Quantum resources in Harrow-Hassidim-Lloyd algorithm

TL;DR

This paper investigates the quantum resources, such as entanglement and coherence, essential for the success of the HHL algorithm in solving linear systems, highlighting their roles and effects under imperfections.

Contribution

It identifies the necessity of bipartite and multipartite entanglement, and quantum coherence, for the HHL algorithm's performance, providing quantitative analysis and effects of imperfections.

Findings

Bipartite entanglement is generated and required at each step.

Multipartite entanglement inversely correlates with performance.

Disorder increases multipartite entanglement but decreases bipartite entanglement and coherence.

Abstract

Quantum algorithms have the ability to reduce runtime for executing tasks beyond the capabilities of classical algorithms. Therefore, identifying the resources responsible for quantum advantages is an interesting endeavour. We prove that nonvanishing quantum correlations, both bipartite and genuine multipartite entanglement, are required for solving nontrivial linear systems of equations in the Harrow-Hassidim-Lloyd (HHL) algorithm. Moreover, we find a nonvanishing l1-norm quantum coherence of the entire system and the register qubit which turns out to be related to the success probability of the algorithm. Quantitative analysis of the quantum resources reveals that while a significant amount of bipartite entanglement is generated in each step and required for this algorithm, multipartite entanglement content is inversely proportional to the performance indicator. In addition, we report that when imperfections chosen from Gaussian distribution are incorporated in controlled rotations, multipartite entanglement increases with the strength of the disorder, albeit error also increases while bipartite entanglement and coherence decreases, confirming the beneficial role of bipartite entanglement and coherence in this algorithm.