Coherence dynamics in quantum algorithm for linear systems of equations
Linlin Ye, Zhaoqi Wu, Shao-Ming Fei
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 . By using the Tsallis relative entropy of coherence and the norm of coherence, we show that the operator coherence of the phase estimation relies on the coefficients obtained by decomposing in the eigenbasis of . We prove that the operator coherence of the inverse phase estimation relies on the coefficients , eigenvalues of and the success probability , and it decreases with the increase of the probability when . Moreover, the variations of coherence deplete with the increase of the success probability and rely on the eigenvalues of as well as the success probability.
AI Impact Assessments
(3 models)Scientific Impact Assessment
Core Contribution
This paper analyzes the quantum coherence dynamics throughout the HHL algorithm for solving linear systems of equations Ax = b. The authors track how coherence evolves after each major step of the algorithm (phase estimation, conditional rotation, inverse phase estimation) using two specific coherence quantifiers: the Tsallis relative α entropy of coherence and the l_{1,p} norm of coherence. The main findings are: (1) coherence after phase estimation depends only on the decomposition coefficients β_i of |b⟩ in the eigenbasis of A; (2) coherence after inverse phase estimation additionally depends on eigenvalues of A and the success probability P_s; (3) for α ∈ (1,2], coherence decreases with increasing success probability; and (4) unlike Grover's and Deutsch-Jozsa algorithms where coherence is always consumed, HHL can exhibit both coherence production and depletion depending on the specific linear system.
Methodological Rigor
The paper follows a straightforward analytical approach: computing density matrices at each algorithmic step and evaluating closed-form expressions for the two coherence measures. The mathematical derivations appear correct and are presented with reasonable detail. However, the methodology is essentially mechanical — substituting known quantum states into known coherence formulas. There is no deep structural insight into *why* coherence behaves as it does, nor any connection to algorithmic performance beyond the observation that coherence variation correlates with success probability.
The examples (2×2 and 4×4 systems) are illustrative but limited. The paper uses a specific value of C = 0.736 derived from a particular circuit design parameter (r=2, t₀=2π), which constrains the generality of numerical results. The analysis assumes ideal, noise-free execution of the HHL algorithm, ignoring practical considerations like finite precision in phase estimation, decoherence, and gate errors. The idealized treatment, while mathematically clean, limits practical relevance.
A notable gap is the absence of any scaling analysis — how does coherence behave as the system dimension N grows? The examples stop at N=4, and no general asymptotic statements are made. This is important because the HHL algorithm's significance lies precisely in its scaling advantages for large N.
Potential Impact
The practical impact of this work is limited. The paper establishes formal relationships between coherence measures and algorithm parameters but does not demonstrate that these relationships provide actionable insights for algorithm design, optimization, or error mitigation. The observation that coherence can be both produced and depleted in HHL (unlike Grover's algorithm) is noted as "novel," but the implications of this finding are not explored in depth.
The work fits within a growing literature on resource-theoretic analysis of quantum algorithms (studying entanglement, coherence, and correlations as resources). However, unlike some prior works (e.g., [67] showing coherence bounds Shor's algorithm performance), this paper does not establish any operational connection between the coherence measures and the algorithm's computational power or efficiency.
Timeliness & Relevance
The study addresses a relevant intersection of quantum resource theory and quantum algorithms. HHL is an important algorithm with applications in quantum machine learning and scientific computing. Understanding the role of quantum resources in algorithms is a timely research direction. However, the specific analysis here is incremental — it applies previously established coherence measures to previously established algorithmic steps without generating fundamentally new understanding.
The paper builds on the approach of [68-70] (coherence dynamics in Grover's algorithm) and extends it to HHL, which is a natural but somewhat predictable extension. Reference [71] (Feng, Chen, Zhao 2023) already studied coherence and entanglement in HHL, though presumably with different coherence measures.
Strengths
1. Comprehensive treatment: The paper systematically tracks coherence through all steps of HHL using two distinct families of coherence measures, providing closed-form expressions.
2. Novel observation: The finding that HHL can exhibit coherence production (not just depletion) contrasts with previous results for other algorithms and is genuinely interesting.
3. Analytical relationships: The explicit dependence on β_i, eigenvalues, and success probability is cleanly derived, particularly the monotonicity results for different ranges of α.
4. Connection between β_i and b: Theorem 5 provides explicit relationships for the 2×2 case.
Limitations
1. Limited depth of insight: The paper computes coherence values but does not explain their operational significance. Why should we care about the specific coherence values at intermediate steps?
2. No scaling analysis: Only 2×2 and 4×4 examples are provided; no general behavior for large N is discussed.
3. Idealized setting: Perfect phase estimation, no noise, no discussion of approximate implementations.
4. No connection to computational advantage: The paper does not establish whether coherence is necessary or sufficient for HHL's quantum speedup.
5. Limited novelty in technique: The approach is a direct application of existing coherence measures to known quantum states — the methodology is not new.
6. Missing comparisons: No comparison with other resource measures (entanglement, discord) within the same framework, which would provide richer context.
7. The claim that coherence "produces" or "depletes" depends on the specific system somewhat weakens the generality of conclusions — one cannot make universal statements about coherence behavior in HHL.
Overall Assessment
This is a technically correct but incremental contribution that extends coherence dynamics analysis from Grover's algorithm to HHL using standard coherence quantifiers. The most interesting finding — that HHL can exhibit both coherence production and depletion — is noteworthy but underdeveloped. The paper would benefit significantly from scaling analysis, operational interpretation of the coherence dynamics, and connections to the algorithm's computational performance.
Generated Apr 17, 2026
Comparison History (40)
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Paper 2 addresses quantum coherence dynamics within the HHL quantum algorithm, connecting fundamental quantum information theory (coherence measures) with a landmark quantum algorithm. This intersection of quantum coherence theory and quantum computing algorithms has broader impact potential across quantum information science. Paper 1 proposes a scheme for negative refraction using EIT in atomic systems, which is more incremental within a well-studied niche area. Paper 2's rigorous mathematical framework and relevance to quantum computing gives it higher potential impact.
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