Quantum algorithm for the lattice Boltzmann method with applications on real quantum devices

Abstract

We introduce a novel quantum algorithm for the lattice Boltzmann method (LBM) based on the one-step simplified LBM. The structure of the algorithm allows for more flexibility in modelling different physics in contrast to earlier quantum algorithms for the LBM, while retaining computational efficiency in terms of the gate and qubit complexity. The new algorithm has potential for full end-to-end quantum utility especially for linear problems. We discuss the implementation of examples in linear acoustics, as well as a nonlinear Navier-Stokes problem that was solved on an IBM QPU in a hybrid simulation loop.

Figures (18)

• Figure 1: A D2Q9 lattice (black lines) and a D2Q17 lattice (adding the red lines) configuration. For the OSSLBM model, the base configuration is D2Q9 with $f_{\alpha}^1$, and the additional $8$ velocities $f_{\alpha}^2$ are interpreted to carry data from the second time-level.
• Figure 2: General representation of the quantum algorithm for the OSSLBM.
• Figure 3: The evolution of the two time-levels in the algorithm. The collision and integration operators act on the two levels in parallel (full superposition), while the propagation operator assigns them different shift steps (partial superposition). Here $\{x_m\}_{t_i}$ refers to the generic set of field variables at a given time $t_i$; these could be for example the pair $(\rho, \mathbf{u} )$.
• Figure 4: An example circuit implementation for the collision step using the SVD to decompose the collision matrix. The size of the register $s$ depends on the model used. In any case, the last qubit of the register here marked with $s_d$ will be used to stored the variables relating to the second time-level. The collision operator then acts on both time-levels in superposition.
• Figure 5: An example implementation for the D2Q9 propagation, where the lattice register is split into $q_1$ (horizontal dimension) and $q_2$ (vertical dimension). The blocks with $\mathcal{F}$ denote the quantum Fourier transform. We use the flag qubit state $s_3 = 0$ to control the two shifts, and reordering operators $X_k$ to shuffle the distribution date accordingly. One can extend this scheme to the D2Q17 model by simply adding extra phase gate blocks ($P_n$) controlled by an additional qubit $s_4$, which is used to store the data $f_{\alpha}^2$ for the second time level. Note that the first reordering operator $X_1$ can be at least partially embedded as a row-ordering of the equilibrium matrix $C$ in the preceeding collision step.