New Quantum Algorithm For Solving Linear System of Equations

TL;DR

This paper introduces a novel quantum algorithm for solving linear systems using a gradient descent approach based on density states, offering improved efficiency and convergence guarantees compared to existing quantum methods.

Contribution

The paper presents a new quantum linear solver algorithm that employs gradient descent and density state formalism, enhancing efficiency and convergence analysis.

Findings

Demonstrates improved complexity over previous quantum algorithms

Provides convergence guarantees for the gradient descent-based quantum solver

Shows potential for more efficient quantum solutions to linear systems

Abstract

Linear equations play a pivotal role in many areas of science and engineering, making efficient solutions to linear systems highly desirable. The development of quantum algorithms for solving linear systems has been a significant breakthrough, with such algorithms rapidly becoming some of the most influential in quantum computing. Subsequent advances have focused on improving the efficiency of quantum linear solvers and extending their core techniques to address various challenges, such as least-squares data fitting. In this article, we introduce a new quantum algorithm for solving linear systems based on the gradient descent method. Inspired by the vector/density state formalism, we represent a point, or vector, as a density state-like entity, enabling us to leverage recent advancements in quantum algorithmic frameworks, such as block encoding, to directly manipulate operators and carry out the gradient descent method in a quantum setting. The operator corresponding to the intermediate solution is updated iteratively, with a provable guarantee of convergence. The quantum state representing the final solution to the linear system can then be extracted by further measurement. We provide a detailed complexity analysis and compare our approach with existing methods, demonstrating significant improvement in certain aspects.