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arXiv:2405.13391

Unitary Quantum Algorithm for the Lattice-Boltzmann Method

David Wawrzyniak, Josef M. Winter, Steffen J. Schmidt, Thomas Indinger, Uwe Schramm, Christian F. Janßen, Nikolaus A. Adams

Published May 22, 2024• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.2405.13391

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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.