Curator's Take
This research provides crucial insights into how quantum coherence behaves during the HHL algorithm, one of the most celebrated quantum algorithms for solving linear systems that could revolutionize fields from machine learning to fluid dynamics. The authors reveal that coherence dynamics are intricately tied to both the mathematical structure of the problem (eigenvalues and coefficients) and the algorithm's success probability, offering a deeper understanding of why HHL works and when it might struggle. These findings are particularly timely as researchers work to implement HHL on near-term quantum devices, where managing coherence is essential for maintaining quantum advantage. Understanding these coherence patterns could help optimize the algorithm's performance and guide the development of more robust quantum linear algebra protocols.
— Mark Eatherly
Summary
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 $α$ 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 $β_{i}$ 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 $β_{i}$, eigenvalues of $A$ and the success probability $P_{s}$, and it decreases with the increase of the probability when $α\in(1,2]$. 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.