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.
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.
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.
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.
Abstract Solving large-scale linear equations is of great significance in many engineering fields, such as weather forecasting and bioengineering. The classical computer solves the linear equations, no matter adopting the elimination method or Kramer’s rule, the time required for solving is in a polynomial relationship with the scale of the equation system. With the advent of the era of big data, the integration of transistors is getting higher and higher. When the size of transistors is close to the order of electron diameter, quantum tunneling will occur, and Moore’s Law will not be followed. Therefore, the traditional computing model will not be able to meet the demand. In this paper, through an in-depth study of the classic HHL algorithm, a small-scale quantum circuit model is proposed to solve a 2×2 linear equations, and the circuit diagram and programming are used to simulate and verify on the Origin Quantum Platform. The fidelity under different parameter values reaches more than 90%. For the case where the matrix to be solved is a sparse matrix, the quantum algorithm has an exponential speed improvement over the best known classical algorithm.
After learning basic quantum computing concepts, it is desirable to reinforce the learning using an important and relatively complex algorithm through which students can observe and appreciate how qubits evolve and interact with each other. Harrow-Hassidim-Lloyd (HHL) quantum algorithm, which can solve linear system problems with exponential speed-up over the classical method and is the basis of many important quantum computing algorithms, is used to serve this purpose. The HHL algorithm is explained analytically followed by a 4-qubit numerical example in bra-ket notation. Matlab code corresponding to the numerical example is available for students to gain a deeper understanding of the HHL algorithm from a pure matrix point of view. A quantum circuit programmed using qiskit is also provided for real hardware execution in IBM quantum computers. After going through the material, students are expected to have a better appreciation of the concepts such as basis transformation, bra-ket and matrix representations, superposition, entanglement, controlled operations, measurement, quantum Fourier transformation, quantum phase estimation, and quantum programming. To help readers review these basic concepts, brief explanations augmented by the HHL numerical examples in the main text are provided in the Appendix.
This work presents a new approach for simulating the HHL linear systems of equations solver algorithm with tensor networks. First, a novel HHL in the qudits formalism, the generalization of qubits, is developed, and then its operations are transformed into an equivalent classical HHL, taking advantage of the non-unitary operations that they can apply. The main novelty of this proposal is to perform a classical simulation of the HHL as efficiently as possible to benchmark the algorithm steps according to its input parameters and the input matrix. The algorithm is applied to three classical simple simulation problems, comparing it with an exact inversion algorithm, and its performance is compared against an implementation of the original HHL simulated in the Qiskit framework, providing both codes. It is also applied to study the sensitivity of the HHL algorithm with respect to its hyperparameter values, reporting the existence of saturation points and maximal performance values. The results show that this approach can achieve a promising performance in computational efficiency to simulate the HHL process without quantum noise, providing a higher bound for its performance.
The HHL quantum algorithm [1] is a procedure that addresses the resolution of linear systems of equations (QLSP). Under certain conditions, the algorithm has a logarithmic order in the number of equations, better than the faster classical method. The algorithm manages to find a quantum state proportional to the solution vector, up to a normalization factor. The disadvantage is that to determine each of the coefficients of the solution vector, the algorithm’s output quantum state must be determined with additional statistical methods, thus losing its exponential advantage. There are certain types of problems in which this disadvantage can be circumvented, the statistical treatment is unavoidable, but for certain cases such as electrical circuits, in which the main interest is to find only one of the currents, (for example the load current), we only need to measure one of the qubits of the solution state. In this article we solve the linear system associated with the currents of an electrical circuit with sinusoidal voltage, using the HHL algorithm, simulated in a Scilab numerical environment. An optimized example of an electrical line transmission wave in a real computer on the IBMQ platform, is also solved