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arXiv:2606.12770

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

quant-phphysics.flu-dynSubmitted Jun 11, 2026• Announced Jun 12, 2026• 0 citations
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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.

Comments
27 pages, 26 figures

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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.
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.
Computational fluid dynamics (CFD) is a cornerstone of classical scientific computing, and there is growing interest in whether quantum computers can accelerate such simulations. The existing proposals for fault-tolerant quantum algorithms for CFD have almost exclusively been based on the Carleman embedding method, used to encode nonlinearities on a quantum computer. Here, we first identify several key challenges that must be addressed in order to establish quantum advantage using this method, including Carleman convergence, time-stepping requirements, condition-number scaling, and data extraction. Building on this analysis, we develop a novel algorithm for the incompressible lattice Boltzmann equation that addresses them, and provide its detailed analysis, including all potential sources of algorithmic complexity, as well as gate count estimates. We find that for an end-to-end problem, a modest quantum advantage may be preserved for selected observables in the high-error-tolerance regime. We establish a lower bound for the Reynolds number scaling of our quantum algorithm in dimension DDD at Kolmogorov microscale resolution with O(Re34(1+D2)×qM)O(\mathrm{Re}^{\frac{3}{4}(1+\frac{D}{2})} \times q_M)O(Re43​(1+2D​)×qM​), where qMq_MqM​ is a multiplicative overhead for data extraction with qM=O(Re38)q_M = O(\mathrm{Re}^{\frac{3}{8}})qM​=O(Re83​) for the drag force. This upper bounds the scaling improvement over classical algorithms by O(Re3D8)O(\mathrm{Re}^{\frac{3D}{8}})O(Re83D​). However, our numerical investigations suggest a lower speedup, with a scaling estimate of O(Re1.936×qM)O(\mathrm{Re}^{1.936} \times q_M)O(Re1.936×qM​) for D=2D=2D=2. Our results give robust evidence that small, but nontrivial, quantum advantages can be achieved in the context of CFD, and motivate the need for additional rigorous end-to-end quantum algorithm development.
Computational fluid dynamics (CFD) is a specialized branch of fluid mechanics that utilizes numerical methods and algorithms to solve and analyze fluid-flow problems. One promising avenue to enhance CFD is the use of quantum computing, which has the potential to resolve nonlinear differential equations more efficiently than classical computers. Here, we try to answer the question of which regimes of nonlinear partial differential equations for fluid dynamics can have an efficient quantum algorithm. We propose a connection between the numerical parameter R , which guarantees efficiency in the truncation of the Carleman linearization, and the physical parameters that describe the fluid flow. This link can be made thanks to the Kolmogorov scale, which determines the minimum size of the grid needed to properly resolve the energy cascade induced by the nonlinear term. Additionally, we introduce the formalism for vector field simulation in different spatial dimensions, providing the discretization of the operators and the boundary conditions.
Numerical simulation of turbulent fluid dynamics needs to either parametrize turbulence—which introduces large uncertainties—or explicitly resolve the smallest scales—which is prohibitively expensive. Here, we provide evidence through analytic bounds and numerical studies that a potential quantum speedup can be achieved to simulate fluid dynamics using quantum computing. Specifically, we provide a lattice Boltzmann formulation of fluid dynamics for which we give evidence that low-order Carleman linearization is much more accurate than previously believed for these systems. This is achieved via a combination of reformulating the Navier-Stokes nonlinearity ( u · ∇ u ) to lattice-Boltzmann nonlinearity ( u 2 ) and accurately linearizing the dynamical equations, which effectively trades nonlinearity for additional degrees of freedom that add negligible expense in the quantum solver. Based on this, we apply a quantum algorithm for simulating the Carleman-linearized lattice Boltzmann equation and provide evidence that its cost scales logarithmically with system size compared with polynomial scaling in the best known classical algorithms. In this paper, we suggest that a quantum advantage may exist for simulating fluid dynamics, paving the way for simulating nonlinear multiscale transport phenomena in a wide range of disciplines using quantum computing.
Nonlinear ordinary and partial differential equations are ubiquitous in science and engineering, yet finding their solutions is often computationally intractable for classical hardware. To determine if quantum computers can offer a practical advantage, one critical challenge that must be solved is determining how to efficiently load exponentially sized matrices onto quantum hardware. In this article, we introduce an alternative linear combination of unitaries (LCU) strategy which relies on an intermediate linear combination of non-unitaries (LCNU) and a systematic embedding procedure. One advantage of this LCU strategy is that it maintains the exact number of terms as in the LCNU. Therefore, this approach offers a data loading framework for matrices that lack an efficient decomposition using the standard LCU alone. Using this approach, we construct a generalized LCNU framework for any Carleman linearized autonomous dynamical system having a polynomial nonlinearity. To demonstrate the effectiveness of our approach, we construct an LCNU for the 3D Carleman linearized lattice Boltzmann equation (LBE). Here, we find that the number of terms in the decomposition scales like Ns∼O(α2Q2)N_s\sim\mathcal{O}(α^2Q^2)Ns​∼O(α2Q2), where ααα is the Carleman truncation order and QQQ is the number of discrete velocities. Importantly, NsN_sNs​ is independent of the number of spatial and temporal discretization points. We then perform a resource estimation of our LCNU's T gate cost when combined with the (1) PREP and SELECT block encoding oracles, and (2) variational quantum linear solver. In the former, the T cost scales like O(α3Q2(log⁡2n)2)\mathcal{O}(α^3Q^2(\log_2n)^2)O(α3Q2(log2​n)2), where nnn is the total number of spatial grid points. The latter requires exactly Ns2(log⁡2(2ntnα)+1)N_s^2(\log_2 (2n_tn^α)+1)Ns2​(log2​(2nt​nα)+1) circuits per iteration for ntn_tnt​ time steps, with a worst case T gate cost of O(α(log⁡2Qn)2)\mathcal{O}(α(\log_2Qn)^2)O(α(log2​Qn)2) among them.
The implementation of quantum algorithms for the simulation of classical fluid dynamics poses a fundamental challenge due to the nonlinearity of the fluid equations. In this work, we provide a pedagogical introduction to quantum computing algorithms for simulating classical fluids, with a special focus on the Carleman-Lattice Boltzmann algorithm, which has captured significant attention in the last couple of years. While this algorithm demonstrates satisfactory convergence to analytical solutions for systems at low-to-moderate Reynolds numbers, it also shows an exponential depth of the corresponding quantum circuit. As a result much further analysis is needed to assess the availability of the Carleman-Lattice Boltzmann method on a quantum computer.