Variational algorithms for linear algebra

TL;DR

This paper introduces variational quantum algorithms tailored for linear algebra tasks that are compatible with noisy intermediate-scale quantum devices, enabling efficient solutions for linear systems and matrix operations.

Contribution

It presents novel variational algorithms, including Hamiltonian morphing and adaptive ansatz, for solving linear algebra problems on NISQ devices, with practical implementation and high fidelity.

Findings

Achieved 99.95% solution fidelity on IBM quantum cloud

Demonstrated effectiveness for sparse matrix problems

Applicable to quantum simulation and optimization tasks

Abstract

Quantum algorithms have been developed for efficiently solving linear algebra tasks. However, they generally require deep circuits and hence universal fault-tolerant quantum computers. In this work, we propose variational algorithms for linear algebra tasks that are compatible with noisy intermediate-scale quantum devices. We show that the solutions of linear systems of equations and matrix-vector multiplications can be translated as the ground states of the constructed Hamiltonians. Based on the variational quantum algorithms, we introduce Hamiltonian morphing together with an adaptive ansatz for efficiently finding the ground state, and show the solution verification. Our algorithms are especially suitable for linear algebra problems with sparse matrices, and have wide applications in machine learning and optimisation problems. The algorithm for matrix multiplications can be also used for Hamiltonian simulation and open system simulation. We evaluate the cost and effectiveness of our algorithm through numerical simulations for solving linear systems of equations. We implement the algorithm on the IBM quantum cloud device with a high solution fidelity of 99.95%.