arxiv: 2604.11113 · v1 · submitted 2026-04-13 · 🪐 quant-ph · physics.flu-dyn

Recognition: unknown

Schr\"odinger-Navier-Stokes Equation for the Quantum Simulation of Navier-Stokes Flows

Luca Cappelli , Sauro Succi , Monica Lacatus , Alessandro Zecchi , Alessandro Roggero

Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:55 UTC · model grok-4.3

classification 🪐 quant-ph physics.flu-dyn

keywords Schrödinger-Navier-Stokesquantum simulationCarleman embeddingHamilton-Jacobi formulationtensor networksNavier-Stokes equationsfluid dynamicsquantum algorithms

0 comments

The pith

The Navier-Stokes equations can be simulated on quantum computers by recasting them in Hamilton-Jacobi form and applying tensor-network Carleman embedding.

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

This paper develops a quantum algorithm for simulating the Navier-Stokes equations that describe classical fluid motion. It starts from a Schrödinger-like wave version of these equations but notes that the dissipation term creates a steep challenge for quantum hardware. The authors therefore adopt the Hamilton-Jacobi formulation of the fluid and embed its dynamics into a tensor network via Carleman linearization, which reduces memory cost. Classical emulation of the resulting algorithm reproduces Kolmogorov-like flows at moderate Reynolds numbers with good accuracy. A sympathetic reader would care because the method includes the full effects of pressure, dissipation, and vorticity, offering a concrete route to quantum simulation of realistic viscous flows.

Core claim

The central claim is that the genuine Navier-Stokes equations, including pressure, dissipation and vorticity, admit a quantum algorithm through their Schrödinger-Navier-Stokes wave formulation once the dissipator is handled by a Hamilton-Jacobi reformulation and the resulting system is linearized with a tensor-network version of Carleman embedding. This combination is shown to converge under classical emulation for Kolmogorov-like flows at moderate Reynolds numbers.

What carries the argument

The tensor-network Carleman embedding of the Hamilton-Jacobi equations (CHJ), which linearizes the nonlinear fluid dynamics while achieving substantial memory savings relative to standard Carleman methods.

If this is right

The algorithm incorporates the complete Navier-Stokes terms of pressure, viscous dissipation, and vorticity without further approximation.
Tensor-network structure yields substantial memory savings compared with direct Carleman linearization of the original equations.
The method converges for Kolmogorov-like flows at moderate Reynolds numbers, as verified by classical emulation.
It supplies the first quantum algorithm based on a wave formulation of the genuine Navier-Stokes equations.

Where Pith is reading between the lines

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

The same Hamilton-Jacobi plus tensor-network route might be tested on other nonlinear dissipative partial differential equations in engineering and physics.
Scaling the CHJ embedding to three spatial dimensions would directly address open questions in quantum simulation of turbulence.
Hardware implementation would require error-mitigation techniques that preserve the low-rank tensor structure throughout the evolution.

Load-bearing premise

The Hamilton-Jacobi reformulation combined with tensor-network Carleman embedding can be realized on quantum hardware with acceptable overhead and without the dissipator destroying the quantum advantage.

What would settle it

If classical emulation of the CHJ algorithm on a Kolmogorov flow at Reynolds number near 100 produces velocity or vorticity statistics that deviate markedly from known reference solutions, the accuracy of the linearization for dissipative flows would be disproved.

Figures

Figures reproduced from arXiv: 2604.11113 by Alessandro Roggero, Alessandro Zecchi, Luca Cappelli, Monica Lacatus, Sauro Succi.

**Figure 3.** Figure 3: FIG. 3. Decay rate of the [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Global relative error of the NSHJ fields ( [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Comparison of the momentum field [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Color map of the momentum field [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Comparison of the momentum field [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. Evolution of relative errors [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. Comparison of the momentum field [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10. Comparison of scaling law in terms of memory required for the standard Carleman and tensor-network approaches. [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗

read the original abstract

The search for quantum-like wave formulations of the Navier-Stokes (Schr\"odinger-Navier-Stokes, SNS for short) equations describing classical dissipative fluids has met with increasing attention in the recent years, due to the large portfolio of potential applications in science and engineering. A SNS formulation of classical fluids was first presented in a largely un-noticed paper by Dietrich and Vautherin back in 1985(Journal de Physique). In this paper, we revisit this specific SNS approach and assess its viability for quantum implementations based on Carleman embedding/linearization techniques. Specifically, we i) Clarify in full mathematical detail why the SNS dissipator presents a steep challenge for quantum computers and propose a way out strategy based on the Hamilton-Jacobi (HJ) formulation of fluid dynamics; ii) Develop a corresponding quantum algorithm using a new technique based on a tensor-network representation of Carleman embedding of the HJ equations (CHJ) which permits substantial memory savings; iii) Emulate the CHJ quantum algorithm on a classical computer and analyse its convergence and accuracy for the specific case of Kolmogorov-like flows at moderate Reynolds numbers. To the best of our knowledge, this is the first quantum algorithm based on a quantum-like wave formulation of the genuine Navier-Stokes equations, including pressure, dissipation and vorticity.

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.

Desk Editor's Note private letter to a colleague

The paper gives a tensor-network Carleman embedding of the Hamilton-Jacobi Navier-Stokes form and shows classical convergence for moderate-Re Kolmogorov flows, but supplies no qubit or gate counts for the quantum version.

read the letter

The main thing here is a concrete proposal for quantum simulation of the full Navier-Stokes equations via a Schrödinger-like wave equation. They take the 1985 Dietrich-Vautherin SNS formulation, recast the fluid dynamics in Hamilton-Jacobi variables to handle the dissipator, then apply a tensor-network version of Carleman linearization that cuts memory use. They classically emulate the resulting algorithm on Kolmogorov flows at moderate Reynolds numbers and report convergence with some accuracy checks. That emulation is the solid part of the work; it is independent of any quantum hardware claim and shows the CHJ construction behaves as expected on classical machines. The novelty claim is that this is the first quantum algorithm covering pressure, vorticity, and dissipation together. The paper does spell out the mathematical steps for the HJ route and the tensor-network savings, which is useful detail. The soft spot is that the quantum resource scaling is left unquantified. No circuit depth, qubit count, or overhead estimate appears for the linear-system solver step once the embedding dimension grows, so the claim that the dissipator problem is solved in practice rests on the reformulation alone rather than on measured or bounded quantum cost. The abstract acknowledges the challenge but the emulation stays classical. This is a proposal paper with reproducible classical results rather than a finished quantum demonstration. It is aimed at people working on quantum algorithms for PDEs and fluid modeling. A serious editor should send it to referees because the emulation provides something concrete to check and the HJ-Carleman route is spelled out enough to be critiqued or extended. It is not ready for immediate use but it is worth the review time.

Referee Report

3 major / 2 minor

Summary. The manuscript revisits the Schrödinger-Navier-Stokes (SNS) formulation of the Navier-Stokes equations, proposes a Hamilton-Jacobi (HJ) reformulation to circumvent the dissipator challenge for quantum implementation, develops a tensor-network Carleman embedding of the HJ equations (CHJ) that achieves memory savings, and classically emulates the resulting quantum algorithm for Kolmogorov-like flows at moderate Reynolds numbers, claiming convergence and accuracy while asserting this is the first quantum algorithm for the genuine NS equations including pressure, dissipation, and vorticity.

Significance. If the CHJ construction translates to quantum hardware with acceptable overhead, the work would open a new route to quantum simulation of dissipative classical fluids. The classical emulation with convergence analysis for moderate-Re Kolmogorov flows is a concrete strength that demonstrates the reformulation's viability in the classical limit; however, the absence of quantified quantum resources leaves the central claim of quantum advantage untested.

major comments (3)

[Abstract] Abstract: the claim that this is 'the first quantum algorithm based on a quantum-like wave formulation of the genuine Navier-Stokes equations, including pressure, dissipation and vorticity' is not supported by any quantitative error bars, circuit-depth analysis, or explicit demonstration that the dissipator term is fully resolved by the HJ route.
[CHJ construction] CHJ construction and emulation section: no explicit qubit count, gate complexity, or circuit-depth bound is provided for the tensor-network representation once the dissipator is restored via the HJ formulation; the memory savings are shown only classically, leaving the translation to quantum linear-system solvers (HHL or variational) unanalyzed.
[Emulation results] Emulation results: while convergence at moderate Re is reported, the impact of the Carleman embedding dimension on quantum circuit resources is not quantified, which is load-bearing for any claim of acceptable overhead or quantum advantage.

minor comments (2)

[CHJ construction] Notation for the tensor-network Carleman embedding could be clarified with an explicit diagram or pseudocode to aid reproducibility.
[Abstract] The abstract would benefit from a single quantitative statement of the observed convergence rate or error norm from the classical emulation.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and indicate planned revisions to strengthen the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that this is 'the first quantum algorithm based on a quantum-like wave formulation of the genuine Navier-Stokes equations, including pressure, dissipation and vorticity' is not supported by any quantitative error bars, circuit-depth analysis, or explicit demonstration that the dissipator term is fully resolved by the HJ route.

Authors: We appreciate this observation. The abstract claim reflects the novelty of combining the SNS formulation with the HJ reformulation to handle the dissipator and the tensor-network Carleman embedding. The classical emulation in the manuscript provides quantitative convergence and accuracy results for Kolmogorov flows, including pressure, dissipation, and vorticity. To address the concern, we will revise the abstract to qualify the claim as the first such proposal with classical validation via the HJ route, add a brief explanation of how the HJ formulation resolves the dissipator, and reference the emulation error metrics. revision: yes
Referee: [CHJ construction] CHJ construction and emulation section: no explicit qubit count, gate complexity, or circuit-depth bound is provided for the tensor-network representation once the dissipator is restored via the HJ formulation; the memory savings are shown only classically, leaving the translation to quantum linear-system solvers (HHL or variational) unanalyzed.

Authors: We agree that explicit quantum resource bounds would strengthen the presentation. The current focus is the classical validation of the CHJ reformulation and its memory savings via tensor networks. We will add a dedicated paragraph in the CHJ section outlining the expected scaling for quantum linear solvers, noting how the tensor-network compression reduces the effective system size for HHL or variational methods, and provide order-of-magnitude estimates based on the Carleman truncation level used in the emulation. revision: partial
Referee: [Emulation results] Emulation results: while convergence at moderate Re is reported, the impact of the Carleman embedding dimension on quantum circuit resources is not quantified, which is load-bearing for any claim of acceptable overhead or quantum advantage.

Authors: The emulation demonstrates convergence with increasing embedding dimension for moderate-Re flows, which is essential to establish viability. We will revise the results section to include a short analysis linking the embedding dimension to the size of the resulting linear system and the potential quantum resources after tensor-network compression, thereby quantifying the path toward acceptable overhead. revision: yes

standing simulated objections not resolved

Complete numerical quantification of qubit counts, gate complexity, and circuit depths for a full quantum implementation, as this requires a separate detailed design of the quantum linear-system solver applied to the CHJ equations.

Circularity Check

0 steps flagged

No significant circularity; derivation chain is self-contained

full rationale

The paper cites the 1985 Dietrich-Vautherin SNS formulation as external prior work, then develops an independent CHJ construction via Hamilton-Jacobi reformulation combined with tensor-network Carleman embedding. It performs classical emulation on Kolmogorov flows to demonstrate convergence at moderate Re. No step reduces a claimed prediction or first-principles result to a fitted parameter, self-definition, or load-bearing self-citation chain. The quantum-algorithm claim rests on the newly developed equations and emulation results rather than renaming or smuggling ansatzes. Resource scaling for quantum hardware is left unquantified, but this is an incompleteness, not a circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The work rests on the standard Navier-Stokes equations as domain assumptions and the mathematical validity of Carleman linearization and Hamilton-Jacobi equivalence for fluids; no new free parameters or invented entities are introduced in the abstract description.

axioms (3)

domain assumption Navier-Stokes equations accurately describe classical dissipative fluid flows including pressure and vorticity
Invoked as the target system to be simulated
domain assumption Hamilton-Jacobi formulation is equivalent to the original fluid equations for the purposes of linearization
Used as the basis for the proposed quantum algorithm
domain assumption Carleman embedding can be represented via tensor networks with substantial memory savings
Central to the new CHJ technique

pith-pipeline@v0.9.0 · 5546 in / 1454 out tokens · 50899 ms · 2026-05-10T15:55:55.462401+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

37 extracted references · 5 canonical work pages

[1]

In the second-order Carleman approximation, all terms of order three and higher are neglected: J ⊗3, J ⊗4 ≈0

Second-order truncation We define the second order lifted state as: J= J1 J2 = J J⊗J .(21) The evolution of the second-order state is obtained by the tensor product of the updated state (20) with itself: ˆJ (2) = ˆJ⊗ ˆJ(22) = (AJ+BJ ⊗2)⊗(AJ+BJ ⊗2) = (A⊗A)J ⊗2 + (A⊗B+B⊗A)J ⊗3 + (B⊗B)J ⊗4. In the second-order Carleman approximation, all terms of order three...
[2]

GPY2cuftHI0vZ4NDt7aX1nENjT0=

Tensor Network Representation The generic order-N C expansion is obtained in a simi- lar manner by taking theN C-th power of Eq. (20). While an explicit arithmetical formulation is possible, an in- tuitive approach based on a tensor network represen- tation proves more convenient. In this representation, first-order state vectorsJ α are depicted as circle...

1985
[3]

Tennie, S

F. Tennie, S. Laizet, S. Lloyd, and L. Magri, Nature Re- views Physics7, 220 (2025)

2025
[4]

S. S. Bharadwaj and K. R. Sreenivasan, S¯ adhana50, 57 (2025)

2025
[5]

Z. Meng, J. Zhong, S. Xu, K. Wang, J. Chen, F. Jin, X. Zhu, Y. Gao, Y. Wu, C. Zhang, N. Wang, Y. Zou, A. Zhang, Z. Cui, F. Shen, Z. Bao, Z. Zhu, Z. Tan, T. Li, P. Zhang, S. Xiong, H. Li, Q. Guo, Z. Wang, C. Song, H. Wang, and Y. Yang, Communications Physics7, 349 (2024)

2024
[6]

Dietrich and D

K. Dietrich and D. Vautherin, Journal de Physique 10.1051/jphys:01985004603031300 (1985)

work page doi:10.1051/jphys:01985004603031300 1985
[7]

Sanavio, E

C. Sanavio, E. Mauri, and S. Succi, IEEE Transactions on Quantum Engineering6, 1 (2025)

2025
[8]

Valiente and N

M. Valiente and N. T. Zinner,Strongly interacting quan- tum systems, Volume 2: Many-body physics, IOP Series in Quantum Technology (IOP Publishing, 2024)

2024
[9]

Ingelmann, S

J. Ingelmann, S. S. Bharadwaj, P. Pfeffer, K. R. Sreeni- vasan, and J. Schumacher, Computers & Fluids281, 106369 (2024)

2024
[10]

S. S. Bharadwaj and K. R. Sreenivasan, Physical Review Research7, 023262 (2025)

2025
[11]

Zylberman, G

J. Zylberman, G. Di Molfetta, M. Brachet, N. F. Loureiro, and F. Debbasch, Physical Review A106, 032408 (2022)

2022
[12]

X. Li, X. Yin, N. Wiebe, J. Chun, G. K. Schenter, M. S. Cheung, and J. M¨ ulmenst¨ adt, Phys. Rev. Res.7, 013036 (2025)

2025
[13]

An end-to-end quantum algorithm for nonlinear fluid dynamics with bounded quantum advantage.arXiv preprint arXiv:2512.03758,

D. Jennings, K. Korzekwa, M. Lostaglio, R. Ashworth, E. Marsili, and S. Rolston, An end-to-end quantum algo- rithm for nonlinear fluid dynamics with bounded quan- tum advantage (2025), arXiv:2512.03758 [quant-ph]

work page arXiv 2025
[14]

M. I. L˘ acatu¸ s and M. M¨ oller, International Journal for Numerical Methods in Engineering127, e70286 (2026)

2026
[15]

Itani and S

W. Itani and S. Succi, Fluids7, 24 (2022)

2022
[16]

Succi, W

S. Succi, W. Itani, K. Sreenivasan, and R. Steijl, Euro- physics Letters144, 10001 (2023)

2023
[17]

Salasnich, S

L. Salasnich, S. Succi, and A. Tiribocchi, International Journal of Modern Physics C35, 2450100 (2024)

2024
[18]

Meng and Y

Z. Meng and Y. Yang, Physical Review Research6, 043130 (2024)

2024
[19]

Verstraete, J

F. Verstraete, J. I. Cirac, and V. Murg, Advances in Physics57, 143 (2008)

2008
[20]

Or´ us, Annals of Physics349, 117 (2014)

R. Or´ us, Annals of Physics349, 117 (2014)

2014
[21]

Sanavio, R

C. Sanavio, R. Scatamacchia, C. de Falco, and S. Succi, Physics of Fluids36, 057143 (2024)

2024
[22]

Sanavio and S

C. Sanavio and S. Succi, AVS Quantum Science6, 023802 (2024)

2024
[23]

Madelung, Zeitschrift f¨ ur Physik40, 322 (1926)

E. Madelung, Zeitschrift f¨ ur Physik40, 322 (1926)

1926
[24]

Khesin, G

B. Khesin, G. Misio lek, and K. Modin, Archive for Ra- tional Mechanics and Analysis234, 549 (2019)

2019
[25]

R. LeVeque,Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time- Dependent Problems, Other Titles in Applied Mathe- matics (Society for Industrial and Applied Mathematics, 2007)

2007
[26]

R. L. Taylor and R. W. Lewis, International Journal for Numerical Methods in Engineering80, 1 (2009)

2009
[27]

Carleman, Acta Mathematica59, 63 (1932)

T. Carleman, Acta Mathematica59, 63 (1932)

1932
[28]

Of course this requirement can be relaxed at the expense of suitably padding the resulting vectors and matrices with zeros
[29]

Camps, L

D. Camps, L. Lin, R. V. Beeumen, and C. Yang, SIAM Journal on Matrix Analysis and Applications45, 801 13 (2024)

2024
[30]

G. H. Low and I. L. Chuang, Quantum3, 163 (2019)

2019
[31]

Gily´ en, Y

A. Gily´ en, Y. Su, G. H. Low, and N. Wiebe, inProceed- ings of the 51st annual ACM SIGACT symposium on theory of computing(2019) pp. 193–204

2019
[32]

A. M. Childs and N. Wiebe, Quant. Inf. Comput.12, 0901 (2012), arXiv:1202.5822 [quant-ph]

work page Pith review arXiv 2012
[33]

Sanavio, W

C. Sanavio, W. A. Simon, A. Ralli, P. Love, and S. Succi, Physics of Fluids37, 037123 (2025)

2025
[34]

Explicit error bounds for Carleman linearization

M. Forets and A. Pouly, Explicit error bounds for carle- man linearization (2017), arXiv:1711.02552 [math.NA]

work page arXiv 2017
[35]

Belmabrouk, Long-time asymptotics for multivariate hawkes processes with long-range interactions (2026), arXiv:2603.05853 [math.PR]

N. Belmabrouk, Long-time asymptotics for multivariate hawkes processes with long-range interactions (2026), arXiv:2603.05853 [math.PR]

work page arXiv 2026
[36]

D’Orazio, S

A. D’Orazio, S. Succi, and C. Arrighetti, Physics of Flu- ids - PHYS FLUIDS15, 2778 (2003). Appendix A: Explicit CHJ Matrices Let us define the discrete divergence operatorD= Dx +D y and Laplace operatorL=D xx +D yy. In a 2D scenario, the expression for the matrix conveying the linear term in Eq. (20) is Aαβ =I+ ∆t   0 0 0 0 −c2 sIνL νD x νDy 0 0νD yy ...

2003
[37]

IV of the main text

Block encoding We now briefly explain how we perform the block en- coding of the update matrix used in Sec. IV of the main text. First we explain how to perform the block encoding of the matrix performing the update of theJ (1) compo- nent M1 =A⊗I⊗I+ B ,(A7) where Bis a square matrix that embedsBin the top right sub-matrix and is zero otherwise. It is hel...

2000