In this paper, we propose a complete formulation of the Lattice Boltzmann Method adapted for quantum computing. The classical collision, based on linear equilibrium distribution functions and streaming steps, are reformulated as linear algebraic operations. The inherently non-unitary collision operator is decomposed using Singular Value Decomposition and the Linear Combination of Unitaries technique. Bounce-back boundary conditions are incorporated directly into the collision matrix, while the streaming step is realized through conditional unitary shift operations on spatial registers, controlled by lattice velocity indices encoded in the distribution function register. This formulation ensures that the streaming step remains purely unitary. The resulting quantum circuit is implemented using Qiskit and validated against Couette flow and Poiseuille flow benchmarks. The simulation accurately reproduces the expected velocity profile, with relative errors below 1 0 − 4 . This work establishes a foundational framework for quantum fluid solvers and provides a pathway toward quantum computational fluid dynamics.
In this paper, we propose a complete formulation of the Lattice Boltzmann Method adapted for quantum computing. The classical collision, based on linear equilibrium distribution functions and streaming steps, are reformulated as linear algebraic operations. The inherently non-unitary collision operator is decomposed using Singular Value Decomposition and the Linear Combination of Unitaries technique. Bounce-back boundary conditions are incorporated directly into the collision matrix, while the streaming step is realized through conditional unitary shift operations on spatial registers, controlled by lattice velocity indices encoded in the distribution function register. This formulation ensures that the streaming step remains purely unitary. The resulting quantum circuit is implemented using Qiskit and validated against Couette flow and Poiseuille flow benchmarks. The simulation accurately reproduces the expected velocity profile, with relative errors below 1 0 − 4 . This work establishes a foundational framework for quantum fluid solvers and provides a pathway toward quantum computational fluid dynamics.
We propose a quantum algorithm for the Lattice Boltzmann (LB) method to simulate fluid flows in the low Reynolds number regime. First, we encode the particle distribution functions (PDFs) as probability amplitudes of the quantum state and demonstrate the need to control the state of the ancilla qubit during the initial state preparation. Second, we express the LB algorithm as a matrix-vector product by neglecting the quadratic non-linearity in the equilibrium distribution function, wherein the vector represents the PDFs, and the matrix represents the collision and streaming operators. Third, we employ classical singular value decomposition (SVD) to decompose the non-unitary collision and streaming operators into a product of unitary matrices. Finally, we show the importance of having a Hadamard gate between the collision and the streaming operations. Our approach has been tested on linear/linearized flow problems such as the advection-diffusion of a Gaussian hill, Poiseuille flow, Couette flow, and lid-driven cavity problems. We provide counts for two-qubit controlled-NOT (CNOT) and single-qubit U gates for test cases involving 9 to 12 qubits, with grid sizes ranging from 24 to 216 points. While the gate count aligns closely with theoretical limits, the high number of two-qubit gates on the order of 107 necessitates careful attention to circuit synthesis.
We propose a quantum algorithm for the lattice Boltzmann (LB) method to simulate fluid flows in the low Reynolds number regime. First, we encode the particle distribution functions (PDFs) as probability amplitudes of the quantum state and demonstrate the need to control the state of the ancilla qubit during the initial state preparation. Second, we express the LB algorithm as a matrix-vector product by neglecting the quadratic non-linearity in the equilibrium distribution function, wherein the vector represents the PDFs, and the matrix represents the collision and streaming operators. Third, we employ classical singular value decomposition to decompose the non-unitary collision and streaming operators into a product of unitary matrices. Finally, we show the importance of having a Hadamard gate between the collision and the streaming operations. Our approach has been tested on linear/linearized flow problems such as the advection-diffusion of a Gaussian hill, Poiseuille flow, Couette flow, and lid-driven cavity problems. We provide counts for two-qubit controlled-NOT and single-qubit U gates for test cases involving 9–12 qubits with grid sizes ranging from 24 to 216 points. While the gate count aligns closely with theoretical limits, the high number of two-qubit gates on the order of 107 necessitates careful attention to circuit synthesis.
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 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 full quantum algorithm for the lattice Boltzmann method for simulating fluid flows, the only such algorithm to implement both the streaming and collision substeps as unitary operators using an efficient number of qubits. We use Hamiltonian simulation as the main route underlying the algorithm, and show that it uses qubits that scale logarithmically in the flow Reynolds number but gates that scale only polynomially. We motivate the discussion by a brief overview of existing attempts at simulating classical fluids on quantum computers and present a pedagogical discussion on assigning quantum operators to classical variables in both the streaming and the collision substeps, after highlighting the incompatibility of the latter with the implementations of the streaming step that exists in the literature. We use the Bhatnagar–Gross–Krook ansatz for the collision term, representing the relaxation toward an equilibrium distribution. For nonlinear collisions, we use Kowalski's framework that links the nonlinear dynamics of a system to the evolution of bosonic modes, assigning a Carleman linearization order to the truncation in the Fock space of the bosons. We present the qubit and gate complexities, in terms of the chosen accuracy and the Reynolds number. In the Appendix, we work out the details of implementing the operators of the truncated bosonic Fock space in terms of single-qubit gates as well as the error scaling for a general polynomial driving function.