# Quantum linear systems algorithms: a primer

## TL;DR

This paper provides a detailed overview of the HHL quantum algorithm for solving linear systems, including its improvements, subroutines, and applications, highlighting its potential exponential speed-up over classical methods.

## Contribution

It offers a comprehensive explanation of the HHL algorithm, its enhancements, and related quantum subroutines, serving as a foundational primer on quantum linear system solvers.

## Key findings

- HHL achieves exponential speed-up over classical algorithms.
- Improvements include variable-time amplitude amplification and Fourier/Chebyshev-based LCUs.
- Discussion of quantum data loading and quantum singular value estimation.

## Abstract

The Harrow-Hassidim-Lloyd (HHL) quantum algorithm for sampling from the solution of a linear system provides an exponential speed-up over its classical counterpart. The problem of solving a system of linear equations has a wide scope of applications, and thus HHL constitutes an important algorithmic primitive. In these notes, we present the HHL algorithm and its improved versions in detail, including explanations of the constituent sub- routines. More specifically, we discuss various quantum subroutines such as quantum phase estimation and amplitude amplification, as well as the important question of loading data into a quantum computer, via quantum RAM. The improvements to the original algorithm exploit variable-time amplitude amplification as well as a method for implementing linear combinations of unitary operations (LCUs) based on a decomposition of the operators using Fourier and Chebyshev series. Finally, we discuss a linear solver based on the quantum singular value estimation (QSVE) subroutine.