In this work, a quantum Taylor series expansion and Least squares-based Fractional-step Lattice Boltzmann Method (TLFS-LBM) is proposed to simulate incompressible flows on non-uniform meshes. Most existing quantum LBMs face two critical challenges: with the relaxation time τ fixed at 1, a fixed mesh resolution corresponds to only a single Reynolds number, and the streaming step, typically implemented through the quantum walk method, is limited to uniform meshes. To address these issues, we retain τ = 1 and decompose the quantum TLFS-LBM into predictor and corrector steps using a fractional-step method. The predictor step is implemented on a quantum circuit, but the resulting viscosity generally differs from the physical viscosity. A diffusion equation is therefore solved classically in the corrector step to restore the correct physical viscosity. Moreover, the streaming step is expressed in the form of A · f eq using Taylor series expansion and least-squares approximation, with singular value decomposition applied to factorize matrix A into unitary matrices for quantum implementation. Since A depends solely on mesh point coordinates, the proposed method is applicable to arbitrary mesh types. Validation through two-dimensional incompressible isothermal and thermal flow simulations confirms that the quantum TLFS-LBM achieves accuracy comparable to its classical counterpart, while extending the applicability of quantum LBM to non-uniform meshes and curved boundaries.
In this work, a quantum Taylor series expansion and Least squares-based Fractional-step Lattice Boltzmann Method (TLFS-LBM) is proposed to simulate incompressible flows on non-uniform meshes. Most existing quantum LBMs face two critical challenges: with the relaxation time τ fixed at 1, a fixed mesh resolution corresponds to only a single Reynolds number, and the streaming step, typically implemented through the quantum walk method, is limited to uniform meshes. To address these issues, we retain τ = 1 and decompose the quantum TLFS-LBM into predictor and corrector steps using a fractional-step method. The predictor step is implemented on a quantum circuit, but the resulting viscosity generally differs from the physical viscosity. A diffusion equation is therefore solved classically in the corrector step to restore the correct physical viscosity. Moreover, the streaming step is expressed in the form of A · f eq using Taylor series expansion and least-squares approximation, with singular value decomposition applied to factorize matrix A into unitary matrices for quantum implementation. Since A depends solely on mesh point coordinates, the proposed method is applicable to arbitrary mesh types. Validation through two-dimensional incompressible isothermal and thermal flow simulations confirms that the quantum TLFS-LBM achieves accuracy comparable to its classical counterpart, while extending the applicability of quantum LBM to non-uniform meshes and curved boundaries.
In this work, an interpolation-supplemented quantum lattice Boltzmann method (IS-QLBM) is proposed to simulate incompressible flows with nonuniform meshes and curved boundaries. Owing to the constraint of lattice uniformity, existing quantum LBMs are restricted to uniform square meshes and regular geometries, which severely limit their practicality. To address this limitation, we introduce an interpolation strategy to reconstruct the distribution functions f at each mesh point from those at neighboring points, thereby formulating the entire streaming step into the form of A·f. The matrix A is then decomposed into unitary matrices via singular value decomposition (SVD) for quantum circuit implementation. Since A depends solely on the mesh point coordinates, IS-QLBM can be applied to arbitrary meshes and geometries. The proposed method is validated using two-dimensional incompressible isothermal and thermal flows. The results show that IS-QLBM agrees very well with its classical counterpart IS-LBM, and that quadratic or higher-order interpolation is recommended to ensure sufficient accuracy. To further improve computational efficiency, we develop a multi-circuit IS-QLBM framework that reduces computational cost to less than one-eighth of the single-circuit implementation without sacrificing accuracy.
Quantum computing shows substantial potential in accelerating simulations and alleviating memory bottlenecks in computational fluid dynamics (CFD), owing to its inherent properties of superposition and entanglement. The lattice Boltzmann method (LBM), being largely algebraic in nature, has inspired the development of various quantum LBMs. However, most existing approaches fix the relaxation time at τ = 1, thereby confining a given mesh resolution to simulations at a single Reynolds number. Although our earlier quantum lattice kinetic scheme (LKS) lifted this restriction, it suffers from instability at high Reynolds numbers. To address this challenge, we propose a quantum fractional-step LBM (FS-LBM). In this framework, the predictor step is implemented on a quantum circuit using the standard LBM formulation, while the corrector step is performed classically. The relaxation time is retained at τ = 1 to ensure seamless compatibility with existing quantum LBMs. Benchmark simulations of representative two- and three-dimensional incompressible isothermal and thermal flows demonstrate that the quantum FS-LBM achieves accuracy and convergence orders consistent with its classical counterpart, while significantly outperforming the quantum LKS in both precision and stability. Notably, this work presents the first quantum LBM simulation of three-dimensional incompressible thermal flows.
Quantum algorithms have been identified as a potential means to accelerate computational fluid dynamics (CFD) simulations, with the lattice Boltzmann method (LBM) being a promising candidate for realizing quantum speedups. Here, we extend the recent quantum algorithm for the incompressible LBM to account for realistic fluid dynamics setups by incorporating walls, inlets, outlets, and external forcing. We analyze the associated complexity cost and show that these modifications preserve the asymptotic scaling, and potential quantum advantage, of the original algorithm. Moreover, to support our theoretical analysis, we provide a classical numerical study illustrating the accuracy, complexity, and convergence of the algorithm for representative incompressible-flow cases, including the driven Taylor-Green vortex, the lid-driven cavity flow, and the flow past a cylinder. Our results provide a pathway to accurate quantum simulation of nonlinear fluid dynamics, and a framework for extending quantum LBM to more challenging flow configurations.
Quantum computing offers a promising pathway to accelerate fluid dynamics simulations. In this work, we propose a hybrid quantum-classical lattice Boltzmann method (LBM) for solving the advection–diffusion equation, as a step toward an efficient and general quantum linear solver framework for fluid flows. Quantum circuits handle the computationally intensive collision and streaming steps, exploiting quantum parallelism and removing parameter constraints, while boundary corrections are managed classically to avoid unnecessary overhead. Various boundary conditions are incorporated within the quantum framework, enhancing the flexibility and applicability of the method. Numerical experiments with Gaussian hill initial conditions under periodic boundaries and zero initial conditions under mixed Dirichlet–Neumann boundaries, each in one and two dimensions, demonstrate the precision of the proposed approach. The computational complexity is analyzed in detail. The collision and streaming steps achieve a lower complexity than their classical LBM counterparts, although overall performance is still limited by the input and output costs. This work represents a critical step toward the practical quantum simulation of complex fluid flows.
Quantum computing (QC) has emerged as a potential computational tool in fluid dynamics. However, a practical QC protocol with end-to-end practical applications to fluid flow is still in nascent. We here propose a simple and practical hybrid QC algorithm based on lattice Boltzmann methods (LBM), which we term HQLBM, for fluid dynamics simulations. The existing QC-based LBMs are limited to simple low-Reynolds-number flow simulations due to their disability to handle the nonlinear collision term in LBM for QC. In this work, a linearized non-equilibrium collision, derived from kinetic theory, is presented for HQLBM simulating practical flows. The present approach ensures the unitary of quantum algorithms while keeping the collision relaxation parameter adjustable for simulating different Reynolds number flows. Most importantly, the linearized collision keeps the accuracy of the original nonlinear one in the lattice Boltzmann equation when it theoretically recovers its macroscopic counterpart—Navier–Stokes equations. The practicality of the proposed method is demonstrated by simulating two typical flows, including lid-driven and natural convection flows in a square cavity at different Reynolds and Rayleigh numbers, respectively. It is shown that the proposed HQLBM has good numerical accuracy compared to the conventional LBM, and lower complexity on comparisons to the QC-based LBM carried out on single-circuit QC even when it is performed on a Qiskit virtual quantum computer. This work offers a practical application of QC-based LBM for complex fluid dynamics problems.
We present a versatile and efficient quantum algorithm based on the Lattice Boltzmann method (LBM) approximate solution of the linear advection-diffusion equation (ADE). We emphasize that the LBM approximation modifies the diffusion term of the underlying exact ADE and leads to a modified equation (mADE). Due to its versatility in terms of operator splitting, the proposed quantum LBM algorithm for the mADE provides a building block for future quantum algorithms to solve the linearized Navier-Stokes equation on quantum computers. We split the algorithm into four operations: initialization, collision, streaming, and calculation of the macroscopic quantities. We propose general quantum building blocks for each operator, which adapt intrinsically from the general three-dimensional case to smaller dimensions and apply to arbitrary lattice-velocity sets. Based on (sub-linear) amplitude data encoding, we propose improved initialization and collision operations with reduced complexity and efficient sampling-based simulation. Quantum streaming algorithms are based on previous developments. The proposed quantum algorithm allows for the computation of successive time steps, requiring full state measurement and reinitialization after every time step. It is validated by comparison with a digital implementation and based on analytical solutions in one and two dimensions. Furthermore, we demonstrate the versatility of the quantum algorithm for two cases with non-uniform advection velocities in two and three dimensions. Various velocity sets are considered to further highlight the flexibility of the algorithm. We benchmark our optimized quantum algorithm against previous methods employed in sampling-based quantum simulators. We demonstrate sampling efficiency, with sampling accelerated convergence requiring fewer shots. • Novel LBM quantum algorithm for solving the linear advection-diffusion equation with variable velocity as a building block. • Modular quantum algorithm for 1 through 3 spatial dimensions and all common lattice-velocity sets. • Significant reduction in quantum-algorithm complexity achieved through efficient encoding strategy leveraging symmetries. • Improved collision quantum-algorithm building block leading to higher success rates for the collision operator. • Benchmarking shows faster convergence and reduced shot requirements compared to previous algorithms.