Time-marching representation based quantum algorithms for the Lattice Boltzmann model of the advection-diffusion equation

TL;DR

The collision-streaming evolution of the LBM is formulated as a compact time-marching scheme and rigorously established its stability under low Mach number conditions, enabling a systematic and fully quantum implementation.

Abstract

This article introduces a novel framework for developing quantum algorithms for the Lattice Boltzmann Method (LBM) applied to the advection-diffusion equation. We formulate the collision-streaming evolution of the LBM as a compact time-marching scheme and rigorously establish its stability under low Mach number conditions. This unified formulation eliminates the need for classical measurement at each time step, enabling a systematic and fully quantum implementation. Building upon this representation, we investigate two distinct quantum algorithmic approaches. The first is a time-marching quantum algorithm realized through sequential evolution operators, for which we provide a detailed implementation-including block-encoding and dilating unitarization-along with a full complexity analysis. The second employs a quantum linear systems algorithm, which encodes the entire time evolution into a single global linear system. We demonstrate that both methods achieve comparable asymptotic time complexities. The proposed algorithms are validated through numerical simulations of benchmark problems in one and two dimensions. This work provides a systematic, measurement-free pathway for the quantum simulation of advection-diffusion processes via the lattice Boltzmann paradigm.

Figures (14)

• Figure 1: A diagram of the D2Q5 lattice Boltzmann discrete velocity model, showing five discrete directions in a 2D grid. The center direction is labeled as 0, with four surrounding directions labeled as 1, 2, 3 and 4. The directions should be symmetrically arranged around the center, representing particle movements in the horizontal and vertical directions.
• Figure 2: A quantum circuit that implements the block-encoding of $E$, denoted by $U_E$.
• Figure 3: A quantum circuit that implements the block-encoding of $A_e^{(1)}$, denoted by $U_{A_e^{(1)}}$.
• Figure 4: A quantum circuit that implements the block-encoding of $E_{05}$, denoted by $U_{E_{05}}$.
• Figure 5: Schematic diagram of the D2Q5 discrete velocity model

Theorems & Definitions (19)

• Lemma 2.1
• proof
• Theorem 3.1
• proof
• Definition 4.1
• Definition 4.2
• Lemma 4.1: Block-encoding of sparse-access matrices
• Lemma 4.2
• Definition 4.3
• Lemma 4.3