Circuit implementation of the globally optimized HHL algorithm and experiments on qiskit

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.