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arxiv: 2606.12770 · v1 · pith:GZQPZLTZnew · submitted 2026-06-11 · 🪐 quant-ph · physics.flu-dyn

Explicit Quantum Circuit Simulation of Nonlinear 1-Dimensional Fluid with Carleman-linearized Boltzmann Method

Keita Kanno , Kazumasa Ueno , Hayato Higuchi , Morimasa Okamoto , Yuya Yoshizuru , Ryoya Ishimaru , Towa Takagi , Kentaro Sakamoto This is my paper

Pith reviewed 2026-06-27 07:00 UTC · model grok-4.3

classification 🪐 quant-ph physics.flu-dyn
keywords quantum circuit simulationCarleman linearizationBoltzmann equationlattice Boltzmann methodquantum singular value transformationnonlinear fluid dynamicsone-dimensional flowTaylor ODE solver
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The pith

Second-order Carleman linearization of the one-dimensional Boltzmann equation lets quantum circuits prepare the final-time state of a nonlinear fluid via a QSVT-based Taylor ODE solver.

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

The paper establishes that the leading nonlinearity in one-dimensional fluid flow can be captured by truncating the Carleman expansion of the Boltzmann equation at second order, converting the problem into a linear system that a quantum computer can solve. It then shows an explicit construction that uses a Taylor-expansion ODE integrator implemented through quantum singular value transformation to prepare the state at a chosen final time. The construction is demonstrated through direct circuit simulation rather than asymptotic analysis alone. A sympathetic reader would care because prior quantum fluid work stayed in the linear regime, and this step supplies a concrete baseline for gate counts, qubit counts, and further extensions while keeping the scaling logarithmic in grid size.

Core claim

By applying second-order Carleman linearization to the one-dimensional Boltzmann equation, the nonlinear fluid problem is turned into a linear system whose time evolution is solved by a Taylor-expansion-based ODE solver realized with quantum singular value transformation; explicit gate-level circuit simulation then prepares the final-time state, and the resulting gate and qubit complexities are shown to scale logarithmically with grid size, with the order of the Carleman truncation controlling the captured nonlinearity.

What carries the argument

second-order Carleman linearization of the one-dimensional Boltzmann equation together with a Taylor-expansion ODE solver implemented via quantum singular value transformation

If this is right

  • Gate and qubit counts scale logarithmically with grid size.
  • Higher-order Carleman expansions capture additional nonlinearity at the cost of larger but still logarithmic resources.
  • The same construction supplies a baseline for reducing computational cost in future work.
  • Extensions to higher dimensions, complex geometries, and extraction of physical quantities become concrete next targets once the one-dimensional nonlinear case is established.

Where Pith is reading between the lines

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

  • The logarithmic scaling suggests that, if the truncation error remains controlled, the method could remain practical even when the spatial grid is refined to resolve finer flow features.
  • Because the circuit is built from explicit elementary gates, one could in principle substitute a different linear-system solver or a different time integrator and measure the change in total gate count directly.
  • If the second-order truncation proves accurate enough for selected benchmark flows, the same linearization step might be reusable as a modular component inside larger quantum CFD pipelines that add forcing terms or boundary conditions.

Load-bearing premise

The second-order truncation of the Carleman expansion produces a sufficiently accurate linear system whose solution on the quantum circuit corresponds to the true nonlinear fluid dynamics at the chosen grid resolution and time step.

What would settle it

Compare the final-time probability distribution or moments obtained from the explicit quantum circuit simulation against the output of a classical nonlinear lattice Boltzmann solver run at identical grid size, time step, and truncation order; systematic deviation beyond statistical error would falsify the claim that the quantum procedure reproduces the target nonlinear dynamics.

Figures

Figures reproduced from arXiv: 2606.12770 by Hayato Higuchi, Kazumasa Ueno, Keita Kanno, Kentaro Sakamoto, Morimasa Okamoto, Ryoya Ishimaru, Towa Takagi, Yuya Yoshizuru.

Figure 1
Figure 1. Figure 1: FIG. 1. Time-evolution matrix [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Block-encoding circuit used in this work (Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Oracle circuit [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Oracle circuit [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Oracle circuit [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Block encoding circuit [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. QSVT circuit for polynomial transformation of the block-encoded [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Continuous BGK reference flow at [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Relative [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Final-time pointwise [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Final-time density profile (top row) and residual relative to the continuous BGK reference (bottom row) at ∆ [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Effect of Taylor truncation order [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Baseline effective condition number [PITH_FULL_IMAGE:figures/full_fig_p014_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Effective condition number [PITH_FULL_IMAGE:figures/full_fig_p015_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Effective condition number [PITH_FULL_IMAGE:figures/full_fig_p016_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. QSVT-based multi-step time evolution for [PITH_FULL_IMAGE:figures/full_fig_p017_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. QSVT [PITH_FULL_IMAGE:figures/full_fig_p019_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. Gate count of the block-encoding circuit [PITH_FULL_IMAGE:figures/full_fig_p020_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. Gate count of the propagator [PITH_FULL_IMAGE:figures/full_fig_p020_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. Qubit count scaling with [PITH_FULL_IMAGE:figures/full_fig_p021_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23. Total gate count per QSVT call as a function of [PITH_FULL_IMAGE:figures/full_fig_p021_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24. Coupling-scale verification [PITH_FULL_IMAGE:figures/full_fig_p024_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: FIG. 25. Density profile (top) and pointwise residual (bot [PITH_FULL_IMAGE:figures/full_fig_p025_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: FIG. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_26.png] view at source ↗
read the original abstract

Quantum computation of fluid dynamics has attracted growing attention as a key application of fault-tolerant quantum computers anticipated in the coming decade, with lattice Boltzmann methods emerging as a particularly promising approach. Explicit and efficient elementary-gate-level circuit simulations, however, have so far been demonstrated only in the linear case. Here we include the leading nonlinearity through second-order Carleman linearization of the one-dimensional Boltzmann equation, and demonstrate, via explicit quantum-circuit simulation, the preparation of the final-time state using a Taylor-expansion-based ODE solver based on the quantum singular value transformation. With this construction, we analyze the gate and qubit complexities, which scale logarithmically with the grid size, the nonlinearity captured by the higher-order Carleman linearization, and the practical utility of higher-order expansions in the Taylor ODE solver. The construction provides a concrete baseline for computational cost reduction and further developments such as extensions to higher dimensions, complex geometries, and the extraction of physical quantities, towards industrially useful quantum CFD.

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 / 0 minor

Summary. The manuscript constructs an explicit quantum circuit for simulating nonlinear one-dimensional fluid dynamics by applying second-order Carleman linearization to the Boltzmann equation and implementing a Taylor-expansion-based ODE solver via quantum singular value transformation (QSVD). It demonstrates preparation of the final-time state through explicit circuit simulation and analyzes the resulting gate and qubit complexities, which are claimed to scale logarithmically with grid size and the order of the Carleman expansion.

Significance. If the second-order Carleman truncation is shown to be sufficiently accurate, the work supplies a concrete, explicitly simulated baseline for quantum-circuit costs in nonlinear CFD with logarithmic scaling, which could inform extensions to higher dimensions and extraction of physical observables. The explicit simulation and complexity analysis are positive features.

major comments (2)
  1. [Abstract] Abstract: the central claim that the quantum circuit prepares a state corresponding to the nonlinear fluid dynamics rests on the second-order Carleman truncation producing a sufficiently accurate linear system, yet no a-priori error bound on the Carleman remainder is supplied and no direct numerical comparison against an independent classical integrator of the original nonlinear equation is reported.
  2. [Abstract] Abstract: the manuscript states the construction and scaling but supplies no numerical verification, error analysis, or comparison against a classical nonlinear solver, so the practical fidelity of the simulated state to the true nonlinear dynamics cannot be assessed from the provided material.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the quantum circuit prepares a state corresponding to the nonlinear fluid dynamics rests on the second-order Carleman truncation producing a sufficiently accurate linear system, yet no a-priori error bound on the Carleman remainder is supplied and no direct numerical comparison against an independent classical integrator of the original nonlinear equation is reported.

    Authors: We agree that the manuscript does not supply an a-priori analytic bound on the Carleman remainder or a direct numerical comparison to a classical nonlinear integrator. The work centers on the explicit quantum-circuit construction and logarithmic complexity scaling for the second-order Carleman-linearized system; validation of truncation accuracy was outside the stated scope. In revision we will add a dedicated numerical section that compares the second-order Carleman solution against an independent classical nonlinear Boltzmann solver on the same 1-D test problems, together with a discussion of observed truncation error. revision: yes

  2. Referee: [Abstract] Abstract: the manuscript states the construction and scaling but supplies no numerical verification, error analysis, or comparison against a classical nonlinear solver, so the practical fidelity of the simulated state to the true nonlinear dynamics cannot be assessed from the provided material.

    Authors: This observation is correct. The explicit circuit simulation and scaling analysis are performed on the Carleman-linearized ODE; no direct fidelity check against the original nonlinear dynamics is reported. We will revise the manuscript to include such numerical verification and error assessment so that readers can evaluate the practical accuracy of the second-order truncation for the chosen test cases. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper constructs an explicit quantum circuit for the Carleman-linearized 1D Boltzmann equation via a Taylor-expansion ODE solver based on QSVD. All steps are presented as direct applications of established primitives (Carleman truncation, QSVD, Taylor series) without any reduction of the central claim to a fitted parameter, self-citation load-bearing premise, or input-by-construction. The derivation remains self-contained; external validation of truncation accuracy is a separate correctness issue, not a circularity defect.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on the domain assumption that Carleman linearization of order two is adequate for the target fluid problem and on standard properties of QSVD and Taylor ODE integration; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Second-order Carleman linearization sufficiently approximates the nonlinear Boltzmann collision term for the 1D fluid problem
    Invoked to convert the nonlinear equation into a linear system that can be solved by quantum linear algebra methods.

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discussion (0)

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Reference graph

Works this paper leans on

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    Figure 4 shows the circuit structure of theA 12 oracle, and Fig

    Second-order block encoding (U A forN C = 2) At second-order Carleman truncation, the block en- coding must implement the full rate matrix (11) with three nontrivial blocks: the diagonal blocksA 11 and A22 =A 11 ⊗I+I⊗A 11, and the off-diagonal block A12 encoding the nonlinear collision coupling fromf (2) tof (1). Figure 4 shows the circuit structure of th...

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    Block encoding ofL The block encodingU L of the time-evolution matrix L(Fig. 1) is implemented using theA-matrix oracles of Sec. II D 2 and II D 3 as controlled subroutines. The block encoding ofLrequires a Taylor-index regis- ter|·⟩ k ofn k =⌈log 2(NK+1)⌉qubits, a time-step register |·⟩m ofn m =⌈log 2(Nt)+1⌉qubits, and a composite row- label register who...

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    Dependence onN x andN t Figure 13 showsκ(L) atN C = 1,N K = 1 as a func- tion ofN t andN x. At fixedN t (right panel),κ(L) de- cays with increasingN x to a constant value. This satu- ration reflects theN x dependence of the spectral radius µ(A) of the rate matrix (Appendix D):µ(A)∝1/N x when the collision rate 1/τdominates (smallN x, via Eq. (18)), andµ(A...

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    Qubit count Table I summarizes the input/output (I/O) qubit count of each level of the algorithm; the total qubit counts including ancillary qubits are visualised in Fig. 22. Both panels confirm the expected logarithmic growth with the grid sizeN x, with the first-order Carleman im- plementation requiring half as many spatial qubits as the second-order on...

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