Quantum-Inspired Computational Fluid Dynamics for Transient Turbulent Compressible Flows

Martin Sevior; Matthew L. Sims-Goh; Melissa Kozul; Muhammad Usman; Richard D. Sandberg; Shang Xian Matthew Lee

REVIEW

Quantum-Inspired Computational Fluid Dynamics for Transient Turbulent Compressible Flows

Shang Xian Matthew Lee, Melissa Kozul, Muhammad Usman, Martin Sevior, Matthew L. Sims-Goh, Richard D. Sandberg

Fluid Dynamics · Submitted 27 Aug 2026

Pith's verdict: First complete compressible QICFD solver with TT arithmetic, but validation is narrow and an unexplained Mach-dependent error undermines generality.

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CitationMartin Sevior, Matthew L. Sims-Goh, Melissa Kozul, Muhammad Usman, Richard D. Sandberg, Shang Xian Matthew Lee. "Quantum-Inspired Computational Fluid Dynamics for Transient Turbulent Compressible Flows." arXiv:2608.26995 (2026). https://pith.science/paper/DD24R7GC

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Editor's note

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This paper does what it says: it builds the first QICFD solver that handles compressible Navier–Stokes entirely in TT format, including division and square-root via Newton–Raphson. That is a genuine step forward from prior incompressible-only work. The TGV validation at Ma=0.8 shows less than 2% error in enstrophy for most of the simulation, which is credible for a first demonstration. The simultaneous simulation trick is neat—20% overhead for two cases is worth noting. The authors are unusually candid: they disclose the concurrent preprint [59], they admit the late-time error source is unclear, and they show their own rank scaling that limits high-Re applicability.

The soft spots are real but proportionate. The validation rests on a single 32^3 TGV compared only with an in-house solver (HiPSTAR). The Ma=0.1 case shows 4.6% error at late times with no explanation. The stress-test gets this right: if the solver is less accurate in the near-incompressible regime, the claim of general compressible correctness is weakened. The rank analysis is honest but damaging: bond dimension reaches 2700 at Re=1600, which the authors themselves say is 'well out of reach.' No code or data are released.

These limitations do not invalidate the paper. It is a first demonstration, not a production solver. The paper earns a serious referee. I would send it to review, asking for analysis of the Ma=0.1 error, ideally a second benchmark (e.g., shock tube or channel flow), and requesting code/data release as a condition for publication. The work is worth engaging with, but the claims need tightening.

Referee report

Major objections (4)

[Section III.A, Fig. 1] The Ma=0.1 TGV case shows up to 4.6% error in enstrophy at late times, and the text states 'it is currently unclear what the source of the error is.' This unexplained discrepancy weakens the claim of general correctness for compressible flows, especially because the low-Mach case is nearly incompressible. The authors should provide at least a hypothesis (e.g., sensitivity to division tolerance, finite-difference stencil artifacts, or TT truncation effects) and preferably a supporting diagnostic (e.g., error per field component, rank evolution for Ma=0.1 vs Ma=0.8). Without analysis, the reader cannot assess whether the solver is robust across Mach numbers.
[Section III.C, Figs. 4–5] The a-priori rank study shows that χ scales as a power law with Re, reaching ~2700 at Re=1600 for σc=1e-12, which the authors acknowledge is 'well out of reach of current QICFD.' Since the solver's runtime scales as O(N χ^4), this fundamentally limits the method to low-Re or highly truncated (high σc) regimes. The paper should explicitly discuss the practical range of Re and accuracy for which the solver is competitive, and clarify the trade-off between σc and error. As it stands, the central motivation (high-fidelity compressible DNS for industrially relevant Re) is contradicted by the paper's own evidence.
[Section III.A] Validation is performed on a single test case (TGV) at a single resolution (32^3 grid, Re=800). While the TGV is a standard benchmark, the claim of being 'the first complete QICFD solver for compressible DNS' demands broader testing—at minimum a second flow configuration (e.g., channel flow, shock-tube problem, or isotropic turbulence) to demonstrate generalizability. The absence of any spatial or temporal convergence study (e.g., varying grid size or time step) further limits confidence in the solver's numerical properties.
[Section II.C, Appendix B] The QICFD solver closely mimics HiPSTAR, which is developed by the same research group (Sandberg et al., Ref. [41]). While this is a natural choice for a baseline, it raises a risk of hidden common implementation assumptions (e.g., identical finite-difference coefficients, boundary treatment, skew-symmetric splitting constants). The paper should explicitly discuss any differences between the two solvers (e.g., precision, iterative tolerances) and whether the comparison could conceal shared biases.

Minor objections (6)

[Fig. 1] The line styles for QICFD and HiPSTAR are indistinguishable in the printed version (both solid black). Consider using different colors or marker styles.
[Section I, introduction] The sentence 'QICFD outperformed classical CFD in terms of accuracy for the same level of memory use' (citing [22]) is ambiguous—outperformed in what metric? Please specify (e.g., error in kinetic energy).
[Table II] The row for 'TTS-TTO contraction' cites reference [58] (CBC algorithm) but the scaling O(N χ^4) is given without justification. A brief explanation of why χ^4 arises (e.g., three intermediate index contractions) would help the reader.
[Section III.B] The claimed runtime increase of 20% for simultaneous simulation (140 s to 170 s) is based on a single measurement. Provide error bars or replicate runs.
[Appendix B, Eq. (B12)–(B18)] The vectorised form of the Navier-Stokes equations is presented, but the relationship between the conservative variable set (ρ, ρu, ρv, ρw, ρe) and the flux vectors is not fully defined. Specify the closure for pressure (given in Appendix B1) earlier to avoid confusion.
[General] Several minor typos: 'preformed' → 'performed' (Fig. 2 caption); 'Forth' → 'Fourth' (Appendix B, line 2); 'sutherland' → 'Sutherland' (Table I).

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The paper

The paper claims

This work introduces the first complete quantum-inspired CFD solver for compressible-flow DNS, in which every arithmetic operation — including the new division and square-root routines — is carried out in the tensor-train format.

The reading

This work claims to have built the first complete quantum-inspired computational fluid dynamics (QICFD) solver: a direct numerical simulation (DNS) of the compressible Navier–Stokes equations in which every arithmetic operation is carried out in the tensor-train (TT) compressed format, never on the full dense grid. The missing pieces — element-wise division and square-root in TT form — are supplied as Newton–Raphson routines and used to update velocities and to evaluate Sutherland's temperature-dependent viscosity. Validation against the classical solver HiPSTAR on the low-resolution Taylor–Green vortex at Mach 0.8 and 0.1 reports below 0.1% error in total kinetic energy and 0.6% in enstrophy before $t=15$, with the same machinery running two differently-initialised cases simultaneously at about 20% extra cost. If right, this extends quantum-inspired CFD to compressible flows — aerospace, turbomachinery, acoustics — where a memory advantage over classical solvers would matter most. The authors are open about the catch: the test case is poorly resolved by classical standards, and their own rank-scaling study shows the needed bond dimension growing as a power law with Reynolds number, reaching values far beyond the current solver.

What carries the argument

The load-bearing object is the tensor train (TT), a chain of small tensors that stores a discretised field of exponential size $2^N$ with only $O(N𝝧^2)$ parameters, where $𝝧$ is the bond dimension (the largest virtual index) and $N$ counts the binary bits per spatial dimension. The two new TT primitives carry the argument: division reciprocates the divisor via Newton–Raphson, with each iteration using only scalar multiplication, element-wise multiplication, and element-wise addition (operations already available in TT), and the square-root routine iterates using that division; both use the same error metrics as stopping criteria. These complete the arithmetic toolkit, so the compressible Navier–Stokes solver, its Sutherland-law viscosity, the skew-symmetric derivative splitting, RK4 stepping, and post-processing (kinetic energy and enstrophy integrals) all run in TT form. The overall cost is set by tensor-train-state / tensor-train-operator contraction, the first application here of a Cholesky-based compression algorithm, giving $O(N𝝧^4)$ runtime and $O(N𝝧^3)$ memory — the scaling law that decides whether the method ever beats classical CFD; in the validated runs intermediate contractions had to be kept at $𝝧=256$ because truncating them to 128 raised the error.

Extended reading notes

Core claim

The central claim is that the last arithmetic obstacle to compressible quantum-inspired CFD has been removed. Earlier QICFD work either assumed constant viscosity or reconstructed the exponentially large dense state; here, division is done by Newton–Raphson reciprocation seeded with an optimal polynomial approximation of $1/x$, and square roots by a second Newton–Raphson iteration that calls the division routine, so the whole solve stays in TT form. That makes Sutherland's law $μ(T)=T^{3/2}(1+S)/(T+S)$ available in TT format and completes the full compressible Navier–Stokes system with fourth-order finite differences, RK4 time stepping, and skew-symmetric derivative splitting. The validation on the Taylor–Green vortex ($Re=800$, $32^3$ grid, Mach 0.8 and 0.1) is framed as a solver-to-solver agreement test on a deliberately under-resolved case: below 0.1% error in total kinetic energy and 0.6% in enstrophy before $t=15$, growing late in the run to about 2.1% (Mach 0.8) and 4.6% (Mach 0.1), a tail the authors say they cannot yet separate into TT numerical error versus chaotic amplification. A simultaneous-simulation mode runs two differently-initialised cases as one TT ensemble with roughly 20% extra runtime instead of double.

Load-bearing premise

The practical advantage over classical CFD depends on compressible turbulent fields staying low-rank in tensor-train form, but the paper's own a-priori encoding study shows the bond dimension growing as a power law with Reynolds number, reaching roughly $𝝧 \approx 2700$ at $Re=1600$, which it describes as well out of reach of the current solver.

Editorial extensions

If this is right

Compressible applications — high-speed aerodynamics, turbomachinery, aeroacoustics — become legitimate QICFD targets, since division and square-root were the operations blocking them.
The end-to-end workflow, from tensor-cross-interpolation initialisation through time stepping to TT-format kinetic-energy and enstrophy post-processing, never constructs the dense field, preserving the memory-scaling rationale in principle.
Simultaneous simulations of differently-initialised cases run as a single TT problem with only about 20% extra runtime, a scaling advantage over performing the same parameter sweep as separate classical runs.
Users can trade accuracy against resources by raising the singular-value cutoff $σ_c$, since higher cutoffs make the bond dimension grow more slowly with Reynolds number at the price of more truncation error.
The new division and square-root routines are general TT primitives, so any tensor-network application needing those operations — not only CFD — inherits them.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable extension of the simultaneous-simulation idea: run a small ensemble of differently-initialised cases and compare ensemble-averaged statistics against the classical reference, since truncation noise may be washable from statistics even when individual snapshots drift; the paper's cited steady-state work points the same way.
The power-law growth of $𝝧$ with $Re$ implies a practical ceiling for TT-based DNS of compressible turbulence: above some Reynolds number the memory advantage inverts into a disadvantage, so the method's most promising territory may be low-$Re$ or statistically stationary flows rather than the high-$Re$ transient cases that originally motivate it.
A concrete next experiment would be to monitor $𝝧(t)$ during a genuine dynamical solve (not only in a-priori encoding) under looser cutoffs and to identify which error source — singular-value truncation or chaotic divergence — first exceeds a tolerance; that would tell practitioners whether to budget accuracy by bond dimension or by integral metrics such as kinetic energy and enstrophy.

Original abstract

Quantum-inspired algorithms are an emerging class of algorithms for computational fluid dynamics (CFD) with potentially favourable scaling for large problems compared to classical methods. However, their applications have been limited to incompressible flows due to arithmetic limitations, which are addressed in this work. This work introduces the first true end-to-end quantum-inspired computational fluid dynamics (QICFD) solver for direct numerical simulation of the compressible Navier-Stokes equations, that is, all arithmetic operations, as well as initialisation and post-processing, are undertaken in the tensor train (TT) format. Importantly, new division and square-root algorithms using TTs enable the use of Sutherland's law for viscosity, and leverages recent improvements in TT arithmetic algorithms for reduced runtime and memory cost. The new QICFD solver is validated by comparison with the classical CFD solver HiPSTAR and by way of a challenging fluid-flow test case, the low resolution Taylor-Green Vortex (TGV) at Mach numbers of 0.8 and 0.1. The TGV test case is a transient turbulent case that is very sensitive to accumulating errors, yet our QICFD solver achieves excellent agreement with the classical CFD reference. This work demonstrates the correctness of the new TT division and square-root algorithms, and that QICFD is capable of compressible flow simulations. The new QICFD solver is also able to perform simultaneous simulations, running multiple TGV-like cases initialised differently in parallel with reduced extra cost. Finally, the demonstrated TGV test case reveals additional challenges of QICFD as well as highlight the need for future advances to make TT methods viable for industrially-relevant conditions.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 7,142 input tokens.

What the abstract itself states, as probabilities: claims first 0.96 · states limitation 0.96 · human participants 0.01 · controlled comparison 0.79 · randomized assignment 0.01 · states quantitative result 0.45 · compares with prior results 0.96 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Quantum-inspired algorithms are an emerging class of algorithms for computational fluid dynamics (CFD) with potentially favourable scaling for large problems compared to classical methods. However, their applications have been limited to incompressible flows due to arithmetic limitations, which are addressed in this work. This work introduces the first true end-to-end quantum-inspired computational fluid dynamics (QICFD) solver for direct numerical simulation of the compressible Navier-Stokes equations, that is, all arithmetic operations, as well as initialisation and post-processing, are undertaken in the tensor train (TT) format. Importantly, new division and square-root algorithms using TTs enable the use of Sutherland's law for viscosity, and leverages recent improvements in TT arithmetic algorithms for reduced runtime and memory cost. The new QICFD solver is validated by comparison with the classical CFD solver HiPSTAR and by way of a challenging fluid-flow test case, the low resolution Taylor-Green Vortex (TGV) at Mach numbers of 0.8 and 0.1. The TGV test case is a transient turbulent case that is very sensitive to accumulating errors, yet our QICFD solver achieves excellent agreement with the classical CFD reference. This work demonstrates the correctness of the new TT division and square-root algorithms, and that QICFD is capable of compressible flow simulations. The new QICFD solver is also able to perform simultaneous simulations, running multiple TGV-like cases initialised differently in parallel with reduced extra cost. Finally, the demonstrated TGV test case reveals additional challenges of QICFD as well as highlight the need for future advances to make TT methods viable for industrially-relevant conditions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Quantum-inspired algorithms are an emerging class of algorithms for computational fluid dynamics (CFD) with potentially favourable scaling for large problems compared to classical methods.	background 1.00	0.06	0.04	hedged possibility 0.86
s1	However, their applications have been limited to incompressible flows due to arithmetic limitations, which are addressed in this work.	gap 0.96	0.11	0.04	asserted 1.00
s2	This work introduces the first true end-to-end quantum-inspired computational fluid dynamics (QICFD) solver for direct numerical simulation of the compressible Navier-Stokes equations, that is, all arithmetic operations, as well as initialisation and post-processing, are undertaken in the tensor train (TT) format.	contribution 0.97	0.95	0.20	asserted 1.00
s3	Importantly, new division and square-root algorithms using TTs enable the use of Sutherland's law for viscosity, and leverages recent improvements in TT arithmetic algorithms for reduced runtime and memory cost.	method 0.64	0.89	0.10	asserted 1.00
s4	The new QICFD solver is validated by comparison with the classical CFD solver HiPSTAR and by way of a challenging fluid-flow test case, the low resolution Taylor-Green Vortex (TGV) at Mach numbers of 0.8 and 0.1.	method 0.84	0.93	0.09	asserted 1.00
s5	The TGV test case is a transient turbulent case that is very sensitive to accumulating errors, yet our QICFD solver achieves excellent agreement with the classical CFD reference.	result 0.98	0.96	0.18	asserted 1.00
s6	This work demonstrates the correctness of the new TT division and square-root algorithms, and that QICFD is capable of compressible flow simulations.	interpretation 0.59	0.95	0.70	asserted 1.00
s7	The new QICFD solver is also able to perform simultaneous simulations, running multiple TGV-like cases initialised differently in parallel with reduced extra cost.	result 0.59	0.93	0.07	asserted 0.99
s8	Finally, the demonstrated TGV test case reveals additional challenges of QICFD as well as highlight the need for future advances to make TT methods viable for industrially-relevant conditions.	limitation 0.56	0.90	0.12	asserted 0.96

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Taylor-Green vortex validation test case results,

**Figure 2.** FIG. 2. (a) Simultaneous simulation validation results show

**Figure 3.** FIG. 3. Runtime and memory use per step for a

**Figure 4.** shows the effects of Re on χ. The simulation parameters are Ma = 0.8, ∆t = 0.001, L = 2π in all dimensions, and 5123 grid size. The Reynolds numbers are Re ∈ {200, 400, 600, 800, 1200, 1600}. For all Re considered, these simulation parameters are considered overresolved in classical CFD. Relative singular value cutoffs σc ∈ {10−6 , 10−9 , 10−12} are used during TTS encoding. For the TGV case, χ of the problem scales as a power law with Re, indicating high Re compressible TGV simulations may prove difficult for QICFD. In contrast, large computational grid simulations of low Re…

**Figure 5.** shows the χ evolution over time for the Ma = 0.8 TGV case at Re ∈ {800, 1600}. χ increases over time until around t = 12 and subsequently decreases. For QICFD, the peak χ informs the maximum resource use for a simulation. The general trend shows that χ starts extremely low, since the TGV initial condition has a low χ structure. χ subsequently increases as the flow becomes more turbulent and disordered, and generally peaks just over half way through the simulation, where turbulent structures are abundant, and finally subsides as the turbulence decays. Since the TGV case passes throug

**Figure 6.** FIG. 6. Finite difference TTO. The TT tensors between

**Figure 7.** FIG. 7. Index labelling of finite difference TTO tensor

Figures reproduced from arXiv:2608.26995 by the authors.

References

Forward citations

Cited by 2 Pith papers

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

Multivariate quantum state preparation with optimized tensor networks
quant-ph 2026-09 accept novelty 7.0 of 10

SCENT uses spectral clustering on pairwise entanglement metrics to discover optimal tree tensor network structures, enabling high-fidelity quantum state preparation of multivariate functions with greatly reduced circu...
Breaking the chain: geometry-native state preparation with ASPIRE
quant-ph 2026-10 conditional novelty 6.0 of 10

A greedy entanglement-guided gate-placement algorithm prepares matrix product states with shallower, higher-fidelity circuits than the sequential staircase when the state has long-ranged correlations.

How the literature uses it

What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 24 sentences without a confident label

Uses the method or tool of 1 paper

The suggested metric for a termination condition is [equation] where $||𝐱 ||$ is the norm of the TTS, and $sum(𝐃)$ is the sum of all elements of the TTS, which can be calculated using the same method as integration of a TTS [ref] .

Quantum-inspired algorithms for multivariate analysis: from interpolation to partial differential equations · Tensor Train Arithmetic Algorithms · 0.90

Cites as support 1 paper

[ref] showed that QICFD dynamical simulations with steady statistics at a low $χ$ settings are still possible, where even if instantaneous snapshots are inaccurate, the overall statistics of the simulation are still correct.

Quantum-Inspired Simulation of 2D Turbulent Rayleigh-B\'enard Convection · Discussion and Conclusion · 0.97

Cites as background 9 papers

Not all operations can be generalised to this form, and previous work has provided routines for basic arithmetic such as addition, multiplication, and applying linear operators [ref] , and more advanced operations such as Fourier transforms, finite differences and integration [ref] .

Fundamental Tensor Operations for Large-Scale Data Analysis in Tensor Train Formats · Introduction · 0.81
Not all operations can be generalised to this form, and previous work has provided routines for basic arithmetic such as addition, multiplication, and applying linear operators [ref] , and more advanced operations such as Fourier transforms, finite differences and integration [ref] .

Quantum-inspired algorithms for multivariate analysis: from interpolation to partial differential equations · Introduction · 0.81
Tensor network methods are an enormously successful tool for quantum simulation, and have played a crucial role in quantum science for decades [ref] .

Stabilizer Tensor Networks with Magic State Injection · Introduction · 0.96

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