Quantum Functional Expansion to Solve Stochastic Differential Equations

Recently, quantum computer started being applied to solve differential equations, which are commonly used to model system behaviors in science and engineering disciplines. However, the existing quantum differential equation solvers involve problem-specific conversion or spatial domain discretization, which affect their generality and scalability. In this paper, we propose a quantum functional expansion (QFE) framework as a generic and scalable approach to solve differential equations on quantum computer. The main idea of QFE is to perform functional expansions on the differential equation first and then encode the solution with quantum states, where the computational basis represents the basis functions of expansion. The variational quantum imaginary-time simulator is then used to solve the converted ordinary differential equations. The QFE framework is demonstrated with two stochastic differential equations. The numerical solutions match very well with the analytical ones even with two to four qubits.