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arXiv:2604.14801 · Physica Scripta

Coherence dynamics in quantum algorithm for linear systems of equations

Linlin Ye, Zhaoqi Wu, Shao-Ming Fei

quant-phSubmitted Apr 16, 2026• Announced Apr 17, 2026• 7 citations
arXivHTMLPDF

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 $α$ 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.

Journal reference
Phys. Scr. 98 (2023) 125104
DOI
10.1088/1402-4896/ad0584
Comments
24 pages, 2 figures

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We propose a hybrid quantum algorithm based on the Harrow-Hassidim-Lloyd (HHL) algorithm for solving a system of linear equations. In this paper, we show that our hybrid algorithm can reduce a circuit depth from the original HHL algorithm by means of a classical information feed-forward after the quantum phase estimation algorithm, and the results of the hybrid algorithm are identical to those of the HHL algorithm. In addition, it is experimentally examined with four qubits in the IBM Quantum Experience setups, and the experimental results of our algorithm show higher accurate performance on specific systems of linear equations than that of the HHL algorithm.
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Abstract Solving linear systems of equations is of vital importance in various fields of science. With the scale of calculation soaring, classical computers are intractable, and the advantage of quantum computing is reflected. Quantum algorithm has exponential acceleration in solving the linear systems A x ⃗ = b ⃗ for sparse and well-conditioned coefficient matrix A with dimension N × N . In this paper, we present a quantum algorithm to solve matrix equations of the form AX = B , where X = ( x ⃗ 1 , x ⃗ 2 , … , x ⃗ n ) , B = ( b ⃗ 1 , b ⃗ 2 , … , b ⃗ n ) , x ⃗ k and b ⃗ k are unit vectors. The problem can be intuitively treated as solving n linear systems simultaneously, which will needs O ( n log N ) complexity by the famous HHL algorithm. We demonstrate that in our algorithm the scale of complexity is O ( log n N ) . In the error analysis part, phase estimation is discussed in details.