This article introduces a novel framework for developing quantum algorithms for the Lattice Boltzmann Method (LBM) applied to the advection-diffusion equation. We formulate the collision-streaming evolution of the LBM as a compact time-marching scheme and rigorously establish its stability under low Mach number conditions. This unified formulation eliminates the need for classical measurement at each time step, enabling a systematic and fully quantum implementation. Building upon this representation, we investigate two distinct quantum algorithmic approaches. The first is a time-marching quantum algorithm realized through sequential evolution operators, for which we provide a detailed implementation-including block-encoding and dilating unitarization-along with a full complexity analysis. The second employs a quantum linear systems algorithm, which encodes the entire time evolution into a single global linear system. We demonstrate that both methods achieve comparable asymptotic time complexities. The proposed algorithms are validated through numerical simulations of benchmark problems in one and two dimensions. This work provides a systematic pathway that avoids full-state measurement and reinitialization at every time step for the quantum simulation of advection-diffusion processes via the lattice Boltzmann paradigm.
This article introduces a novel framework for developing quantum algorithms for the Lattice Boltzmann Method (LBM) applied to the advection-diffusion equation. We formulate the collision-streaming evolution of the LBM as a compact time-marching scheme and rigorously establish its stability under low Mach number conditions. This unified formulation eliminates the need for classical measurement at each time step, enabling a systematic and fully quantum implementation. Building upon this representation, we investigate two distinct quantum algorithmic approaches. The first is a time-marching quantum algorithm realized through sequential evolution operators, for which we provide a detailed implementation-including block-encoding and dilating unitarization-along with a full complexity analysis. The second employs a quantum linear systems algorithm, which encodes the entire time evolution into a single global linear system. We demonstrate that both methods achieve comparable asymptotic time complexities. The proposed algorithms are validated through numerical simulations of benchmark problems in one and two dimensions. This work provides a systematic pathway that avoids full-state measurement and reinitialization at every time step for the quantum simulation of advection-diffusion processes via the lattice Boltzmann paradigm.
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.
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.
We present a quantum algorithm for computational fluid dynamics based on the Lattice-Boltzmann method. Our approach involves a novel encoding strategy and a modified collision operator, assuming full relaxation to the local equilibrium within a single time step. Our quantum algorithm enables the computation of multiple time steps in the linearized case, specifically for solving the advection-diffusion equation, before necessitating a full state measurement. Moreover, our formulation can be extended to compute the non-linear equilibrium distribution function for a single time step prior to measurement, utilizing the measurement as an essential algorithmic step. However, in the non-linear case, a classical postprocessing step is necessary for computing the moments of the distribution function. We validate our algorithm by solving the one dimensional advection-diffusion of a Gaussian hill. Our results demonstrate that our quantum algorithm captures non-linearity.
The Lattice Boltzmann method (LBM) is a well-established mesoscopic approach for simulating fluid dynamics by evolving particle distribution functions on discrete lattices. While the LBM is highly parallelizable on classical hardware, its translation to quantum algorithms is impeded by the collision process, which is intrinsically nonlinear and irreversible. Several existing quantum formulations implement this process through repeated quantum tomography and state preparation at every timestep, leading to significant overheads. We introduce a quantum LBM based on a denoising-type collision operator that avoids tomography-based updates. The collision dynamics are reformulated as an orthogonal projection onto the linearized manifold of equilibrium distributions around a reference state. This geometric approach filters non-equilibrium components while preserving lattice symmetries and approximating nonlinear terms needed to recover hydrodynamic behavior. A complete pipeline is presented with efficient gate-level realizations, incorporating encoding of distributions, collision, streaming, boundary conditions, and measurement of physical quantities such as hydrodynamic forces. In addition, we outline an approach for implementing projector-based operators deterministically without postselection, paving the way to fully coherent multi-timestep LBM simulations. Numerical experiments for advection-diffusion and flow problems demonstrate that the method reproduces macroscopic behaviors with high accuracy, with performance depending on the choice of reference state.
We present a quantum algorithm for computational fluid dynamics based on the Lattice-Boltzmann method. Our approach involves a novel encoding strategy and a modified collision operator, assuming full relaxation to the local equilibrium within a single time step. Our quantum algorithm enables the computation of multiple time steps in the linearized case, specifically for solving the advection-diffusion equation, before necessitating a full state measurement. Moreover, our formulation can be extended to compute the non-linear equilibrium distribution function for a single time step prior to measurement, utilizing the measurement as an essential algorithmic step. However, in the non-linear case, a classical postprocessing step is necessary for computing the moments of the distribution function. We validate our algorithm by solving the one dimensional advection-diffusion of a Gaussian hill. Our results demonstrate that our quantum algorithm captures non-linearity.
We present a quantum algorithm for computational fluid dynamics based on the Lattice-Boltzmann method. Our approach involves a novel encoding strategy and a modified collision operator, assuming full relaxation to the local equilibrium within a single time step. Our quantum algorithm enables the computation of multiple time steps in the linearized case, specifically for solving the advection-diffusion equation, before necessitating a full state measurement. Moreover, our formulation can be extended to compute the non-linear equilibrium distribution function for a single time step prior to measurement, utilizing the measurement as an essential algorithmic step. However, in the non-linear case, a classical postprocessing step is necessary for computing the moments of the distribution function. We validate our algorithm by solving the one dimensional advection-diffusion of a Gaussian hill. Our results demonstrate that our quantum algorithm captures non-linearity.