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arxiv: 2605.28135 · v1 · pith:AIX5CQ64new · submitted 2026-05-27 · 🪐 quant-ph · physics.flu-dyn

A Demonstration of Quantum Circuit Implementation for Obstacle Flow Using Carleman-Linearized Lattice Boltzmann Method

Kazumasa Ueno , Keita Kanno , Yasunori Lee This is my paper

Pith reviewed 2026-06-29 12:24 UTC · model grok-4.3

classification 🪐 quant-ph physics.flu-dyn
keywords quantum computinglattice Boltzmann methodCarleman linearizationquantum singular value transformationcomputational fluid dynamicsblock-encodingfluid simulation
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The pith

Quantum circuits for fluid flow around an obstacle achieve logarithmic scaling in qubits and gates with lattice size.

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

This paper implements a quantum algorithm for two-dimensional fluid flow around an obstacle by applying Carleman linearization to the lattice Boltzmann method, turning the nonlinear problem into a linear system. The linear system is solved on a quantum computer through block-encoding of the system matrix followed by quantum singular value transformation. Inflow, outflow, and no-slip boundary conditions are expressed as sparse matrix operations and embedded using index-value encoding. The resulting circuits require numbers of qubits and gates that grow only logarithmically with the number of lattice points. This scaling is presented as evidence that quantum computers could eventually address fluid-dynamics problems that remain expensive on classical hardware.

Core claim

We implement a quantum algorithm for two-dimensional linearized fluid flow around an obstacle, using block-encoding of the linear-system matrix and quantum singular value transformation (QSVT) to solve it. Inflow, outflow, and no-slip boundary conditions are formulated as sparse matrix operations and efficiently embedded into quantum circuits using index-value encoding. We demonstrate logarithmic scaling of the required numbers of qubits and gates with respect to the number of lattice points, suggesting the potential feasibility of quantum-computational fluid dynamics simulations.

What carries the argument

Block-encoding of the Carleman-linearized lattice Boltzmann matrix combined with quantum singular value transformation to solve the linear system, with boundary conditions incorporated as sparse matrix operations via index-value encoding.

If this is right

  • Quantum treatment of computational fluid dynamics becomes potentially feasible because qubit and gate counts do not grow linearly with lattice resolution.
  • Non-periodic boundary conditions can be handled inside quantum circuits for fluid problems without destroying the logarithmic scaling.
  • The same linearization-plus-QSVT pipeline applies to other two-dimensional flow configurations once the matrix is constructed.
  • High-Reynolds-number regimes remain accessible provided the Carleman truncation order is chosen appropriately.

Where Pith is reading between the lines

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

  • If the logarithmic scaling survives noise and error correction on actual hardware, quantum computers could eventually simulate flows at resolutions unreachable by classical machines.
  • The approach could be extended to three-dimensional flows or time-dependent problems by increasing the dimension of the linearized operator while preserving the encoding strategy.
  • Practical utility will depend on how the truncation error in Carleman linearization behaves at the Reynolds numbers of interest.

Load-bearing premise

Carleman linearization of the lattice Boltzmann method gives a sufficiently accurate model of the nonlinear fluid dynamics for obstacle flow, and boundary conditions can be embedded as sparse matrix operations without prohibitive overhead or errors.

What would settle it

Execute the quantum circuit on successively larger lattices and check whether measured qubit and gate counts follow the claimed logarithmic dependence, or compare the circuit output for a fixed lattice against a classical solver applied to the identical linearized system.

Figures

Figures reproduced from arXiv: 2605.28135 by Kazumasa Ueno, Keita Kanno, Yasunori Lee.

Figure 1
Figure 1. Figure 1: High-level overview of the quantum lattice Boltzmann method. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Unit discrete velocity vectors of the D2Q9 lattice. The vectors [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Halfway bounce-back boundary condition at a stationary wall. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Quantum circuit for UA(1) . OsetBC. This oracle computes one of the following three boundary-condition types: • Bounce-back (|BC⟩ = |11⟩) for ◦ nodes at the left boundary (nx = 0) with a left-pointing post-collision velocity. ◦ nodes at the top (ny = Ny − 1) or bottom (ny = 0) boundaries with velocities pointing into the wall. ◦ nodes adjacent to the obstacle with velocities pointing into it. Since inflow … view at source ↗
Figure 5
Figure 5. Figure 5: Quantum circuit for OsetBC. Each depicted gate actually consists of multiple multi-controlled NOT gates (controlled on each boundary condition). where, recalling Eq. (2.20), Cq ∗q =  1 − 1 τ  δq ∗q + wq ∗ τ  1 + cq · cq ∗ c 2 s  . (2.39) When Cq ∗q < 0, a controlled-Z gate needs to be additionally applied to correct the sign. For the outflow boundary (|BC⟩ = |10⟩, taken here as the right boundary), the… view at source ↗
Figure 6
Figure 6. Figure 6: Quantum circuit for Ocollision. Each depicted gate actually consists of multiple multi-controlled rotation gates. Ostreaming. This oracle first swaps the pre/post-collision velocity registers |q⟩ and |q ∗ ⟩. The streaming operations are then applied to the pre-collision velocity register |q ∗ ⟩ q (encoding the post-collision velocity q ∗ after the swap) and the node register, controlled on BC: 17 [PITH_FU… view at source ↗
Figure 7
Figure 7. Figure 7: Quantum circuit for Ostreaming. 18 [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Quantum circuit for UL. Here, O−hA(1) is the inner part of the block-encoding circuit UA(1) in [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: A quantum circuit for the QSVT-based QLSA. The ancilla register [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Flow configuration for the 2D channel with a rectangular obstacle. The gray [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Spectral properties of the matrix L as a function of the total simulation time T = Nt · h, for various grid sizes Nx (= Ny). 3.2 Simulation results of flow around an obstacle We validate the proposed QLBM through two complementary approaches. First, we per￾form a state-vector simulation of the full quantum circuit on a small problem instance and compare it with a classical time-stepping simulation to conf… view at source ↗
Figure 12
Figure 12. Figure 12: State-vector simulation results of the QLBM [PITH_FULL_IMAGE:figures/full_fig_p027_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Accuracy of the Clenshaw-based solution with respect to the linear reference as [PITH_FULL_IMAGE:figures/full_fig_p030_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Velocity fields at t = 25 (nt = 50) for Nx = Ny = 32, T = 64. 28 [PITH_FULL_IMAGE:figures/full_fig_p030_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Quantum gate counts of UL [PITH_FULL_IMAGE:figures/full_fig_p031_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Qubit counts of UL. The ancilla qubit counts are identical for all Nt values. 29 [PITH_FULL_IMAGE:figures/full_fig_p031_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Estimated total T-gate count for a single QSVT-based linear system solve as [PITH_FULL_IMAGE:figures/full_fig_p033_17.png] view at source ↗
read the original abstract

Fluid simulations, especially at high Reynolds numbers, are computationally expensive on classical computers, making them promising application targets for quantum computing. Recent studies have combined the lattice Boltzmann method (LBM) with Carleman linearization to design quantum algorithms for computational fluid dynamics (CFD). However, practical quantum-circuit implementations of these algorithms that incorporate non-periodic boundary conditions have not been fully explored. In this work, we implement a quantum algorithm for two-dimensional linearized fluid flow around an obstacle, using block-encoding of the linear-system matrix and quantum singular value transformation (QSVT) to solve it. Inflow, outflow, and no-slip boundary conditions are formulated as sparse matrix operations and efficiently embedded into quantum circuits using index-value encoding. We demonstrate logarithmic scaling of the required numbers of qubits and gates with respect to the number of lattice points, suggesting the potential feasibility of quantum-computational fluid dynamics simulations.

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 / 1 minor

Summary. The manuscript presents a quantum circuit implementation for 2D obstacle flow simulation via the Carleman-linearized lattice Boltzmann method. Boundary conditions are encoded as sparse matrices and embedded using index-value encoding; the linear system is solved with block-encoding plus QSVT. The central claim is a demonstration of logarithmic scaling in both qubit count and gate count with respect to the number of lattice points N.

Significance. If the reported gate scaling holds after accounting for all factors in the QSVT implementation, the work would supply a concrete circuit-level example of handling non-periodic boundaries in a quantum CFD algorithm, extending prior Carleman-LBM literature. The absence of supporting numerical data or condition-number analysis currently limits its immediate utility as a feasibility demonstration.

major comments (2)
  1. [Abstract and scaling results] Abstract and scaling results: the claim of logarithmic gate scaling with N does not address the QSVT query complexity O(κ/ε), where κ is the condition number of the Carleman-linearized operator. No analysis or data is supplied showing that κ remains independent of lattice resolution or Reynolds number, so the reported gate-count scaling does not follow from the construction.
  2. [Results section] Results section: the abstract states that logarithmic scaling is demonstrated, yet the manuscript supplies no numerical verification, error bounds, circuit-depth measurements, or comparison against classical solvers, leaving the central empirical claim without concrete supporting data.
minor comments (1)
  1. Clarify whether the reported gate counts include the full QSVT circuit (including block-encoding overhead) or only the query complexity; add explicit statements on the assumed precision ε and any N-dependence of κ.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major point below and will revise the manuscript accordingly to strengthen the presentation of our scaling claims and supporting analysis.

read point-by-point responses

  1. Referee: [Abstract and scaling results] Abstract and scaling results: the claim of logarithmic gate scaling with N does not address the QSVT query complexity O(κ/ε), where κ is the condition number of the Carleman-linearized operator. No analysis or data is supplied showing that κ remains independent of lattice resolution or Reynolds number, so the reported gate-count scaling does not follow from the construction.

    Authors: We agree that the overall complexity of the algorithm includes the QSVT factor O(κ/ε). The logarithmic scaling we report applies specifically to the qubit count and the gate count for the block-encoding construction and the index-value encoding of the sparse boundary matrices, both of which are O(log N) with respect to the number of lattice points. The dependence of the condition number κ on lattice resolution and Reynolds number is not analyzed in the current work. We will add a dedicated discussion of this point, including references to known bounds for Carleman-linearized operators, in the revised manuscript. revision: yes

  2. Referee: [Results section] Results section: the abstract states that logarithmic scaling is demonstrated, yet the manuscript supplies no numerical verification, error bounds, circuit-depth measurements, or comparison against classical solvers, leaving the central empirical claim without concrete supporting data.

    Authors: The demonstration is analytical and derives directly from the explicit circuit construction: the index-value encoding embeds the boundary conditions with O(log N) qubits and gates, and the block-encoding of the Carleman-linearized operator likewise scales logarithmically. No numerical circuit simulations are provided because the manuscript focuses on the feasibility of the circuit-level implementation rather than empirical runtime benchmarks. We will incorporate theoretical error bounds from the QSVT analysis and a brief comparison of asymptotic scaling against classical methods in the revised results section. revision: partial

Circularity Check

0 steps flagged

No circularity; scaling follows from standard sparse block-encoding construction

full rationale

The paper implements block-encoding of the Carleman-linearized LBM matrix (with embedded boundaries as sparse operations) and applies QSVT. Logarithmic qubit/gate scaling with lattice points N is the direct, standard consequence of index encoding an O(N)-sized sparse matrix, not a fitted parameter, self-definition, or self-citation chain that reduces the claim to its own inputs. No quoted equations or steps exhibit the enumerated circular patterns. The derivation remains self-contained against external quantum linear-algebra benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the accuracy of Carleman linearization for the target flow and the efficient embeddability of boundary conditions; no free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption Carleman linearization of LBM yields a linear system whose solution accurately represents the fluid flow around an obstacle
    Invoked to justify the quantum algorithm for the nonlinear problem.
  • domain assumption Inflow, outflow, and no-slip boundaries can be formulated as sparse matrix operations efficiently embedded via index-value encoding
    Stated directly in the abstract as the mechanism for handling boundaries.

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Forward citations

Cited by 2 Pith papers

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

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

    quant-ph 2026-06 unverdicted novelty 7.0

    Explicit quantum-circuit simulation of nonlinear 1D fluid via second-order Carleman-linearized Boltzmann equation and QSVD Taylor ODE solver, with logarithmic scaling analysis.

  2. Efficient and Expressive Boundary Conditions in Quantum Lattice Boltzmann Methods

    quant-ph 2026-05 unverdicted novelty 6.0

    New boundary condition approach for QLBM using one coherent operation on the full boundary, claimed to use fewer resources asymptotically and practically for bounce-back and specular reflection.

Reference graph

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