# Quantum gradient descent for linear systems and least squares

## TL;DR

This paper introduces a quantum method for performing gradient descent on affine functions, enabling faster solutions to linear systems and least squares problems with potential quantum speedups over classical algorithms.

## Contribution

It presents the first quantum approach for gradient descent on affine functions, with applications to linear systems and least squares, reducing computational costs.

## Key findings

- Quantum gradient descent can be performed efficiently for affine functions.
- The method reduces the cost of solving positive semidefinite linear systems.
- Quantum algorithms demonstrate potential for significant speedups in large-scale linear algebra problems.

## Abstract

Quantum machine learning and optimization are exciting new areas that have been brought forward by the breakthrough quantum algorithm of Harrow, Hassidim and Lloyd for solving systems of linear equations. The utility of {classical} linear system solvers extends beyond linear algebra as they can be leveraged to solve optimization problems using iterative methods like gradient descent. In this work, we provide the first quantum method for performing gradient descent when the gradient is an affine function. Performing $\tau$ steps of the gradient descent requires time $O(\tau C_S)$ for weighted least squares problems, where $C_S$ is the cost of performing one step of the gradient descent quantumly, which at times can be considerably smaller than the classical cost. We illustrate our method by providing two applications: first, for solving positive semidefinite linear systems, and, second, for performing stochastic gradient descent for the weighted least squares problem with reduced quantum memory requirements. We also provide a quantum linear system solver in the QRAM data structure model that provides significant savings in cost for large families of matrices.