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arXiv:2505.01271 · Physical review. E

Improved quantum lattice Boltzmann method for advection-diffusion equations with a linear collision model

Li Xu, Ming Li, Lei Zhang, Hai Xia Sun, Jun Ping Yao

Published Apr 11, 2025• 15 citations
arXivHTMLPDF

Abstract

Quantum computing has made tremendous progress in recent years, providing potentialities for breaking the bottleneck of computing power in the field of scientific computing, like computational fluid dynamics. To reduce computational costs and achieve an acceleration, we propose an ancilla-free quantum lattice Boltzmann method for advection-diffusion equations that fully leverages the parallelism of quantum computing. More significantly, there is no need to perform quantum state tomography in each previous loop, if the macroscopic variables for a certain loop is needed. The nonunitary collision operators are replaced by the unique local unitary operations, and the removal of ancilla qubit greatly diminishes the complexity of the quantum circuit. The numerical simulations of the D_{1}Q_{3} and D_{2}Q_{5} models have confirmed the feasibility of the proposed algorithm.

DOI
10.1103/physreve.111.045305

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physics.comp-phCross-listedarXiv:2610.40851•Nov 2, 2026
Quantum computing has made tremendous progress in recent years, providing potentialities for breaking the bottleneck of computing power in the field of scientific computing, like computational fluid dynamics. To reduce computational costs and achieve an acceleration, we propose an ancilla-free quantum lattice Boltzmann method for advection-diffusion equations that fully leverages the parallelism of quantum computing. More significantly, there is no need to perform quantum state tomography in each previous loop, if the macroscopic variables for a certain loop is needed. The nonunitary collision operators are replaced by the unique local unitary operations, and the removal of ancilla qubit greatly diminishes the complexity of the quantum circuit. The numerical simulations of the D_{1}Q_{3} and D_{2}Q_{5} models have confirmed the feasibility of the proposed algorithm.
The Quantum Lattice Boltzmann Method (QLBM) is one of the most promising approaches for realizing the potential of quantum computing in simulating computational fluid dynamics. Many recent works mostly focus on classical simulation, and rely on full state tomography. Several key algorithmic issues like observable readout, data encoding, and impractical circuit depth remain unsolved. As a result, these are not directly realizable on any quantum hardware. We present a series of novel algorithmic advances which allow us to implement the QLBM algorithm, for the first time, on a quantum computer. Hardware results for the time evolution of a 2D Gaussian initial density distribution subject to a uniform advection-diffusion field are presented. Furthermore, 3D simulation results are presented for particular non-uniform advection fields, devised so as to avoid the problem of diminishing probability of success due to repeated post-selection operations required for multiple timesteps. We demonstrate the evolution of an initial quantum state governed by the advection-diffusion equation, accounting for the iterative nature of the explicit QLBM algorithm. A tensor network encoding scheme is used to represent the initial condition supplied to the advection-diffusion equation, significantly reducing the two-qubit gate count affording a shorter circuit depth. Further reductions are made in the collision and streaming operators. Collectively, these advances give a path to realizing more practical, 2D and 3D QLBM applications with non-trivial velocity fields on quantum hardware.
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.
The Quantum Lattice Boltzmann Method (QLBM) has emerged as one of the most promising quantum computing approaches for the numerical simulation of problems in computational fluid dynamics (CFD). The dynamics is formulated in terms of mesoscopic particle distribution functions governed by a discrete Boltzmann transport equation, comprising local streaming and collision operations. In this work, the resulting macroscopic behavior corresponds to the advection-diffusion equation, which we adopt as a canonical model problem for transport phenomena. Building upon recent progress in QLBM implementations, we advance towards more realistic problem settings that better reflect conventional CFD requirements. We address, for the first time, transport under the action of non uniform velocity fields on quantum hardware. We implement our demonstration using IonQ's trapped-ion systems including Forte generation systems and a 64-qubit Barium development system similar to the forthcoming IonQ Tempo line. We identify the density readout and subsequent reloading of the fluid density as a potential bottleneck of the current algorithm and discuss several approaches to mitigate this bottleneck. We identify the use of MPS shadow tomography as a promising method to efficiently scale the readout to large system with complex density distributions. Lastly, we introduce and simulate a novel method to implement wall boundaries for advection-diffusion in QLBM, and discuss the prospects of scaling to higher-complexity 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.
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.
Classical computational fluid dynamics (CFD) methods encounter significant computational bottlenecks in large systems and at high Reynolds number flows due to the rapid increase in degrees of freedom, even on flagship supercomputers. As quantum computers emerge, the Quantum Lattice Boltzmann Method (QLBM) has been proposed as a quantum-native alternative for solving fluid dynamics problems. However, until recently, nearly all studies have relied on classical quantum simulators rather than real quantum hardware. This study presents a benchmark of a one-dimensional QLBM implementation executed on three different IBM Quantum hardware. We design and test a quantum circuit for the advection-diffusion equation using amplitude encoding, a Bhatnagar-Gross-Krook (BGK)-type collision operator, and a streaming step with periodic boundary conditions. Our numerical experiments evaluate transpiled two-qubit gate count, total circuit depth, runtime, and accuracy metrics including maximum absolute error (L∞L_\inftyL∞​ norm) and root-mean-square error (RMSE). We find that the present collision and streaming operators dominate the overall cost and become the primary scalability bottleneck on current hardware. To mitigate the prohibitive cost of full quantum state tomography on hardware-based benchmarking exercises, we introduce a sampler-based measurement strategy that reconstructs macroscopic density fields directly from computational-basis measurements. This approach substantially reduces measurement overhead while preserving the expected advection-diffusion dynamics on hardware. These results provide a quantitative assessment of QLBM performance in the noisy intermediate-scale quantum (NISQ) regime and identify the principal algorithmic and experimental limitations that must be addressed for QLBM to become viable for fluid dynamics.