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

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.

💡 Research Summary

This paper presents a comprehensive framework for constructing measurement‑free quantum algorithms that simulate the lattice Boltzmann method (LBM) for the advection‑diffusion equation (ADE). The authors first reformulate the classic collision‑streaming LBM with the Bhatnagar‑Gross‑Krook (BGK) collision operator into a compact “time‑marching” representation. By introducing a combined matrix M(t) = P·A(t), where A(t) encodes the local BGK relaxation and equilibrium distribution and P is a permutation matrix implementing streaming, they show that under low‑Mach‑number (Ma ≪ 1) and under‑relaxation (τ* > 1) conditions the operator norm ‖M(t)‖ ≤ 1. This stability proof (Theorem 3.1) guarantees that repeated application of the quantum evolution operator does not cause numerical blow‑up, a crucial property for any quantum time‑stepping scheme.

Based on this representation, two distinct quantum algorithms are developed:

1. Sequential Time‑Marching Quantum Algorithm
   The algorithm applies the time‑marching operator M(t) sequentially for T time steps. The authors construct a block‑encoding of M(t) using sparse‑matrix techniques (s = 5 for the D2Q5 lattice) and the linear‑combination‑of‑unitaries (LCU) method to convert the non‑unitary BGK collision into a unitary operation. The block‑encoding requires O(polylog (N)) depth and O(log N) ancilla qubits, with a constant‑order success probability determined by τ*. Each step involves a controlled‑rotation and a phase‑estimation subroutine to implement the LCU. The total runtime scales as O(T·polylog (N)/ε), where ε is the desired precision, offering a quantum speed‑up over the classical O(T·N) cost while preserving the full spatial resolution.

2. Quantum Linear‑Systems Algorithm (QLSA) Approach
   Here the entire time evolution is assembled into a single linear system (I − M) ψ = ψ₀, where ψ encodes the concatenated distribution functions over all time steps. Because ‖M‖ ≤ 1, the condition number κ of (I − M) remains bounded, allowing the use of modern QLSA techniques (e.g., HHL, quantum singular‑value transformation) with runtime O(polylog (N,T)/ε). This method eliminates intermediate measurements entirely; the final quantum state directly yields the solution at the final time. The trade‑off is a larger Hilbert space (dimension ≈ N·(q + 1) × T) and a modest increase in ancilla qubits, but the overall asymptotic complexity matches that of the sequential algorithm.

Both algorithms are analyzed in detail: the authors provide explicit constructions of the block‑encoding matrices, discuss the dilation required to embed non‑unitary steps into unitary circuits, and derive the gate counts and depth. They also address practical aspects such as implementation of various boundary conditions (Dirichlet, Neumann, reflective) via additional unitary layers, and the handling of variable diffusion coefficients and multi‑velocity regimes.

Numerical validation is performed on benchmark problems: a one‑dimensional advection‑diffusion test and a two‑dimensional rotating‑flow scenario. Classical LBM solutions are compared against the quantum‑simulated results obtained from a state‑of‑the‑art quantum circuit simulator. The error scales as O(Ma² + Δt²), confirming the theoretical convergence analysis. Gate‑count measurements show that the sequential algorithm requires slightly more gates per time step, while the QLSA approach incurs a larger ancilla overhead but fewer overall circuit layers.

In summary, the paper delivers a rigorous, measurement‑free quantum formulation of LBM, proves its stability, and offers two viable quantum algorithmic pathways with comparable asymptotic complexities. By removing the need for intermediate measurements and re‑initializations, the work paves the way for scalable quantum simulations of advection‑diffusion and, more broadly, fluid‑dynamic processes using the lattice Boltzmann paradigm.