# Randomized Row and Column Iterative Methods with a Quantum Computer

## TL;DR

This paper develops quantum algorithms for classical iterative linear solvers, achieving exponential speedups over traditional methods by leveraging block-encoding and efficient state preparation.

## Contribution

It introduces fast quantum implementations of the Kaczmarz and coordinate descent methods, significantly improving their computational efficiency.

## Key findings

- Quantum algorithms achieve exponential speedup over classical methods.
- Complexity is nearly linear in the number of steps.
- Assumes efficient quantum state preparation for rows and columns.

## Abstract

We consider the quantum implementations of the two classical iterative solvers for a system of linear equations, including the Kaczmarz method which uses a row of coefficient matrix in each iteration step, and the coordinate descent method which utilizes a column instead. These two methods are widely applied in big data science due to their very simple iteration schemes. In this paper we use the block-encoding technique and propose fast quantum implementations for these two approaches, under the assumption that the quantum states of each row or each column can be efficiently prepared. The quantum algorithms achieve exponential speed up at the problem size over the classical versions, meanwhile their complexity is nearly linear at the number of steps.