# Quantum linear systems algorithms: a primer

arXiv: https://arxiv.org/abs/1802.08227 · HTML: https://mlsanity.com/paper/1802.08227/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Feb 2018.

## Built on by (most cited first, 14 of 14)

- [A Primer on Quantum Machine Learning](https://mlsanity.com/paper/2511.15969/) (2025)
- [Quantum Machine Learning For Classical Data](https://mlsanity.com/paper/2105.03684/) (2021)
- [Hybrid Quantum-Classical Learning for Multiclass Image Classification](https://mlsanity.com/paper/2508.18161/) (2025)
- [Quantum diffusion map for nonlinear dimensionality reduction](https://mlsanity.com/paper/2106.07302/) (2021)
- [An Introduction to Quantum Computing for Statisticians and Data Scientists](https://mlsanity.com/paper/2112.06587/) (2021)
- [Q-Newton: Hybrid Quantum-Classical Scheduling for Accelerating Neural Network Training with Newton's Gradient Descent](https://mlsanity.com/paper/2405.00252/) (2024)
- [Towards identifying possible fault-tolerant advantage of quantum linear system algorithms in terms of space, time and energy](https://mlsanity.com/paper/2502.11239/) (2025)
- [Quantum Deep Learning Still Needs a Quantum Leap](https://mlsanity.com/paper/2511.01253/) (2025)
- [Quantum Levenberg--Marquardt Algorithm for optimization in Bundle Adjustment](https://mlsanity.com/paper/2203.02311/) (2022)
- [A Quantum-Powered Photorealistic Rendering](https://mlsanity.com/paper/2211.03418/) (2022)
- [An Exponential Separation Between Quantum and Quantum-Inspired Classical Algorithms for Linear Systems](https://mlsanity.com/paper/2411.02087/) (2024)
- [Sublinear-Time Algorithms for Diagonally Dominant Systems and Applications to the Friedkin-Johnsen Model](https://mlsanity.com/paper/2509.13112/) (2025)
- [HHL with a Coherent Fourier Oracle: A Proof-of-Concept Quantum Architecture for Joint Melody-Harmony Generation](https://mlsanity.com/paper/2604.20882/) (2026)
- [When AI meets quantum information: A comprehensive review](https://mlsanity.com/paper/2607.00365/) (2026)

## Abstract (arXiv)

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.