A Novel Variational Quantum Algorithm for Solving High-Dimensional Partial Differential Equations in Climate Modeling

Abstract

This paper proposes a variational quantum algorithm to address the large-scale and low computational efficiency issues of high-dimensional partial differential equations (PDEs) in climate science. In this method, parameterized quantum circuits encode the discrete solutions of PDEs into quantum states. Then, through repeated quantum measurements, we can obtain the expectation values of the operators. By iteratively adjusting the circuit parameters, a hybrid quantum-classical optimization loop can reduce the error in the solution. The Schrödinger equation and the Poisson equation are benchmark partial differential equations for numerical experiments, and they are four-dimensional and five-dimensional, respectively. According to the above results, the convergence speed is relatively fast; after 100 iterations, the median relative errors of all test cases are less than 1.6% and 3.1%, respectively, with the final target loss ranging between 0.014 and 0.045. The above comparison indicates that in high-dimensional and ill-posed problems, quantum solvers can achieve the same or better accuracy and robustness compared to traditional methods. Resource analysis indicates that quantum algorithms use less memory and fewer sub-exponential growth gates and runtime in larger systems. According to the above findings, variational quantum solvers can be used for complex scientific computing problems. In addition, they hope that future climate models will be more accurate and efficient.