Explicit near-optimal quantum algorithm for solving the advection-diffusion equation

Quantum computing has the potential to speed up simulations of classical dynamics by leveraging quantum superposition and entanglement to parallelize the processing of high-dimensional data. Yet, due to the intrinsically unitary nature of quantum mechanics, modeling dissipative dynamics has required significant development. Quantum algorithms for dissipative dynamics are usually based on the transformation of nonunitary initial-value problems into a system of linear equations which then can be solved by a quantum linear system algorithm such as the quantum singular value transformation. However, to reduce the number of calls to the initialization oracle, it is better to use the Linear Combination of Hamiltonian Simulations (LCHS) algorithm. We propose an explicit LCHS quantum circuit, explain how to achieve high success probability and an optimal linear scaling with time for this method, and test it on modeling the advection-diffusion equation on a classical emulator of fault-tolerant quantum computers.