Researchers are currently working on computational solutions based on quantum systems to accelerate the speed of complex mathematical models. This work presented how to formulate complex computational problems as a quantum system of linear equations and find solutions using Quantum Linear System Algorithm (QLSA), also called Quantum Harrow-Hassidim-Lloyd (HHL) algorithm. This paper showed experimental evaluation of multiple problem statements (curve-fitting functions, interpolating polynomials) as a quantum system of linear equations that involve computation of Vandermonde matrices as co-efficient matrices on IBM Quantum Information Software Kit for Quantum Computation (QISKit) platform. Along with a few examples demonstrating its evaluation on diagonal, Hermitian, and Non-Hermitian matrices as co-efficient matrices. The fidelity is used as a measure of performance for comparing the accuracy of quantum results with respect to existing classical solutions on IBM QISKit and drawing conclusions from the experimental results. Experimental evaluation shows that the fidelity depends on the sparsity of the input matrices and therefore the results vary depending on those matrices.
Researchers are currently working on computational solutions based on quantum systems to accelerate the speed of complex mathematical models. This work presented how to formulate complex computational problems as a quantum system of linear equations and find solutions using Quantum Linear System Algorithm (QLSA), also called Quantum Harrow-Hassidim-Lloyd (HHL) algorithm. This paper showed experimental evaluation of multiple problem statements (curve-fitting functions, interpolating polynomials) as a quantum system of linear equations that involve computation of Vandermonde matrices as co-efficient matrices on IBM Quantum Information Software Kit for Quantum Computation (QISKit) platform. Along with a few examples demonstrating its evaluation on diagonal, Hermitian, and Non-Hermitian matrices as co-efficient matrices. The fidelity is used as a measure of performance for comparing the accuracy of quantum results with respect to existing classical solutions on IBM QISKit and drawing conclusions from the experimental results. Experimental evaluation shows that the fidelity depends on the sparsity of the input matrices and therefore the results vary depending on those matrices.
One of the most significant developments in quantum computing is the Harrow-Hassidim-Lloyd (HHL) method, which can solve linear equation systems at exponential speedup. Because linear systems are essential to many scientific fields, including physics, engineering, and machine learning, this approach has great potential to revolutionize computational paradigms. The HHL algorithm is thoroughly examined in this work, with particular attention paid to its theoretical framework, real-world application utilizing IBM's Qiskit platform, and the difficulties in simulating quantum algorithms on noisy intermediate-scale quantum (NISQ) devices. Using parameters like fidelity, time complexity, and scalability, the research further evaluates the HHL algorithm's performance in comparison to traditional methods. According to the findings, quantum simulations work well for small-scale matrices like 2x2 and 4x4, but expanding the approach to bigger systems is still difficult because of hardware and software constraints. Finally, the research emphasizes the key directions for advancing quantum hardware and algorithms to overcome current scalability challenges, enabling broader applicability of the HHL algorithm in solving complex linear systems.
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.
HHL algorithm is a basic quantum algorithm, which is mainly used to solve linear equations. It has a wide range of potential applications in science, engineering, finance, economics and other fields. Although the current scale of quantum computers is small, we can still demonstrate basic quantum algorithms. In this paper, taking the HHL quantum algorithm as an example, the quantum circuit of the HHL quantum algorithm corresponding to 4 qubits and 7 qubits is given, and the simulation verification is carried out on the IBM qiskit development platform. The results show that when the input matrix is a second-order matrix, the fidelity is very high, but when the input is a fourth-order matrix, the probability that the result is close to the real value is very high only when the sparsity of the matrix is relatively strong. However, when the sparsity of the input matrix is very weak, the result will be very different from the real value.
We propose a hybrid quantum algorithm based on the Harrow-Hassidim-Lloyd (HHL) algorithm for solving a system of linear equations. In this paper, we show that our hybrid algorithm can reduce a circuit depth from the original HHL algorithm by means of a classical information feed-forward after the quantum phase estimation algorithm, and the results of the hybrid algorithm are identical to those of the HHL algorithm. In addition, it is experimentally examined with four qubits in the IBM Quantum Experience setups, and the experimental results of our algorithm show higher accurate performance on specific systems of linear equations than that of the HHL algorithm.
In this paper, we study the error propagation and generation in the Harrow-Hassidim-Lloyd (HHL) quantum algorithm runs on IBM-Q hardware with the help of a MATLAB simulator. HHL is a quantum algorithm that can provide exponential speedup over the fastest classical algorithm (conjugate gradient method) in solving systems of linear equations (SLE). However, without error correction, it cannot give correct results even in a 2-variable system due to its complexity. In this study, an HHL quantum circuit for a 2-variable SLE is implemented in IBM-Q and the error is extracted after each stage of the circuit and compared to a MATLAB simulator. We identified three major sources of errors, namely single-qubit flipping, gate infidelity, and error propagation. We also found that at the ancillary bit rotation stage, the error becomes large but the encoded solution still has high fidelity. However, the solution is mostly lost after the inverse quantum phase estimation which is necessary to extract the solution efficiently. Therefore, it is suggested that error correction resources, if limited, should be added to the second half of the circuit.
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.