be assessed.

    Authors: We agree that explicit error bounds connecting the artificial-compressibility parameter ε to the deviation from the incompressible Stokes solution are required for a complete assessment. The current manuscript derives complexity for the regularized system but does not state the regularization error estimate in §4 or reference it in the abstract. In the revised version we will add a lemma in §4 establishing ||u_ε − u|| ≤ Cε (under standard assumptions on the domain and data) and will update both the abstract and the complexity statement to cite this bound, thereby clarifying the regime in which the exponential speedup applies to the original incompressible problem. revision: yes

  2. Referee: [§3] §3 (regularization and circuit construction): the mapping via Schrödingerisation assumes that the regularized system remains sufficiently close to the incompressible limit, but the manuscript does not quantify how the free parameter in the artificial-compressibility term propagates into the final circuit depth or solution error, which is load-bearing for the exponential-speedup claim.

    Authors: We acknowledge that §3 does not quantify the dependence of circuit depth and total error on the compressibility parameter. The Schrödingerisation step produces a unitary whose Lipschitz constant grows as 1/ε, which affects both the number of Trotter steps and the overall query complexity. In the revision we will insert a short analysis in §3 deriving the scaling of circuit depth with 1/ε and showing that the composite error (regularization + discretization + quantum simulation) remains controlled when ε is chosen proportionally to the target accuracy. This will make explicit that the exponential speedup in dimensionality is retained for any fixed accuracy in the incompressible limit. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation chain begins from the Stokes equations, applies artificial compressibility regularization to obtain a unified system, maps it explicitly to a quantum circuit via Schrödingerisation, and then performs a complexity analysis on that circuit construction. No step reduces by definition to its own inputs, renames a fitted quantity as a prediction, or relies on a load-bearing self-citation whose validity is presupposed by the present work. The exponential speedup claim is presented as following from the circuit depth and gate count analysis, which is independent of the target result itself. External corroboration via Qiskit numerics is cited separately. The paper is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; full mathematical details unavailable. The artificial compressibility parameter is expected to be a free parameter whose value affects accuracy.

free parameters (1)
  • artificial compressibility parameter
    Regularization parameter introduced to convert the saddle-point problem into a form amenable to Schrödingerisation; its specific value or selection method is not stated in the abstract.
axioms (1)
  • domain assumption Schrödingerisation maps the regularized Stokes system to an explicit quantum circuit whose complexity can be rigorously bounded.
    Core mapping step asserted in the abstract.

pith-pipeline@v0.9.1-grok · 5680 in / 1076 out tokens · 30160 ms · 2026-07-02T08:29:46.260597+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

34 extracted references · 25 canonical work pages

  1. [1]

    Ge, Zhihao and Feng, Minfu and He, Yinnian , TITLE =. J. Math. Anal. Appl. , FJOURNAL =. 2009 , NUMBER =. doi:10.1016/j.jmaa.2009.01.039 , URL =

    work page doi:10.1016/j.jmaa.2009.01.039 2009

  2. [2]

    Zhang, Xiangxiong , TITLE =. J. Comput. Phys. , FJOURNAL =. 2017 , PAGES =. doi:10.1016/j.jcp.2016.10.002 , URL =

    work page doi:10.1016/j.jcp.2016.10.002 2017

  3. [3]

    Si, Zhiyong and Wang, Yunxia and Feng, Xinlong , TITLE =. Math. Model. Anal. , FJOURNAL =. 2015 , NUMBER =. doi:10.3846/13926292.2015.1091394 , URL =

    work page doi:10.3846/13926292.2015.1091394 2015

  4. [4]

    Wang, Xiuli and Zhai, Qilong and Zhang, Ran , TITLE =. J. Comput. Appl. Math. , FJOURNAL =. 2016 , PAGES =. doi:10.1016/j.cam.2016.04.031 , URL =

    work page doi:10.1016/j.cam.2016.04.031 2016

  5. [5]

    Chandrashekar, Praveen , TITLE =. Int. J. Adv. Eng. Sci. Appl. Math. , FJOURNAL =. 2016 , NUMBER =. doi:10.1007/s12572-015-0160-z , URL =

    work page doi:10.1007/s12572-015-0160-z 2016

  6. [6]

    Yin, Zhe and Jiang, Ziwen and Xu, Qiang , TITLE =. J. Appl. Math. , FJOURNAL =. 2012 , PAGES =. doi:10.1155/2012/761242 , URL =

    work page doi:10.1155/2012/761242 2012

  7. [7]

    arXiv preprint , year =

    Lin Lin , title =. arXiv preprint , year =. 2201.08309 , archivePrefix =

    work page arXiv

  8. [8]

    , TITLE =

    Feynman, Richard P. , TITLE =. Found. Phys. , FJOURNAL =. 1986 , NUMBER =. doi:10.1007/BF01886518 , URL =

    work page doi:10.1007/bf01886518 1986

  9. [9]

    An, Dong and Liu, Jin-Peng and Lin, Lin , TITLE =. Phys. Rev. Lett. , FJOURNAL =. 2023 , NUMBER =. doi:10.1103/physrevlett.131.150603 , URL =

    work page doi:10.1103/physrevlett.131.150603 2023

  10. [10]

    Jin, Shi and Liu, Nana and Yu, Yue , TITLE =. Phys. Rev. Lett. , FJOURNAL =. 2024 , NUMBER =. doi:10.1103/physrevlett.133.230602 , URL =

    work page doi:10.1103/physrevlett.133.230602 2024

  11. [11]

    Jin, Shi and Liu, Nana and Yu, Yue , TITLE =. J. Comput. Phys. , FJOURNAL =. 2025 , PAGES =. doi:10.1016/j.jcp.2025.114138 , URL =

    work page doi:10.1016/j.jcp.2025.114138 2025

  12. [12]

    Jin, Shi and Liu, Nana and Yu, Yue , TITLE =. Phys. Rev. A , FJOURNAL =. 2023 , NUMBER =. doi:10.1103/physreva.108.032603 , URL =

    work page doi:10.1103/physreva.108.032603 2023

  13. [13]

    2024 , eprint=

    Schr\"odingerization based Quantum Circuits for Maxwell's Equation with time-dependent source terms , author=. 2024 , eprint=

    2024

  14. [14]

    Hu, Junpeng and Jin, Shi and Liu, Nana and Zhang, Lei , journal =. Quantum. doi:10.22331/q-2024-12-12-1563 , url =

    work page doi:10.22331/q-2024-12-12-1563 2024

  15. [15]

    Jin, Shi and Li, Xiantao and Liu, Nana and Yu, Yue , TITLE =. SIAM J. Sci. Comput. , FJOURNAL =. 2024 , NUMBER =. doi:10.1137/23M1563451 , URL =

    work page doi:10.1137/23m1563451 2024

  16. [16]

    Jin, Shi and Liu, Nana and Ma, Chuwen , TITLE =. SIAM J. Numer. Anal. , FJOURNAL =. 2025 , NUMBER =. doi:10.1137/24M164272X , URL =

    work page doi:10.1137/24m164272x 2025

  17. [17]

    Jin and N

    S. Jin and N. Liu , title =. Proceedings of the Royal Society A , volume =

  18. [18]

    Hamiltonian simulation for hyperbolic partial differential equations by scalable quantum circuits , author =. Phys. Rev. Res. , volume =. 2024 , month =. doi:10.1103/PhysRevResearch.6.033246 , url =

    work page doi:10.1103/physrevresearch.6.033246 2024

  19. [19]

    Tavelli, Maurizio and Dumbser, Michael , TITLE =. J. Comput. Phys. , FJOURNAL =. 2016 , PAGES =. doi:10.1016/j.jcp.2016.05.009 , URL =

    work page doi:10.1016/j.jcp.2016.05.009 2016

  20. [20]

    and Araújo, Ismael C

    Vale, Rafaella and Azevedo, Thiago Melo D. and Araújo, Ismael C. S. and Araujo, Israel F. and da Silva, Adenilton J. , journal=. Circuit Decomposition of Multicontrolled Special Unitary Single-Qubit Gates , year=

  21. [21]

    and Chuang, Isaac L

    Nielsen, Michael A. and Chuang, Isaac L. , year=. Quantum Computation and Quantum Information: 10th Anniversary Edition , publisher=

  22. [22]

    Theory of Trotter Error with Commutator Scaling , author =. Phys. Rev. X , volume =. 2021 , month =. doi:10.1103/PhysRevX.11.011020 , url =

    work page doi:10.1103/physrevx.11.011020 2021

  23. [23]

    Feng, Fang and Yang, Hui and Zhu, Shengfeng , TITLE =. Commun. Comput. Phys. , FJOURNAL =. 2025 , NUMBER =

    2025

  24. [24]

    Hao, Tiantian and Shao, Feng and Wei, Dongyi and Zhang, Zhifei , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2025 , NUMBER =. doi:10.1093/imrn/rnaf283 , URL =

    work page doi:10.1093/imrn/rnaf283 2025

  25. [25]

    Ambainis, Andris , TITLE =. 29th. 2012 , ISBN =

    2012

  26. [26]

    and Kothari, Robin and Somma, Rolando D

    Childs, Andrew M. and Kothari, Robin and Somma, Rolando D. , title =. SIAM Journal on Computing , volume =. 2017 , doi =

    2017

  27. [27]

    2023 , issn =

    Rapid solution of logical equivalence problems by quantum computation algorithm , journal =. 2023 , issn =. doi:https://doi.org/10.1016/j.asoc.2022.109844 , url =

    work page doi:10.1016/j.asoc.2022.109844 2023

  28. [28]

    Liu, Nana and Thompson, Jayne and Weedbrook, Christian and Lloyd, Seth and Vedral, Vlatko and Gu, Mile and Modi, Kavan , TITLE =. Phys. Rev. A , FJOURNAL =. 2016 , NUMBER =. doi:10.1103/physreva.93.052304 , URL =

    work page doi:10.1103/physreva.93.052304 2016

  29. [29]

    , TITLE =

    Shor, Peter W. , TITLE =. SIAM Rev. , FJOURNAL =. 1999 , NUMBER =. doi:10.1137/S0036144598347011 , URL =

    work page doi:10.1137/s0036144598347011 1999

  30. [30]

    Quantum Algorithm for Linear Systems of Equations , author =. Phys. Rev. Lett. , volume =. 2009 , month =

    2009

  31. [31]

    Client-friendly continuous-variable blind and verifiable quantum computing , author =. Phys. Rev. A , volume =. 2019 , month =

    2019

  32. [32]

    A numerical method for solving incompressible viscous flow problems , journal =

    Alexandre Joel Chorin , abstract =. A numerical method for solving incompressible viscous flow problems , journal =. 1967 , issn =. doi:https://doi.org/10.1016/0021-9991(67)90037-X , url =

    work page doi:10.1016/0021-9991(67)90037-x 1967

  33. [33]

    Mardal, Kent Andre and Tai, Xue-Cheng and Winther, Ragnar , TITLE =. SIAM J. Numer. Anal. , FJOURNAL =. 2002 , NUMBER =. doi:10.1137/S0036142901383910 , URL =

    work page doi:10.1137/s0036142901383910 2002

  34. [34]

    Brown and Ricardo Cortez and Michael L

    David L. Brown and Ricardo Cortez and Michael L. Minion , abstract =. Accurate Projection Methods for the Incompressible Navier–Stokes Equations , journal =. 2001 , issn =. doi:https://doi.org/10.1006/jcph.2001.6715 , url =

    work page doi:10.1006/jcph.2001.6715 2001