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arXiv:1309.1656 · International Journal of Modern Physics C

Matrix-valued quantum lattice Boltzmann method

Christian B. Mendl

Published Jan 23, 2015• 5 citations
arXivHTMLPDF

Abstract

We devise a lattice Boltzmann method (LBM) for a matrix-valued quantum Boltzmann equation, with the classical Maxwell distribution replaced by Fermi–Dirac functions. To accommodate the spin density matrix, the distribution functions become 2 × 2 matrix-valued. From an analytic perspective, the efficient, commonly used BGK approximation of the collision operator is valid in the present setting. The numerical scheme could leverage the principles of LBM for simulating complex spin systems, with applications to spintronics.

DOI
10.1142/s0129183115501132

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We devise a lattice Boltzmann method (LBM) for a matrix-valued quantum Boltzmann equation, with the classical Maxwell distribution replaced by Fermi–Dirac functions. To accommodate the spin density matrix, the distribution functions become 2 × 2 matrix-valued. From an analytic perspective, the efficient, commonly used BGK approximation of the collision operator is valid in the present setting. The numerical scheme could leverage the principles of LBM for simulating complex spin systems, with applications to spintronics.
In the existing quantum algorithms for the lattice Boltzmann method, resolving the unitarity problem of the BGK collision operator from first principles remains a fundamental challenge, and a systematically constructed unitary alternative that reproduces the BGK relaxation structure to leading order has not yet been established in the literature. In this work, we propose a novel quantum lattice Boltzmann method (QLBM) based on the rotation collision operator, in which the non-unitary BGK relaxation is replaced by a unitary rotation in the amplitude space of the distribution function, and the corresponding gate-level quantum circuits are also constructed. Specifically, the rotation collision operator is developed for both the D2Q5 model of the convection-diffusion equation and the D2Q9 model of the incompressible Navier--Stokes equations. It is worth noting that, for the D2Q5 model, the equilibrium-state preparation circuit can be precompiled and reused, and the resulting unitary blocks can be sequentially composed once the state-dependent collision parameters are specified. Finally, some numerical experiments are performed to validate the developed QLBM, demonstrating its accuracy and effectiveness.
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 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 introduce a novel quantum algorithm for the lattice Boltzmann method (LBM) based on the one-step simplified LBM. The structure of the algorithm allows for more flexibility in modelling different physics in contrast to earlier quantum algorithms for the LBM, while retaining computational efficiency in terms of the gate and qubit complexity. The new algorithm has potential for full end-to-end quantum utility especially for linear problems. We discuss the implementation of examples in linear acoustics, as well as a nonlinear Navier-Stokes problem that was solved on an IBM QPU in a hybrid simulation loop.
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.
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.