Improved circuit implementation of the HHL algorithm and its simulations on QISKIT

Abstract

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 $$2\times 2$$ 2 × 2 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 $$4\times 4$$ 4 × 4 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.

Implementation of the HHL Algorithm for Solving the Poisson Equation on Quantum Simulators

“…If A is not positive definite, the HHL algorithm may not work correctly. The specific mathematical transformations, operations, and algorithms used in the HHL approach are well explained in the work of M. Zhang et al [23].…”