Coherence dynamics in quantum algorithm for linear systems of equations

TL;DR

This paper investigates the dynamics of quantum coherence in the HHL quantum algorithm for solving linear systems, analyzing how coherence measures depend on eigenvalues, coefficients, and success probability.

Contribution

It introduces a detailed analysis of coherence evolution in the HHL algorithm using Tsallis entropy and $l_{1,p}$ norm, revealing dependencies on eigenvalues and success probability.

Findings

Operator coherence depends on eigenbasis coefficients and eigenvalues.

Coherence decreases as success probability increases.

Coherence variations are influenced by eigenvalues and success probability.

Abstract

Quantum coherence is a fundamental issue in quantum mechanics and quantum information processing. We explore the coherence dynamics of the evolved states in HHL quantum algorithm for solving the linear system of equation $A\overrightarrow{x}=\overrightarrow{b}$. By using the Tsallis relative $\alpha$ entropy of coherence and the $l_{1,p}$ norm of coherence, we show that the operator coherence of the phase estimation $P$ relies on the coefficients $\beta_{i}$ obtained by decomposing $|b\rangle$ in the eigenbasis of $A$. We prove that the operator coherence of the inverse phase estimation $\widetilde{P}$ relies on the coefficients $\beta_{i}$, eigenvalues of $A$ and the success probability $P_{s}$, and it decreases with the increase of the probability when $\alpha\in(1,2]$. Moreover, the variations of coherence deplete with the increase of the success probability and rely on the eigenvalues of $A$ as well as the success probability.