Abstract In 2019, Yonghae Lee et al. combined the circuit implementation of the HHL quantum algorithm with a classical computer, and designed a hybrid HHL quantum circuit optimization algorithm to reduce experimental errors caused by decoherence and so on. However, the optimization is achieved only in the auxiliary quantum coding phase, and no quantum resource reduction is done on the quantum phase estimation and inverse quantum phase estimation stages. At the same time, the circuit optimization illustration on the 2th-order linear equation system just has the result and no specific process. In this paper, based on the idea of the hybrid HHL algorithm and the general quantum circuit implementation framework of HHL, a global optimization HHL algorithm is proposed. The feasibility of the global optimization HHL algorithm is verified by IBM's qiskit. The detail circuit optimization illustrations on the 4th-order linear equations show that the global optimization HHL algorithm can effectively reduce quantum resources without losing the fidelity of the results. Thus the global optimization HHL algorithm can further avoid some result errors than the existing implementation methods.
Abstract In 2019, Yonghae Lee et al. combined the circuit implementation of the HHL quantum algorithm with a classical computer, and designed a hybrid HHL quantum circuit optimization algorithm to reduce experimental errors caused by decoherence and so on. However, the optimization is achieved only in the auxiliary quantum coding phase, and no quantum resource reduction is done on the quantum phase estimation and inverse quantum phase estimation stages. At the same time, the circuit optimization illustration on the 2th-order linear equation system just has the result and no specific process. In this paper, based on the idea of the hybrid HHL algorithm and the general quantum circuit implementation framework of HHL, a global optimization HHL algorithm is proposed. The feasibility of the global optimization HHL algorithm is verified by IBM's qiskit. The detail circuit optimization illustrations on the 4th-order linear equations show that the global optimization HHL algorithm can effectively reduce quantum resources without losing the fidelity of the results. Thus the global optimization HHL algorithm can further avoid some result errors than the existing implementation methods.
In 2019, Yonghae Lee et al. combined the circuit implementation of the Harrow-Hassidim-Lloyd (HHL) algorithm with a classical computer, and designed a hybrid HHL algorithm to reduce experimental errors caused by decoherence and so on. However, the improvement is achieved only in the auxiliary quantum coding phase, and no quantum resource reduction is done on the quantum phase estimation and inverse quantum phase estimation stages. At the same time, the circuit improvement illustration on a [Formula: see text] linear system just has the result and no specific process. In this paper, based on the idea of the hybrid HHL algorithm and a generic circuit of HHL algorithm, an improved circuit implementation of the HHL algorithm is proposed. The feasibility of the improved circuit implementation of the HHL algorithm is verified by IBM's qiskit. The improved circuit illustrations on a [Formula: see text] linear system show that the improved circuit implementation of the HHL algorithm can effectively reduce quantum resources without losing the fidelity of the results. Thus the improved circuit implementation of the HHL algorithm can further avoid some result errors than the existing implementation methods.
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.
The limited capabilities of current quantum hardware significantly constrain the scale of experimental demonstrations of most quantum algorithmic primitives. This makes it challenging to perform benchmarking of the current hardware using useful quantum algorithms, i.e., application-oriented benchmarking. In particular, the Harrow-Hassidim-Lloyd (HHL) algorithm is a critical quantum linear algebra primitive, but the majority of the components of HHL are far out of the reach of noisy intermediate-scale quantum devices, which has led to the proposal of hybrid classical-quantum variants. The goal of this work is to further bridge the gap between proposed near-term friendly implementations of HHL and the kinds of quantum circuits that can be executed on noisy hardware. Our proposal adds to the existing literature of hybrid quantum algorithms for linear algebra that are more compatible with the current scale of quantum devices. Specifically, we propose two modifications to the Hybrid HHL algorithm proposed by Lee et al., leading to our algorithm Hybrid HHL++ : (1) propose a novel algorithm for determining a scaling factor for the linear system matrix that maximizes the utility of the amount of ancillary qubits allocated to the phase estimation component of HHL, and (2) introduce a heuristic for compressing the HHL circuit. We demonstrate the efficacy of our work by running our modified Hybrid HHL on Quantinuum System Model H-series trapped-ion quantum computers to solve different problem instances of small-scale portfolio optimization problems, leading to the largest experimental demonstrations of HHL for an application to date.
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.
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.
Abstract Quantum computing is a new computing mode that follows the laws of quantum mechanics to control quantum information units for computation. In terms of computational efficiency, due to the existence of quantum mechanical superposition, some known quantum algorithms can process problems faster than traditional general-purpose computers. HHL algorithm is an algorithm for solving linear system problems. Compared with classical algorithms in solving linear equations, it has an exponential acceleration effect in certain cases and as a sub-module, it is widely used in some machine learning algorithms to form quantum machines learning algorithms. However, there are some limiting factors in the use of this algorithm, which affect the overall effect of the algorithm. How to improve it to make the algorithm perform better has become an important issue in the field of quantum computing. This paper summarizes the optimization and improvement of HHL algorithm since it was proposed, and the application of HHL algorithm in machine learning, and discusses some possible future improvements of some subroutines in HHL algorithm.