Variational quantum algorithm for the Poisson equation

The Poisson equation has wide applications in many areas of science and engineering. Although there are some quantum algorithms that can efficiently solve the Poisson equation, they generally require a fault-tolerant quantum computer, which is beyond the current technology. We propose a variational quantum algorithm (VQA) to solve the Poisson equation, which can be executed on noisy intermediate-scale quantum devices. In detail, we first adopt the finite-difference method to transform the Poisson equation into a linear system. Then, according to the special structure of the linear system, we find an explicit tensor product decomposition, with only $(2{log}_{2}n+1)$ items, of its coefficient matrix under a specific set of simple operators, where $n$ is the dimension of the coefficient matrix. This implies that the proposed VQA needs fewer quantum measurements, which dramatically reduces the required quantum resources. Additionally, we design observables to efficiently evaluate the expectation values of the simple operators on a quantum computer. Numerical experiments demonstrate that our algorithm can solve the Poisson equation.