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arxiv: 2604.14801 · v1 · submitted 2026-04-16 · 🪐 quant-ph

Coherence dynamics in quantum algorithm for linear systems of equations

Linlin Ye , Zhaoqi Wu , Shao-Ming Fei This is my paper

Pith reviewed 2026-05-10 11:11 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum coherenceHHL algorithmlinear systems of equationsphase estimationTsallis entropysuccess probabilityquantum algorithms
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The pith

In the HHL quantum algorithm, the coherence of the inverse phase estimation operator decreases with rising success probability for Tsallis parameter alpha between 1 and 2.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates the dynamics of quantum coherence during the execution of the HHL algorithm for solving linear systems Ax = b on a quantum computer. Using the Tsallis relative alpha entropy and l1,p norm as coherence measures, it analyzes how coherence in the phase estimation and inverse phase estimation steps depends on the input state decomposition and algorithm parameters. A sympathetic reader would care because controlling coherence could improve the efficiency and resource use of quantum linear system solvers. The analysis reveals specific dependencies and monotonic behaviors in coherence as success probability changes.

Core claim

The operator coherence of the phase estimation P relies on the coefficients beta_i from decomposing the input state |b> in the eigenbasis of A. The operator coherence of the inverse phase estimation ~P relies on beta_i, the eigenvalues of A, and the success probability P_s; it decreases with increasing P_s when alpha is in (1,2]. The variations of coherence decrease with increasing success probability and also depend on the eigenvalues of A and P_s.

What carries the argument

Tsallis relative alpha-entropy of coherence and l_{1,p} norm of coherence applied to the phase estimation operator P and its inverse ~P within the standard HHL quantum circuit.

If this is right

  • The coherence in phase estimation depends solely on the beta_i coefficients of the input vector.
  • Coherence in inverse phase estimation decreases monotonically with success probability for alpha in (1,2].
  • Coherence variations become smaller as success probability increases, with dependence on matrix eigenvalues.
  • These relations hold under the assumption of the standard HHL structure and clean eigenbasis decomposition.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Optimizing the success probability in HHL could serve as a way to reduce unwanted coherence effects in quantum computations.
  • Similar coherence analysis might be extended to other quantum algorithms that rely on phase estimation.
  • These findings could guide the choice of coherence measures or circuit parameters in designing more stable quantum linear solvers.
  • If the success probability can be tuned independently, it provides a control knob for coherence management.

Load-bearing premise

The analysis assumes the standard HHL circuit with phase estimation and its inverse, where the input state has a clean decomposition into the eigenbasis of A and the success probability is well-defined and independent.

What would settle it

An experiment or simulation showing that the coherence of the inverse phase estimation does not decrease with increasing success probability for alpha=1.5, or that it depends on factors other than beta_i, eigenvalues, and P_s, would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.14801 by Linlin Ye, Shao-Ming Fei, Zhaoqi Wu.

Figure 1
Figure 1. Figure 1: The y-axis stands for the values of coherence. Subfigures a (b) is for the case that the coherence based on the l1,p norm (Tsallis relative α entropy). The operator coherence of P (green, Eqs. (42) and (43)), R (blue dashed, Eqs. (44) and (45)), Pe (black dotted, Eqs. (46) and (47)) and the variations of coherence based on the l1,p norm and the Tsallis relative α entropy (red dot-dashed, Eqs. (48) and (49)… view at source ↗
Figure 2
Figure 2. Figure 2: The y-axis stands for the values of coherence. Subfigures a (b) is for the case that the coherence based on the l1,p norm (Tsallis relative α entropy). The operator coherence of P (green, Eqs. (51) and (52)), R (blue dashed, Eqs. (53) and (54)), Pe (black dotted, Eqs. (55) and (56)) and the variations of coherence based on the l1,p norm and the Tsallis relative α entropy (red dot-dashed, Eqs. (57) and (58)… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper explores coherence dynamics in the HHL quantum algorithm for solving linear systems Ax=b. Using the Tsallis relative α-entropy of coherence and the l_{1,p} norm of coherence, it claims that the operator coherence of the phase estimation operator P depends only on the coefficients β_i from the decomposition of |b⟩ in the eigenbasis of A. For the inverse operator ~P, coherence depends on β_i, the eigenvalues of A, and the success probability P_s; the authors prove that this coherence decreases with increasing P_s when α ∈ (1,2], and that coherence variations diminish with P_s while depending on the eigenvalues and P_s.

Significance. If the claimed monotonicity and dependences are established with rigorous, circuit-consistent derivations, the work could illuminate how quantum coherence evolves through the steps of a canonical quantum linear solver. This might inform resource accounting or coherence management in near-term implementations of HHL and related algorithms. The choice of Tsallis and l_{1,p} measures provides a concrete, quantifiable link between coherence and algorithmic success probability.

major comments (2)
  1. [Abstract (proof of monotonicity for ~P)] Abstract and the proof that operator coherence of ~P decreases with P_s (for α ∈ (1,2]): the coherence is expressed as a function of β_i, eigenvalues of A, and P_s treated as an independent variable. In the standard HHL circuit, however, P_s is fixed by the controlled-rotation angles, which are chosen as functions of the estimated eigenvalues (typically sin(θ/2) = C/λ_i). Varying P_s therefore simultaneously modifies the unitary ~P and the post-selected state, so the partial dependence on P_s alone does not describe a physically realizable variation along the algorithm's execution path.
  2. [Derivations and circuit analysis] The manuscript states that proofs are given for the stated dependences on β_i, eigenvalues, and P_s, yet no explicit derivations, error bounds, or circuit diagrams for the phase estimation P and its inverse ~P are supplied. Without these, it is impossible to verify whether the claimed relations hold under the standard HHL assumptions or whether post-hoc choices were made in the coherence calculations.
minor comments (1)
  1. [Notation] The notation ~P for the inverse phase estimation should be introduced with an explicit definition or reference to the standard HHL circuit decomposition to avoid ambiguity with other inverse operations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their insightful comments on our manuscript. We respond to each major comment below, providing clarifications and indicating the revisions we will make.

read point-by-point responses

  1. Referee: Abstract and the proof that operator coherence of ~P decreases with P_s (for α ∈ (1,2]): the coherence is expressed as a function of β_i, eigenvalues of A, and P_s treated as an independent variable. In the standard HHL circuit, P_s is fixed by the controlled-rotation angles chosen as functions of the estimated eigenvalues. Varying P_s modifies the unitary ~P and the post-selected state, so the partial dependence does not describe a physically realizable variation.

    Authors: We appreciate this observation. Our derivation shows the mathematical dependence of the coherence on P_s for the inverse operator ~P, which is valid regardless of how P_s is realized in the circuit. In practice, selecting different values for the constant C in the rotation angles (sin(θ/2) = C/λ_i) results in different P_s, and our result demonstrates that higher success probability corresponds to lower coherence in ~P for α in (1,2]. We will update the abstract and discussion to emphasize that this functional dependence guides the choice of algorithm parameters rather than describing intra-circuit dynamics. The proof itself remains unchanged as it correctly follows from the coherence definitions. revision: partial

  2. Referee: The manuscript states that proofs are given for the stated dependences on β_i, eigenvalues, and P_s, yet no explicit derivations, error bounds, or circuit diagrams for the phase estimation P and its inverse ~P are supplied. Without these, it is impossible to verify whether the claimed relations hold under the standard HHL assumptions.

    Authors: We regret that the derivations were not presented with sufficient explicitness in the submitted version. The dependences follow from substituting the matrix elements of P and ~P into the formulas for Tsallis relative α-entropy of coherence and l_{1,p} norm of coherence. We will include detailed derivations in the main text or as an appendix, along with circuit diagrams illustrating the phase estimation and its inverse. Where approximations are used in phase estimation, we will add error bounds. This will ensure the claims are verifiable under standard HHL assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations apply standard coherence measures to HHL circuit states

full rationale

The paper computes operator coherence of P and ~P via the Tsallis relative α-entropy and l_{1,p} norm applied to the standard HHL decomposition |b⟩ = ∑ β_i |u_i⟩ and the controlled rotations that define P_s. These are direct algebraic consequences of the definitions and the known eigenbasis expansion; no parameters are fitted then relabeled as predictions, no self-citations bear the central claim, and no ansatz is smuggled. The monotonicity statements for α ∈ (1,2] are obtained by differentiating the explicit coherence expressions with respect to P_s while holding β_i and λ_i fixed, which is a valid (if idealized) mathematical step rather than a definitional reduction. The analysis remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claims rest on the standard mathematical definitions of the Tsallis relative alpha entropy of coherence and the l1,p norm of coherence, plus the known circuit structure and success-probability definition of the HHL algorithm. No free parameters are fitted, no new entities are postulated, and no ad-hoc axioms beyond domain-standard quantum information theory are invoked.

axioms (3)
  • standard math Tsallis relative alpha entropy of coherence satisfies the required properties for an operator coherence measure
    Invoked when applying the measure to the phase-estimation operators P and ~P.
  • standard math l1,p norm of coherence is a valid coherence monotone
    Used alongside the Tsallis measure to characterize coherence dynamics.
  • domain assumption The HHL algorithm proceeds via phase estimation P followed by inverse phase estimation ~P with well-defined success probability P_s
    The entire analysis is conditioned on the standard HHL circuit and its success probability.

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