Variational quantum algorithm for the Poisson equation based on the banded Toeplitz systems

For solving the Poisson equation it is usually possible to discretize it into solving the corresponding linear system $Ax=b$.Variational quantum algorithms (VQAs) for the discreted Poisson equation have been studied before. We give a VQA based on the banded Toeplitz systems for solving the Poisson equation with respect to the structural features of matrix $A$. In detail, we decompose the matrix $A$ and $A^2$ into a linear combination of the corresponding banded Toeplitz matrix and sparse matrices with only a few non-zero elements. For the one-dimensional Poisson equation with different boundary conditions and the $d$-dimensional Poisson equation with Dirichlet boundary conditions, the number of decomposition terms is less than the work in [Phys. Rev. A 108, 032418 (2023)]. Based on the decomposition of the matrix, we design quantum circuits that evaluate efficiently the cost function.Additionally, numerical simulation verifies the feasibility of the proposed algorithm. In the end, the VQAs for linear systems of equations and matrix-vector multiplications with $K$-banded Teoplitz matrix $T_n^K$ are given, where $T_n^K\in R^{n\times n}$ and $K\in O({\rm ploy}\log n)$.