Quantum Search in Superposed Quantum Lattice Gas Automata and Lattice Boltzmann Systems

As the scope of Computational Fluid Dynamics (CFD) grows to encompass ever larger problem scales, so does the interest in whether quantum computing can provide an advantage. In recent years, Quantum Lattice Gas Automata (QLGA) and Quantum Lattice Boltzmann Methods (QLBM) have emerged as promising candidates for quantum-native implementations of CFD solvers. Though the progress in developing QLGA and QLBM algorithms has been significant, it has largely focused on the development of models rather than applications. As a result, the zoo of QLGA and QLBM algorithms has grown to target several equations and to support many extensions, but the practical use of these models is largely limited to quantum state tomography and observable measurement. This limitation is crucial in practice, because unless very specific criteria are met, such measurements may cancel out any potential quantum advantage. In this paper, we propose an application based on discrete optimization and quantum search, which circumvents flow field measurement altogether. We propose methods for simulating many different lattice configurations simultaneously and describe how the usage of amplitude estimation and quantum search can provide an asymptotic quantum advantage. Throughout the paper, we provide detailed complexity analyses of gate-level implementations of our circuits and consider the benefits and costs of several encodings.