Abstract The application of quantum algorithms to classical problems is generally accompanied by significant bottlenecks when transferring data between quantum and classical states, often negating any intrinsic quantum advantage. Here we address this challenge for a well-known algorithm for linear systems of equations, originally proposed by Harrow, Hassidim and Lloyd (HHL), by adapting it into a predictor–corrector instead of a direct solver. Rather than seeking the solution at the next time step, the goal now becomes determining the change between time steps. This strategy enables the intelligent omission of computationally costly steps commonly found in many classical algorithms, while simultaneously mitigating the notorious readout problems associated with extracting solutions from a quantum state. Random or regularly performed skips instead lead to simulation failure. We demonstrate that our methodology secures a useful polynomial advantage over a conventional application of the HHL algorithm. The practicality and versatility of the approach are illustrated through applications in various fields such as smoothed particle hydrodynamics, plasma simulations, and reactive flow configurations. Moreover, the proposed algorithm is well suited to run asynchronously on future heterogeneous hardware infrastructures and can effectively leverage the synergistic strengths of classical as well as quantum compute resources.
Abstract The application of quantum algorithms to classical problems is generally accompanied by significant bottlenecks when transferring data between quantum and classical states, often negating any intrinsic quantum advantage. Here we address this challenge for a well-known algorithm for linear systems of equations, originally proposed by Harrow, Hassidim and Lloyd (HHL), by adapting it into a predictor–corrector instead of a direct solver. Rather than seeking the solution at the next time step, the goal now becomes determining the change between time steps. This strategy enables the intelligent omission of computationally costly steps commonly found in many classical algorithms, while simultaneously mitigating the notorious readout problems associated with extracting solutions from a quantum state. Random or regularly performed skips instead lead to simulation failure. We demonstrate that our methodology secures a useful polynomial advantage over a conventional application of the HHL algorithm. The practicality and versatility of the approach are illustrated through applications in various fields such as smoothed particle hydrodynamics, plasma simulations, and reactive flow configurations. Moreover, the proposed algorithm is well suited to run asynchronously on future heterogeneous hardware infrastructures and can effectively leverage the synergistic strengths of classical as well as quantum compute resources.
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.
Recent advancements in quantum computing provide opportunities to address the challenges of computational resource limitations in computational fluid dynamics (CFD). This work presents a hybrid quantum-classical CFD framework that leverages quantum linear algorithms to practical flow simulations. By transforming nonlinear fluid dynamics problems into linear systems, the framework employs quantum linear algorithms to compute solutions, effectively integrating quantum and classical computing approaches. Furthermore, this framework employs subspace methods to map the original large-scale linear systems to small ones solved by quantum linear algorithms, thereby enabling the solution of large-scale problems using the currently limited quantum resources. We applied the famous Harrow–Hassidim–Lloyd (HHL) algorithm and variational quantum linear solver to simulate complex flows, including unsteady flows around a cylinder, simulation of an aircraft with turbulence model, and combustion flows. The performance and quantum resource consumption of these algorithms were evaluated in practical flow scenarios, demonstrating their effectiveness with an average relative error lower than 0.001%. We conducted quantum resource and convergence analyses to adapt this framework for use with near-term quantum computers. Our framework supports simulations of up to 9.4 × 106 grid cells, marking a significant advancement from previous quantum approaches limited to simple flows. This paper not only offers a way for employing quantum linear algorithms in solving complex flow problems but also provides insights into optimizing quantum algorithms for CFD applications, pushing forward the practical utilization of quantum CFD.
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.
Efficiently solving large-scale sparse linear systems poses a significant challenge in computational science, especially in fields such as physics, engineering, machine learning, and finance. Traditional classical algorithms face scalability issues as the size of these systems increases, leading to performance degradation. On the other hand, quantum algorithms, like the Harrow-Hassidim-Lloyd (HHL) algorithm, offer exponential speedups for solving linear systems, yet they are constrained by the current state of quantum hardware and sensitivity to matrix condition numbers. This paper introduces a hybrid classical-quantum algorithm that combines CUDA-accelerated preconditioning techniques with the HHL algorithm to solve sparse linear systems more efficiently. The classical GPU parallelism is utilised to preprocess and precondition the matrix, reducing its condition number, while quantum computing is employed to solve the preconditioned system using the HHL algorithm. Additionally, the algorithm integrates machine learning models, particularly reinforcement learning, to dynamically optimise system parameters, such as block sizes and preconditioning stratgies, based on real-time performance data. Our experimental results show that the proposed approach not only surpasses traditional methods in speed and scalability but also mitigates some of the inherent limitations of quantum algorithms. This work pushes the boundaries of efficient computing and provides a foundation for future advancements in hybrid computational frameworks.
Quantum computing has generated significant expectations within the scientific community. However, the development of noise-resilient quantum computers capable of addressing a broad class of practical problems is expected to require at least another decade. During this ongoing evolution, rigorous step-by-step tutorials are essential not only to clarify fundamental concepts but also to disseminate them beyond the core quantum computing community to the wider engineering audience interested in applications. In many engineering fields, the efficient solution of linear systems of equations is a central computational task. For this reason, although it remains an open question whether the Harrow–Hassidim–Lloyd (HHL) algorithm will ultimately play a decisive role in future fault-tolerant application-scale quantum (FASQ) computers, clear and rigorous tutorials on this paradigmatic algorithm remain of fundamental importance. While well-structured introductions to elementary implementations of HHL already exist, certain simplifying assumptions—appropriate for pedagogical clarity—may limit their extension to more general or application-oriented settings. In this work, the HHL algorithm is presented through a detailed analytical treatment, followed by a complete example including a generic module for eigenvalue inversion. A corresponding Jupyter Notebook implementation in Qiskit is provided to support practical understanding. After working through the material, readers are expected to gain a clearer appreciation of how quantum computing concepts operate in practice and why quantum approaches are of interest for combinatorial optimization and machine learning problems.
Quantum computation offers a promising alternative to classical computing methods in many areas of numerical science, with algorithms that make use of the unique ways in which quantum computers store and manipulate data often achieving dramatic improvements in performance over their classical counterparts. The potential efficiency of quantum computers is particularly important for numerical simulations, where the capabilities of classical computing systems are often insufficient for the analysis of real-world problems. In this work, we study problems involving the solution of matrix equations, for which there is limited research on efficient, general, and fully implemented quantum procedures. We develop a generalization of the Harrow/Hassidim/Lloyd algorithm by providing an alternative unitary for eigenphase estimation. This unitary, which we have adopted from research in the area of quantum walks, has the advantage of being well defined for any arbitrary matrix equation, thereby allowing the solution procedure to be directly implemented on quantum hardware for any well-conditioned system. The procedure is most useful for sparse matrix equations, as it allows for the inverse of a matrix to be applied with O(sN log(N)) complexity, where N is the number of unknowns, and s is an upper bound on the number of nonzero elements in any row of the system matrix. This efficiency is competitive with the most sophisticated available classical solvers, and is independent of the structure of the input matrix. Hence, this procedure represents an important step towards the achievement of quantum advantage over classical algorithms for practical problems in numerical science.