Quantum linear system solvers: A survey of algorithms and applications
Mauro E. S. Morales, Lirandë Pira, Philipp Schleich, Kelvin Koor, Pedro C. S. Costa, Dong An, Alán Aspuru‐Guzik, Lin Lin, Patrick Rebentrost, Dominic William Berry
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.
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.
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(logN), in contrast to classical approaches, which in the general case exhibit a complexity of O(N3), 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.
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.
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.
Systems of linear equations are ubiquitous across science, engineering, machine learning, and even finance. While classical methods can be prohibitively slow for large-scale problems, quantum linear systems algorithms offer the potential for exponential speedup in certain parameter regimes. However, a significant gap persists between this theoretical promise and practical implementation, as the advantages are often obscured by the substantial quantum resources and high sensitivity to noise inherent in current quantum hardware. One way to bridge this gap is through the use of Iterative Refinement, a classical post-processing scheme that can exponentially improve the accuracy to which a linear system of equations can be solved using low-precision arithmetic. In the context of quantum linear systems algorithms, such as the HHL algorithm proposed by Harrow, Hassidim, and Loyd, Iterative Refinement can greatly reduce the quantum resources required to calculate an accurate solution in terms of tomography cost, circuit volume, and fault-tolerant overhead. Here, we compute and benchmark highly precise solutions to linear systems of equations of up to eight variables by running HHL with Iterative Refinement on NISQ quantum computers. We also present our open-source implementation, emphasizing that our circuit is not tailored to specific problem instances, as most available implementations are.
Abstract This study explores the application of the Harrow–Hassidim–Lloyd (HHL) algorithm for solving linear systems within the context of information security and quantum computing. Linear systems are fundamental in scientific, engineering, financial, and economic applications, impacting tasks such as differential equation solving, regression analysis, and Fourier transforms. In the quantum domain, effective implementation of the HHL algorithm relies on proper utilization of Pauli matrices and the choice of computational bases, including the computational basis and Hadamard basis. This work highlights how basis selection influences algorithmic efficiency and solution accuracy, offering insights for optimizing quantum computation workflows. By examining the HHL algorithm, we demonstrate its potential for accelerating linear system solutions and enhancing information processing capabilities, which may contribute to more secure and efficient computational methodologies in diverse application areas.