Quantum lattice Boltzmann method for several time steps: A local Carleman linearization algorithm

TL;DR

This work advances quantum LBM by introducing a locality-preserving Carleman linearization encoding that enables multi-step quantum simulations on 2D lattices. By decoupling the collision from nonlocal registers and employing a shift-based permutation to map Carleman variables, the authors achieve a local collision implementation with a per-step scaling of $O(\\log_2^3(N) + Q^4)$ and a realistic per-step success probability around $10^{-2}$. The approach is validated on quantum emulators, showing accurate reproduction of classical LBM results for small lattices and promising qualitative agreement for larger problems, though shot noise and the non-unitarity of the collision operator (mitigated by LCU) remain key challenges. Overall, the paper demonstrates a viable path toward multi-time-step QLBM, highlighting both its potential and the practical hurdles to achieving quantum advantage in this nonlinear, mesoscopic fluid modeling context.

Abstract

This article presents a novel encoding for quantum Lattice Boltzmann method algorithm using Carleman linearization. In contrast to previous articles \cite{Sanavio2024LatticeBC,sanavio2025carleman}, the encoding used allows for local collision rules while keeping a higher probability to obtain the right result, which is of the order of $10^{-2}$. The algorithm scales as $O(log_2^3(N)+Q^4)$ each time step with $N$ the number of lattice sites of the 2D lattice and $Q$ the number of channels with a constant number of qubits when using dynamical circuits.

Figures (13)

• Figure 1: Scheme of channels for D2Q9. Each number represents the index used to map each channel.
• Figure 2: Quantum circuit scheme for QLBM using Carleman. The circuit is composed of the state preparation (SP), collision (CL), propagation (PG) and permutation (PM) operators. The dashed lines encompass the section of the circuit that is repeated $T$ time steps. Notice that the black dots in the quantum gates represent several conditions over the given register.
• Figure 4: Distribution of channels for $f$ at $x_1=0$ after 5 time steps using $L_x=2$ and their relative error $\epsilon$ compared to the classical simulation. The notation used here is the standard for LBM D2Q9 following the scheme in Fig \ref{['fig:d2q9']}.
• Figure 5: Distribution of channels for $f$ at $x_1=0$ after 2 time steps using $L_x=2$ and their relative error compared to the classical simulation. The notation used here is the standard for LBM D2Q9 following the scheme in Fig \ref{['fig:d2q9']}.
• Figure 6: Distribution of channels for $g$ at $x_1,x_2=0$ after 2 time steps using $L_x=2$ and their relative error compared to the classical simulation. The channel index refers to the elements $c=(c_1,c_2)$ with the standard for LBM D2Q9 following the scheme in Fig \ref{['fig:d2q9']}, where the channels have been arranged in an array with row ordering.