The Poisson equation has many applications across the broad areas of science and engineering. Most quantum algorithms for the Poisson solver presented so far, either suffer from lack of accuracy and/or are limited to very small sizes of the problem, and thus have no practical usage. Here we present an advanced quantum algorithm for solving the Poisson equation with high accuracy and dynamically tunable problem size. After converting the Poisson equation to the linear systems through the finite difference method, we adopt the Harrow-Hassidim-Lloyd (HHL) algorithm as the basic framework. Particularly, in this work we present an advanced circuit that ensures the accuracy of the solution by implementing non-truncated eigenvalues through eigenvalue amplification as well as by increasing the accuracy of the controlled rotation angular coefficients, which are the critical factors in the HHL algorithm. We show that our algorithm not only increases the accuracy of the solutions, but also composes more practical and scalable circuits by dynamically controlling problem size in the NISQ devices. We present both simulated and experimental results, and discuss the sources of errors. Finally, we conclude that overall results on the quantum hardware are dominated by the error in the CNOT gates.
This paper assesses the numerical accuracy of the Harrow-Hassidim-Lloyd (HHL) algorithm in solving a finite-difference approximation of the Poisson equation, with the Thomas algorithm used as a comparative benchmark. In the one-dimensional setting with homogeneous boundary conditions, the HHL algorithm exhibits numerical accuracy comparable to the Thomas algorithm; however, the HHL algorithm exhibits increased sensitivity to variations in input parameters for simulations with nonhomogeneous boundary conditions. While the current implementation of the HHL algorithm is limited to solving the one-dimensional Poisson equation, we propose a hybrid framework to extend the applicability of the HHL algorithm to solving the two-dimensional Poisson equation. Our results show that although both algorithms perform similarly for small systems of 8 × 8 mesh nodes, the HHL algorithm shows increased input-parameter sensitivity for systems of 16 × 16 mesh nodes, indicating that further development is needed to scale the hybrid approach effectively.
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.
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 simulation of electromagnetic (EM) structures is still in its infancy. In this paper, we investigate the possibility of applying quantum computing to solve the finite element equations for EM problems. Specifically, we propose to leverage a milestone in quantum computing, i.e., the Harrow/Hassidim/Lloyd (HHL) algorithm, to solve the finite element equations in the EM domain. To do this, we first present a systematic method to reformulate the finite element equations into quantum computation format and apply the HHL algorithm to solve the new equations. Taking advantage of the special properties of finite element matrices in EM problems, we then devise a stepwise algorithm to efficiently determine the suitable values for the hyperparameters of HHL. The proposed method can in theory solve the EM finite element equations in logarithmic time relative to the size of the finite element matrix. This is exponentially faster than that achievable by classical computation as the matrix size increases. A two-dimensional EM example is used to illustrate how the proposed method can be used to find the solutions to EM problems through quantum computation.
Solving systems of linear equations is important both in classical and quantum computing. Harrow-Hassidim-Lloyd algorithm (HHL algorithm), a quantum algorithm for solving systems of linear equations, has achieved exponentially improved performance compared with corresponding classical algorithm. We will show in this article that the HHL algorithm is actually a duality quantum algorithm with non- unitary transformation from initial to the final state. We present a detailed realization of the HHL algorithm in a duality quantum computing formalism that allows the construction of non- unitary operations. The divider structure, combinor structure, and the total quantum circuit for the HHL algorithm in the duality quantum computing formalism are given explicitly.