Quantum Lattice Boltzmann Method for Multiple Time Steps Without Reinitialization for Linear Advection-Diffusion Problems

TL;DR

This work presents a fully quantum extension of the Quantum Lattice Boltzmann Method (QLBM) that enables multiple time steps of linear advection-diffusion simulations without any mid-circuit measurements or state reinitialization. The authors implement a unitary collision via tailored rotations, conditioned streaming, and a re-prep mechanism that uses time qubits to cyclically prepare the state for successive steps, with a decay model that quantifies amplitude loss per step. Verified on 1D (D1Q2, D1Q3) and 2D (D2Q9) advection-diffusion problems against classical LBM, the approach shows good agreement and convergence, while highlighting sampling-noise and the need for increased shots for longer runs. The method offers a pathway to extract surface or scalar quantities without reconstructing the full flow field, potentially improving scalability for large grids, though it trades off with increased circuit depth and coherence-time demands; future work targets amplitude amplification and nonlinear extensions.

Abstract

To simulate highly-resolved flow fields, we extend the Quantum Lattice Boltzmann Method (QLBM) to be able to simulate multiple time steps without state extraction or reinitialization. We adjust and extend given QLBM approaches from the literature to completely remove the need to measure or reinitialize the flow field in between the simulation time steps. Therefore, our algorithm does not require to sample the entire flow field at any time. We solve the linear advection-diffusion problem and derive all necessary equations and build the corresponding quantum circuit diagrams, including details on the QLBM blocks and explicitly drawing the circuit gates. We discuss the general decay of a QLBM step and how that effects our algorithm. The new algorithm is verified on 1D and 2D test cases using the shot method of IBMs Qiskit package. We show excellent agreement and convergence between our QLBM and classical LBM methods. The conclusion section includes a discussion on the advantages of our algorithm as well as limitations and to what extent it is more efficient.

Figures (15)

• Figure 1: Quantum circuit for a single quantum lattice Boltzmann time step.
• Figure 2: Our quantum circuit extension of the QLBM routine for multiple quantum lattice Boltzmann time steps without state extraction or reinitialization.
• Figure 3: Full quantum circuit of the QLBM simulation with the extension of the QLBM routine for $T$ time steps without state extraction or reinitialization from figure \ref{['fig:QLBM_circ_mult_steps']}.
• Figure 4: Quantum circuit for the collision step in figure \ref{['fig:QLBM_circ_mult_steps']} for the D1Q2 scheme (fig. \ref{['fig:D1Q2_coll_circ']}) and a the D1Q3 scheme (fig. \ref{['fig:D1Q3_coll_circ']}) for spatially constant diffusion and flow velocity. For spatially varying diffusion or flow velocities, multiple $RY$ gates with controls on the grid qubits $\ket{q_\text{grid}}$ need to be used.
• Figure 5: Quantum circuit for the collision step in figure \ref{['fig:QLBM_circ_mult_steps']} for the D2Q9 scheme for spatially constant diffusion and flow velocity. Each $RY$-gate distributes the velocity distribution functions into their corresponding velocity direction subspace according to the procedure in table \ref{['tab:D2Q9_distributing_f']} in section \ref{['sec:appendix_coll_distr_D2Q9']} the appendix. The argument $\theta$ for each $RY$-gate has to be chosen according to equation \ref{['eq:theta_coll_keep_mult_subspaces']}. For additionally spatially varying diffusion or flow velocities, multiple $RY$ gates with controls on the grid qubits $\ket{q_\text{grid}}$ need to be used.