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

Kurzfassung

Scale resolving fluid flow simulations can be a real challenge for complex flows but may be necessary in order to accurately model fluid flow phenomena. While the computational cost may be reduced by modeling the behaviour on small scales by approximations instead of resolving all scales, the results may not be able to model the fluid sufficiently accurate. Quantum computing offers a solution to deal with the required high resolution and its resulting large amount of data, that classically may not be feasible to deal with. To do so, amplitude encoding is used to store the fluid properties of each cell in the complex amplitudes of the quantum states of qubits. An existing approach of the Quantum Lattice Boltzmann method (QLBM) from the literature is extended to a fully quantum algorithm without the need of reinitialization and the advection-diffusion equation (ADE) is solved. The Quantum Lattice Boltzmann method using the D1Q2 scheme as first proposed by Budinski is extended. The literature that build on this approach need the information of the full grid after each time step to reinitialize the state for the next time step. In contrast to that, our algorithm can perform all time steps of the simulation from start to end without needing to extract the information of the state at any time. This is achieved by efficiently adding qubits for the time steps to be calculated. With this method, we achieve the following advantages: First, we avoid the computationally expensive complete extraction and reinitialization of a full complex flow field in between the time steps. With reinitialization after every time step, the sampling count increases linearly with the number of time steps. However in our algorithm, the loss of probability amplitude to the ancillas can be compensated by amplitude amplification. Additionally, when mapping the grid to scalar or reduced quantities, we can evaluate such quantities without the need of sampling the entire grid at any time. Our proposed extension of the QLBM is a fully quantum algorithm. Compared to other QLBM algorithms that require the sampling of the state and reinitialize at each time step, our algorithm avoids the need of these samplings during the simulation. With adding additional time-qubits, we can perform multiple time steps without the reinitialization process. The amplitudes of the grid states decay with multiple time steps but the decay factor is small enough such that the number of sampling to resolve the grid at the end of the simulation is more efficient than with the reinitialization method. The number of additional time-qubits scale only logarithmically with the number of total time steps of the simulation and the number of gates scale polynomial with the number of time-qubits, so poly-logarithmically per time step and linear in the number of total time steps.