Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

A quantum linear system solver

Steven Herbert

Published Dec 8, 2025• 0 citations
Publisher

Abstract

Abstract In a sense, gate-based quantum computers are just machines that apply (extremely) large matrices to input states (vectors), and so it is natural to ask whether linear algebra applications can benefit from quantum computing. In the case of solving linear systems, this question has been answered with an emphatic ‘yes’. In 2009, Aram Harrow, Avinatan Hassidim, and Seth Lloyd proposed a quantum algorithm to solve linear systems exponentially faster than the best classical counterpart. The quantum linear systems solver is ubiquitously known by the initials of its inventors, ‘HHL’. As well as detailing HHL, this chapter also explores some of the quantum machine learning applications that have been proposed based on HHL along with results showing that these algorithms are often ‘dequantised’ in the problem space of interest – i.e., there exists an efficient classical ‘quantum-inspired’ algorithm.

DOI
10.1093/9780191964381.003.0013

Related papers

Quantum linear system solvers: A survey of algorithms and applications

Mauro E. S. Morales, Lirandë Pira, Philipp Schleich +7 more

Solving linear systems of equations plays a fundamental role in numerous computational problems from different fields of science. The widespread use of numerical methods to solve these systems motivates investigating the feasibility of solving linear systems problems using quantum computers. In this work, we provide a survey of the main advances in quantum linear systems algorithms, together with some applications. We summarize and analyze the main ideas behind some of the algorithms for the quantum linear systems problem in the literature. The analysis begins by examining the Harrow-Hassidim-Lloyd (HHL) solver. We note its limitations and reliance on computationally expensive quantum methods, then highlight subsequent research efforts which aimed to address these limitations and optimize runtime efficiency and precision via various paradigms. We focus in particular on the post-HHL enhancements which have paved the way towards optimal lower bounds with respect to error tolerance and condition number. By doing so, we propose a taxonomy that categorizes these studies. Furthermore, by contextualizing these developments within the broader landscape of quantum computing, we explore the foundational work that have inspired and informed their development, as well as subsequent refinements. Finally, we discuss the potential applications of these algorithms in differential equations, quantum machine learning, and many-body physics.

arXiv:2411.02522•Reviews of Modern Physics•Jan 8, 2026•4 citations

Quantum linear systems algorithms: a primer

Danial Dervovic, Mark Herbster, Peter Mountney +3 more

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.

arXiv:1802.08227•Feb 22, 2018•62 citations

COMPARATIVE ANALYSIS OF CLASSICAL AND QUANTUM ALGORITHMS FOR SOLVING LINEAR SYSTEMS OF EQUATIONS: THEORETICAL INSIGHTS AND BENCHMARKING CHALLENGES

Journal of Theoretical and Applied Information Technology

Modern computers utilize a model based on a simple Turing machine concept. This study contains an extensive comparative review of classical and quantum algorithm approaches to solving a system of linear equations. The study details standard classical approaches like Gaussian Elimination or Conjugate Gradient Method and compares these with the algorithm of quantum theory algorithm HHL (Harrow-Hassidim-Lloyd). It considers the logic behind each of them, their benchmarks, scalability, implementation difficulties and practical use. Benchmarking results illustrate the fact that a classical approach continues to do especially good for small to moderate sized problems. However, quantum computation has the potential for an exponential speedup using algorithms like HHL in well-conditioned sparse matrices. The current limits of the hardware are discussed and future research directions needed to be able to utilize quantum computing for larger scale linear algebra problems are presented.

Journal of Theoretical and Applied Information Technology•Jun 30, 2025•0 citations

The HHL algorithm: Implementation and research directions

Varsha Sambhaje, Anju Chaurasia

Linear systems of equations lie at the heart of numerous scientific and engineering challenges. In cutting-edge arena like artificial intelligence, machine learning and neuro-computation, these systems serve as a fundamental tool for mathematical modeling. Classical algorithms for solving linear systems have been extensively developed and forms the backbone of diverse applications across various scientific disciplines. While classical algorithms exist for solving linear systems, they often encounter limitations termed “NP-completeness” as data complexity increases. The emerging field of quantum computing offers a revolutionary approach to deal with these kinds of problems. The Harrow–Hassidim–Lloyd (HHL) algorithm tackles these challenges and opens new avenues for research. This study delves into the contemporary effectiveness of the HHL algorithm to address systems of linear equations. By examining recent research in quantum machine learning, we aim to assess the HHL algorithm’s potential to revolutionize the process of optimizing hyperparameters for machine learning models, resulting in increased efficiency and cost savings. This paper meticulously analyzes the HHL algorithm and explores its evolution from conception to the latest advancements. A comprehensive examination of the HHL algorithm, including its evolution over time, is thoroughly explored. The investigation delves into the potential challenges and limitations that might hinder the practical deployment of the HHL algorithm. Identifying these roadblocks will pave the way for future research and development efforts.

International Journal of Quantum Information•May 31, 2024•2 citations

Solving Linear Systems of Equations with the Quantum HHL Algorithm: A Tutorial on the Physical and Mathematical Foundations for Undergraduate Students

Lucas Queiroz Galvão, de Souza, Anna Beatriz M., A. Santos +2 more

Quantum computing enables the efficient resolution of complex problems, often outperforming classical methods across various applications. In 2009, Harrow, Hassidim and Lloyd proposed an algorithm for solving linear systems of equations, demonstrating exponential speedup (under ideal conditions) with a complexity of $poly(\log N)$, in contrast to classical approaches, which in the general case exhibit a complexity of $O(N^3)$, although they can achieve $O(N)$ in specific cases involving sparse matrices. This algorithm holds promise for advancements in machine learning, the solution of differential equations, linear regression, and cryptographic analysis. However, its structure is intricate, and there is a notable lack of detailed instructional materials in the literature. In this context, this paper presents a tutorial addressing the physical and mathematical foundations of the HHL algorithm, aimed at undergraduate students, explaining its theoretical construction and its implementation for solving linear equation systems. After discussing the underlying mathematical and physical concepts, we present numerical examples that illustrate the evolution of the quantum circuit. Finally, the algorithm's complexity, limitations, and future prospects are analyzed. The examples are compared with their classical simulations, allowing for an operational assessment of the algorithm's performance.

arXiv:2509.16640•Sep 20, 2025•0 citations

Quantum Machine Learning

Santanu Pattanayak

In this chapter and the next one, we will explore the exciting areas of quantum machine learning and quantum deep learning. Machine learning and deep learning have seen great success in recent years because of the increase in the computational power at our disposal and because of the high-end research in these fields. Quantum machine learning presents an exciting opportunity to increase the computational efficiency of the existing machine learning algorithms as well as presents a way to tackle some of the more computationally intractable problems. In this chapter, we start with the Harrow-Hassidim-Lloyd algorithm, popularly known as HHL, which acts as the matrix inversion routine in the quantum computing domain. Hence, HHL will be the default choice for algorithms such as linear regression and least square support vector machines. Subsequently, we touch upon quantum linear regression and support vector machines in detail in this chapter. We will then move on to implementing quantum routines such as quantum dot product and quantum Euclidean distances since they are integral to several machine learning algorithms such as the k-means clustering and nearest neighbor methods. In this regard, we will implement the k-means clustering method in detail. Also, we will discuss how Grover’s algorithm can be used to optimize quantum objectives by illustrating its usage in the cluster assignment in the k-means algorithm. Principal component analysis is an important machine learning technique, and we will walk through its quantum implementation in great detail as well.

Apress eBooks•Jan 1, 2021•9 citations
Abstract In a sense, gate-based quantum computers are just machines that apply (extremely) large matrices to input states (vectors), and so it is natural to ask whether linear algebra applications can benefit from quantum computing. In the case of solving linear systems, this question has been answered with an emphatic ‘yes’. In 2009, Aram Harrow, Avinatan Hassidim, and Seth Lloyd proposed a quantum algorithm to solve linear systems exponentially faster than the best classical counterpart. The quantum linear systems solver is ubiquitously known by the initials of its inventors, ‘HHL’. As well as detailing HHL, this chapter also explores some of the quantum machine learning applications that have been proposed based on HHL along with results showing that these algorithms are often ‘dequantised’ in the problem space of interest – i.e., there exists an efficient classical ‘quantum-inspired’ algorithm.
Solving linear systems of equations plays a fundamental role in numerous computational problems from different fields of science. The widespread use of numerical methods to solve these systems motivates investigating the feasibility of solving linear systems problems using quantum computers. In this work, we provide a survey of the main advances in quantum linear systems algorithms, together with some applications. We summarize and analyze the main ideas behind some of the algorithms for the quantum linear systems problem in the literature. The analysis begins by examining the Harrow-Hassidim-Lloyd (HHL) solver. We note its limitations and reliance on computationally expensive quantum methods, then highlight subsequent research efforts which aimed to address these limitations and optimize runtime efficiency and precision via various paradigms. We focus in particular on the post-HHL enhancements which have paved the way towards optimal lower bounds with respect to error tolerance and condition number. By doing so, we propose a taxonomy that categorizes these studies. Furthermore, by contextualizing these developments within the broader landscape of quantum computing, we explore the foundational work that have inspired and informed their development, as well as subsequent refinements. Finally, we discuss the potential applications of these algorithms in differential equations, quantum machine learning, and many-body physics.
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.
Modern computers utilize a model based on a simple Turing machine concept. This study contains an extensive comparative review of classical and quantum algorithm approaches to solving a system of linear equations. The study details standard classical approaches like Gaussian Elimination or Conjugate Gradient Method and compares these with the algorithm of quantum theory algorithm HHL (Harrow-Hassidim-Lloyd). It considers the logic behind each of them, their benchmarks, scalability, implementation difficulties and practical use. Benchmarking results illustrate the fact that a classical approach continues to do especially good for small to moderate sized problems. However, quantum computation has the potential for an exponential speedup using algorithms like HHL in well-conditioned sparse matrices. The current limits of the hardware are discussed and future research directions needed to be able to utilize quantum computing for larger scale linear algebra problems are presented.
Linear systems of equations lie at the heart of numerous scientific and engineering challenges. In cutting-edge arena like artificial intelligence, machine learning and neuro-computation, these systems serve as a fundamental tool for mathematical modeling. Classical algorithms for solving linear systems have been extensively developed and forms the backbone of diverse applications across various scientific disciplines. While classical algorithms exist for solving linear systems, they often encounter limitations termed “NP-completeness” as data complexity increases. The emerging field of quantum computing offers a revolutionary approach to deal with these kinds of problems. The Harrow–Hassidim–Lloyd (HHL) algorithm tackles these challenges and opens new avenues for research. This study delves into the contemporary effectiveness of the HHL algorithm to address systems of linear equations. By examining recent research in quantum machine learning, we aim to assess the HHL algorithm’s potential to revolutionize the process of optimizing hyperparameters for machine learning models, resulting in increased efficiency and cost savings. This paper meticulously analyzes the HHL algorithm and explores its evolution from conception to the latest advancements. A comprehensive examination of the HHL algorithm, including its evolution over time, is thoroughly explored. The investigation delves into the potential challenges and limitations that might hinder the practical deployment of the HHL algorithm. Identifying these roadblocks will pave the way for future research and development efforts.
Quantum computing enables the efficient resolution of complex problems, often outperforming classical methods across various applications. In 2009, Harrow, Hassidim and Lloyd proposed an algorithm for solving linear systems of equations, demonstrating exponential speedup (under ideal conditions) with a complexity of poly(log⁡N)poly(\log N)poly(logN), in contrast to classical approaches, which in the general case exhibit a complexity of O(N3)O(N^3)O(N3), although they can achieve O(N)O(N)O(N) in specific cases involving sparse matrices. This algorithm holds promise for advancements in machine learning, the solution of differential equations, linear regression, and cryptographic analysis. However, its structure is intricate, and there is a notable lack of detailed instructional materials in the literature. In this context, this paper presents a tutorial addressing the physical and mathematical foundations of the HHL algorithm, aimed at undergraduate students, explaining its theoretical construction and its implementation for solving linear equation systems. After discussing the underlying mathematical and physical concepts, we present numerical examples that illustrate the evolution of the quantum circuit. Finally, the algorithm's complexity, limitations, and future prospects are analyzed. The examples are compared with their classical simulations, allowing for an operational assessment of the algorithm's performance.
In this chapter and the next one, we will explore the exciting areas of quantum machine learning and quantum deep learning. Machine learning and deep learning have seen great success in recent years because of the increase in the computational power at our disposal and because of the high-end research in these fields. Quantum machine learning presents an exciting opportunity to increase the computational efficiency of the existing machine learning algorithms as well as presents a way to tackle some of the more computationally intractable problems. In this chapter, we start with the Harrow-Hassidim-Lloyd algorithm, popularly known as HHL, which acts as the matrix inversion routine in the quantum computing domain. Hence, HHL will be the default choice for algorithms such as linear regression and least square support vector machines. Subsequently, we touch upon quantum linear regression and support vector machines in detail in this chapter. We will then move on to implementing quantum routines such as quantum dot product and quantum Euclidean distances since they are integral to several machine learning algorithms such as the k-means clustering and nearest neighbor methods. In this regard, we will implement the k-means clustering method in detail. Also, we will discuss how Grover’s algorithm can be used to optimize quantum objectives by illustrating its usage in the cluster assignment in the k-means algorithm. Principal component analysis is an important machine learning technique, and we will walk through its quantum implementation in great detail as well.