Efficient Variational Quantum Linear Solver for Structured Sparse Matrices

TL;DR

This paper introduces a new basis for variational quantum linear solvers that leverages the structure of sparse matrices, significantly reducing the complexity of quantum circuit implementations for solving PDE-related linear systems.

Contribution

The authors propose an alternative basis that exploits matrix sparsity, enabling more efficient quantum algorithms for structured sparse matrices in VQLS, with logarithmic scaling of tensor product terms.

Findings

Reduced number of tensor product terms scales logarithmically with matrix size.

Efficient quantum circuits are designed using unitary completion for non-unitary basis operators.

Comparison shows advantages over existing methods like unitary dilation and Bell basis measurement.

Abstract

We develop a novel approach for efficiently applying variational quantum linear solver (VQLS) in context of structured sparse matrices. Such matrices frequently arise during numerical solution of partial differential equations which are ubiquitous in science and engineering. Conventionally, Pauli basis is used for linear combination of unitary (LCU) decomposition of the underlying matrix to facilitate the evaluation the global/local VQLS cost functions. However, Pauli basis in worst case can result in number of LCU terms that scale quadratically with respect to the matrix size. We show that by using an alternate basis one can better exploit the sparsity and underlying structure of matrix leading to number of tensor product terms which scale only logarithmically with respect to the matrix size. Given this new basis is comprised of non-unitary operators, we employ the concept of unitary completion to design efficient quantum circuits for computing the global/local VQLS cost functions. We compare our approach with other related concepts in the literature including unitary dilation and measurement in Bell basis, and discuss its pros/cons while using VQLS applied to Heat equation as an example.