A novel quantum Fourier ordinary differential equation solver for solving linear and nonlinear partial differential equations

In this work, a novel quantum Fourier ordinary differential equation (ODE) solver is proposed to solve both linear and nonlinear partial differential equations (PDEs). Traditional quantum ODE solvers transform a PDE into an ODE system via spatial discretization and then integrate it, thereby converting the task of solving the PDE into computing the integral for the driving function f(x). These solvers rely on the quantum amplitude estimation algorithm, which requires the driving function f(x) to be within the range of [0, 1] and necessitates the construction of a quantum circuit for the oracle R that encodes f(x). This construction can be highly complex, even for simple functions like f(x) = x. An important exception arises for the specific case of f(x) = sin2(mx + c), which can be encoded more efficiently using a set of Ry rotation gates. To address these challenges, we expand the driving function f(x) as a Fourier series and propose the quantum Fourier ODE Solver. This approach not only simplifies the construction of the oracle R but also removes the restriction that f(x) must lie within [0,1]. The proposed method was evaluated by solving several representative linear and nonlinear PDEs, including the Navier–Stokes (N–S) equations. The results show that the quantum Fourier ODE solver produces results that closely match both analytical and reference solutions.