Quantum algorithms for one-dimensional advection-diffusion equation

We demonstrate the application of two hybrid quantum-classical algorithms, the Quantum Linear System Algorithm (QLSA) and the Variational Quantum Algorithm (VQA), for the numerical solution of a one-dimensional advection-diffusion problem. The QLSA solves the system of linear equations which follows from the discretization of the flow problem, where the input matrices are prepared classically and the solution to the matrix equation is computed on a quantum simulator. The VQA evaluates a cost function by parameterized quantum circuits, while a classical optimization is performed to find the cost minimum that corresponds to the solution at the next time step. For both methods, the first-order Euler scheme is used to advance in time. We show that both algorithms can successfully solve the given problem and compare the accuracy of the results and their dependence on the number of qubits.