Resource-efficient quantum algorithm for linear systems of equations

TL;DR

This paper introduces the Shadow Quantum Linear Solver (SQLS), a resource-efficient quantum algorithm that combines variational methods and classical shadows to solve linear systems with fewer qubits and exponential advantages over existing approaches.

Contribution

The paper presents the SQLS, a novel hybrid quantum algorithm that reduces resource requirements and demonstrates practical application to solving the 2D Laplace Equation.

Findings

SQLS requires logarithmic qubits in system size.

Exponential advantage in circuit execution per cost function evaluation.

Successful application to discretized Laplace Equation.

Abstract

Finding the solution to linear systems is at the heart of many applications in science and technology. Over the years a number of algorithms have been proposed to solve this problem on a digital quantum device, yet most of these are too demanding to be applied to the current noisy hardware. In this work, an original algorithmic procedure to solve the Quantum Linear System Problem (QLSP) is presented, which combines ideas from Variational Quantum Algorithms (VQA) and the framework of classical shadows. The result is the Shadow Quantum Linear Solver (SQLS), a quantum algorithm solving the QLSP avoiding the need for large controlled unitaries, requiring a number of qubits that is logarithmic in the system size. In particular, our heuristics show an exponential advantage of the SQLS in circuit execution per cost function evaluation when compared to other notorious variational approaches to solving linear systems of equations. We test the convergence of the SQLS on a number of linear systems, and results highlight how the theoretical bounds on the number of resources used by the SQLS are conservative. Finally, we apply this algorithm to a physical problem of practical relevance, by leveraging decomposition theorems from linear algebra to solve the discretized Laplace Equation in a 2D grid for the first time using a hybrid quantum algorithm.