Quantum iterative methods for solving differential equations with application to computational fluid dynamics

Quantum methods are proposed for solving differential equations that are based on a gradual improvement of the solution via an iterative process, and are targeted at applications in fluid dynamics. First, the Jacobi iteration is implemented on a quantum register that utilizes a linear combination of unitaries (LCU) approach to store the trajectory information. Second, quantum methods are extended to Gauss-Seidel iterative methods. Additionally, a quantum-suitable resolvent decomposition is proposed based on the Woodbury identity. From a technical perspective, tools are developed and utilized for the block encoding of specific matrices as well as their multiplication. The approach is benchmarked on paradigmatic fluid dynamics problems. These results stress that instead of inverting large matrices, one can program quantum computers to perform multigrid-type computations and leverage corresponding advances in scientific computing.