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.
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.
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.
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.
The quantum lattice Boltzmann method (QLBM) holds promise for efficient fluid simulations, yet the treatment of realistic boundary conditions, particularly solid wall boundary with arbitrary geometry, remains a critical open challenge. We propose a QLBM algorithm that incorporates wall boundaries as well as inlet/outlet velocity conditions. Building upon the existing QLBM with ensemble transformations, we adopt a density-matrix encoding that offers greater flexibility in circuit design and enables the implementation of non-unitary operations through Kraus operators. For boundary enforcement, we design a quantum implementation of the half-way bounce-back scheme and introduce a component exchange step prior to the streaming step, which naturally enforces the no-slip condition without modifying the transport operation. Inlet and outlet conditions are imposed via controlled SWAP operations combined with ancilla registers prepared in the prescribed velocity states. Our QLBM is quantitatively validated through simulations of two-dimensional flows past a backward-facing step, a cylinder, and an obstacle with complex geometry, demonstrating both accuracy and extensibility of our boundary algorithm.
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∞ 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.
We present a mass-weighted meshless formulation for the shifted incompressible lattice Boltzmann method (LBM) whose streaming step is exactly unitary, combining with the Carleman embedding. In traditional meshless LBM, on a scattered node set the semi-Lagrangian streaming step is an interpolation, which is not unitary; a generic block encoding of this matrix therefore requires a normalization proportional to its sparsity and logarithmically many ancillas. To overcome the non-unitarity, we instead express the interpolation as the exponential exp(a Dx + b Dy) of mimetic derivative operators. Endowing each node with its Voronoi mass and applying a mass-weighted similarity transform renders these derivative operators skew-Hermitian by enforcing moment constraints together with low-wave number trigonometric differentiation conditions, so that the interpolation becomes exactly unitary. Such unitarization of interpolation leave the local collision step unchanged up to a rescaling of its quadratic term, where the same weights define the amplitude encoding of the distribution functions. For quasi-uniform node sets, the mass-weighting diagonal remains within a unit-order normalization that degrades only mildly as the Carleman truncation order grows, thereby mitigating the cost of block encoding. Numerical experiments on the Taylor Green vortex confirm that the method reproduces the full meshless LBM, with the velocity error decreasing as the Carleman order increases.
Abstract This work presents a fully quantum implementation of the Quantum Lattice Boltzmann Method (QLBM) for simulating one-dimensional heat conduction with internal heat generation. Building on the classical Lattice Boltzmann framework, QLBM reformulates the collision and streaming steps as unitary operations on quantum circuits, enabling site-wise parallel updates on near-term quantum hardware. In this formulation, classical distribution functions are encoded into quantum amplitudes, and the evolution of the system is performed entirely through quantum gates, including a collision operator and a streaming step realized as a qubit permutation. The model is validated by simulating a canonical diffusion problem with a localized heat source, and results are benchmarked against both classical LBM and an analytical solution obtained via ODE integration. The impact of quantum sampling noise is systematically assessed by varying the number of measurement shots, revealing that higher shot counts significantly improve fidelity to classical results. Root-mean-square error (RMSE) analyses confirm that the fully quantum model accurately captures the spatiotemporal behavior of the system, demonstrating the viability of QLBM for modeling nonunitary physics in thermally driven systems.