MiS Preprint Repository

Solving a Poisson equation is generally reduced to solve a linear system with a coefficient matrix $A$ of entries $a_{ij}$, $i,j=1,2,...,n$, from the discretized Poisson equation. Although the Variational Quantum Algorithms (VQAs) are promising algorithms to solve the discretized Poisson equation, they generally require that $A$ be decomposed into a sum of $O[\text{poly}(\text{log}_2n)]$ simple operators in order to evaluate efficiently the loss function. An tensor product decomposition of $A$ with $2\text{log}_2n+1$ terms has been explored in previous works. In this paper, based on the decomposition of sparse Hamiltonians we greatly reduce the number of the terms. We first write the loss function in terms of the operator $\sigma_x\otimes A$ with $\sigma_x$ denoting the standard Pauli operator. Then for one-dimensional Poisson equations with different boundary conditions and for $d$-dimensional Poisson equations with Dirichlet boundary conditions, we decompose $\sigma_x\otimes A$ into a sum of at most 7 and $(4d+1)$ hermitian, one-sparse and self-inverse operators, respectively. We design explicitly the quantum circuits to evaluate efficiently the loss functions. The decomposition method and the design of quantum circuits can also be easily extended to linear systems with hermitian and sparse coefficient matrices satisfying $a_{i,i+c}=a_{c}$ for $c=0,1,\cdots,n-1$ and $i=0,\cdots,n-1-c$.