Simulating non-trivial incompressible flows with a quantum lattice Boltzmann algorithm

TL;DR

The paper extends the quantum lattice Boltzmann method (LBM) for incompressible flows by incorporating nontrivial geometries (walls, inlets, outlets) and driving forces using Carleman embedding, enabling a linear quantum processing framework. It analyzes how boundary and driving extensions modify the Carleman-based linear system, showing that block-encoding overhead scales as a multiplicative factor around $4^{N_C}$ with additional ancilla costs, while preserving the original asymptotic quantum advantage. Classical numerical experiments on driven Taylor-Green vortex, lid-driven cavity, and flow past a cylinder validate Carleman convergence and provide insights into error behavior and condition numbers under these extensions. The results suggest that, within the studied 2D, low-Re regime and small Carleman orders ($N_C\le 3$), the end-to-end quantum LBM remains a promising route toward quantum-accelerated CFD, though substantial work remains to realize exponential speedups in realistic 3D, high-Re flows.

Abstract

Quantum algorithms have been identified as a potential means to accelerate computational fluid dynamics (CFD) simulations, with the lattice Boltzmann method (LBM) being a promising candidate for realizing quantum speedups. Here, we extend the recent quantum algorithm for the incompressible LBM to account for realistic fluid dynamics setups by incorporating walls, inlets, outlets, and external forcing. We analyze the associated complexity cost and show that these modifications preserve the asymptotic scaling, and potential quantum advantage, of the original algorithm. Moreover, to support our theoretical analysis, we provide a classical numerical study illustrating the accuracy, complexity, and convergence of the algorithm for representative incompressible-flow cases, including the driven Taylor-Green vortex, the lid-driven cavity flow, and the flow past a cylinder. Our results provide a pathway to accurate quantum simulation of nonlinear fluid dynamics, and a framework for extending quantum LBM to more challenging flow configurations.

Figures (8)

• Figure 1: Carleman truncation error for the forced Taylor-Green vortex flow with $\beta=0.75$. Left: Dependence of $\epsilon_C$ on the Reynolds number $\mathrm{Re}$ and truncation order $N_C=1,2,3$ for one advection time. Right: Time evolution of $\epsilon_{\mathrm{rel}}$ over $10$ advection times for $N_C=1,2$ and $\mathrm{Re}=50$.
• Figure 2: Spatial form of the classical LBM solution, the Carleman LBM solution and their associated error for the forced Taylor-Green vortex flow after $10$ advection times for $\beta=0.75$ and $\mathrm{Re}=50$. Left: Classical LBM solution. Middle: Linear Carleman approximation (bottom) and error (top). Right: Quadratic Carleman approximation (bottom) and error (top).
• Figure 3: Carleman truncation error for the lid-driven cavity flow with $\beta=1$. Left: Dependence of $\epsilon_C$ on the Reynolds number $\mathrm{Re}$ and truncation order $N_C=1,2$ for one advection time. Right: Time evolution of $\epsilon_{\mathrm{rel}}$ over $10$ advection times for $N_C=1,2$ and $\mathrm{Re}=25$.
• Figure 4: Spatial form of the classical LBM and Carleman LBM solutions (red) and their associated error (blue) for the lid-driven cavity flow after $10$ advection times for $\beta=1$ and $\mathrm{Re}=25$. Left: the classical LBM solution. Middle: linear Carleman approximation. Right: the quadratic Carleman approximation.
• Figure 5: Carleman truncation error for the flow past a cylinder with $\beta=0.75$. Left: Dependence of $\epsilon_C$ on the Reynolds number $\mathrm{Re}$ and truncation order $N_C=1,2$ for one advection time. Right: Time evolution of $\epsilon_{\mathrm{rel}}$ over $5$ advection times for $N_C=1,2$ and $\mathrm{Re}=6$.