Variational quantum algorithms for Poisson equations based on the decomposition of sparse Hamiltonians

TL;DR

This paper introduces a novel decomposition method for sparse Hamiltonians to improve variational quantum algorithms for solving Poisson equations, reducing the number of terms and designing efficient quantum circuits.

Contribution

It presents a new sparse Hamiltonian decomposition technique that significantly reduces the number of terms needed for variational quantum algorithms solving Poisson equations.

Findings

Decomposes $\sigma_xigotimes A$ into at most 7 and (4d+1) Hermitian operators for 1D and d-dimensional cases.

Designs explicit quantum circuits for efficient loss function evaluation.

Method extends to linear systems with specific Hermitian and sparse matrices.

Abstract

Solving a Poisson equation is generally reduced to solving 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 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. A 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 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 the one-dimensional Poisson equations with different boundary conditions and for the $d$-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 function. 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 $i=0,\cdots,n-1-c$.