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arxiv: 2604.16190 · v1 · submitted 2026-04-17 · 🪐 quant-ph

Coherence dynamics in Simon's quantum algorithm

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

Pith reviewed 2026-05-10 07:48 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum coherenceSimon's algorithmTsallis entropyl1,p normcoherence dynamicsoracle operatorqubit registersdimension dependence
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The pith

Coherence in Simon's quantum algorithm increases with the dimension N and is produced overall for N greater than 4.

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

The paper examines how quantum coherence evolves step by step through Simon's algorithm for finding hidden periods in a function. It tracks coherence separately in the two registers using Tsallis relative alpha-entropy and the l1,p norm. Both registers show coherence that grows larger as the state-space dimension N increases. The oracle step that encodes the function leaves the amount of coherence exactly the same. Across the full run of the algorithm, net coherence is created when N exceeds 4 and is lost when N is smaller than 4.

Core claim

The coherences of the first register and the second register both rely on the dimension N of the state spaces of the n-qubit systems and increase with the increase of N. The oracle operator O does not change the coherence. In overall the coherence is in production when N>4 and in depletion when N<4.

What carries the argument

Coherence measures via Tsallis relative alpha-entropy and l1,p norm applied to the two registers during the unitary steps of Simon's algorithm.

If this is right

  • Coherence of both registers increases as the state-space dimension N grows.
  • The oracle operator leaves the coherence of the state unchanged at every step.
  • The full algorithm produces a net gain in coherence when N exceeds 4.
  • The full algorithm produces a net loss in coherence when N is less than 4.

Where Pith is reading between the lines

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

  • The size-dependent production or depletion of coherence may influence how well the algorithm distinguishes periods for small versus large systems.
  • The transition at N=4 offers a concrete point for numerical checks on small quantum simulators.
  • Similar tracking of coherence could be applied to other period-finding or hidden-subgroup algorithms to see if the same N threshold appears.

Load-bearing premise

The chosen mathematical definitions accurately capture the quantum coherence that matters for the algorithm, and Simon's algorithm is implemented in its standard n-qubit circuit form.

What would settle it

Compute the two coherence measures on the state after each gate in Simon's algorithm for n=1 (N=2) and for n=3 (N=8), then check whether the net change from start to finish is negative for N=2 and positive for N=8.

read the original abstract

Quantum coherence plays a pivotal role in quantum algorithms. We study the coherence dynamics of the evolved states in Simon's quantum algorithm based on Tsallis relative $\alpha$ entropy and $l_{1,p}$ norm. We prove that the coherences of the first register and the second register both rely on the dimension $N$ of the state spaces of the $n$ qubit systems, and increase with the increase of $N$. We show that the oracle operator $O$ does not change the coherence. Moreover, we study the coherence dynamics in the Simon's quantum algorithm and prove that in overall the coherence is in production when $N>4$ and in depletion when $N<4$.

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 / 2 minor

Summary. The manuscript analyzes coherence dynamics in Simon's quantum algorithm using the Tsallis relative α-entropy and the l_{1,p} norm of coherence. It claims to prove that coherence in both the first and second registers depends on the dimension N=2^n of the n-qubit systems and increases with N, that the oracle operator O leaves coherence invariant, and that overall coherence undergoes production for N>4 and depletion for N<4.

Significance. If the derivations hold, the work offers a quantitative resource-theoretic view of coherence evolution through the standard steps of Simon's algorithm. The reported oracle invariance and the N-dependent production/depletion threshold could illuminate how coherence scales with system size in query algorithms. Agreement between two distinct coherence measures would strengthen the robustness of the conclusions.

major comments (2)
  1. [Coherence dynamics section (around the proofs of N-dependence and threshold)] The abstract and main claims assert explicit proofs of N-dependence, oracle invariance, and the production/depletion threshold at N=4, yet the manuscript does not display the explicit state vectors after each gate, the intermediate coherence expressions, or the algebraic steps that yield these results. Without these derivations (including any error analysis or closed-form expressions for the coherence change ΔC(N)), the central claims cannot be verified.
  2. [Section on overall coherence dynamics] The switch from depletion to production at exactly N=4 is presented as a proven result, but it is unclear whether this threshold follows analytically from the sign of ΔC or is identified numerically/post-hoc. The manuscript should derive the condition under which the net coherence change changes sign as a function of N for both measures.
minor comments (2)
  1. [Definitions of coherence measures] Clarify the range of the parameter α in the Tsallis relative entropy and confirm that the reported N-dependence holds for the chosen α (or is independent of it).
  2. [Results for registers] Specify the precise definition of the l_{1,p} norm used and whether p is fixed or varied; include a brief comparison of the numerical values obtained from both measures at each algorithm step.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and have revised the manuscript to improve the explicitness of the derivations.

read point-by-point responses

  1. Referee: [Coherence dynamics section (around the proofs of N-dependence and threshold)] The abstract and main claims assert explicit proofs of N-dependence, oracle invariance, and the production/depletion threshold at N=4, yet the manuscript does not display the explicit state vectors after each gate, the intermediate coherence expressions, or the algebraic steps that yield these results. Without these derivations (including any error analysis or closed-form expressions for the coherence change ΔC(N)), the central claims cannot be verified.

    Authors: We agree that the intermediate derivations were not presented with sufficient detail. In the revised manuscript we now include the explicit state vectors after each gate, the intermediate expressions for both the Tsallis relative α-entropy and the l_{1,p} norm of coherence, the full algebraic steps establishing N-dependence and oracle invariance, and the closed-form expression for ΔC(N) together with a brief discussion of numerical precision. revision: yes

  2. Referee: [Section on overall coherence dynamics] The switch from depletion to production at exactly N=4 is presented as a proven result, but it is unclear whether this threshold follows analytically from the sign of ΔC or is identified numerically/post-hoc. The manuscript should derive the condition under which the net coherence change changes sign as a function of N for both measures.

    Authors: The threshold is obtained analytically. We derive the net change ΔC(N) in closed form for both measures, then solve ΔC(N)=0 to obtain the transition point N=4. The sign of ΔC(N) is positive for N>4 (production) and negative for N<4 (depletion). This analytic condition, rather than a numerical observation, is now stated explicitly in the revised section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are direct applications of chosen measures

full rationale

The paper defines coherence via Tsallis relative α-entropy and l_{1,p} norm, then computes these quantities explicitly on the initial, post-oracle, and final states of the standard Simon circuit for n qubits (dimension N=2^n). The N-dependence, oracle invariance, and net production/depletion threshold at N=4 emerge as algebraic consequences of those definitions applied to the known superposition and measurement steps; no equation reduces to a fitted parameter, self-referential definition, or load-bearing self-citation. The central claims remain independent of the inputs once the two coherence quantifiers and the circuit are fixed.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard Hilbert-space formulation of n-qubit quantum mechanics, the usual definition of Simon's oracle, and the mathematical definitions of the two coherence measures; no new physical entities or free parameters are introduced.

axioms (2)
  • domain assumption Standard n-qubit Hilbert space and unitary evolution for Simon's algorithm
    The paper assumes the conventional circuit model and oracle definition of Simon's algorithm without deriving them.
  • domain assumption Tsallis relative α entropy and l_{1,p} norm are valid and complete quantifiers of quantum coherence
    The analysis adopts these two measures as given and does not prove they capture all relevant coherence properties.

pith-pipeline@v0.9.0 · 5406 in / 1484 out tokens · 68052 ms · 2026-05-10T07:48:39.868375+00:00 · methodology

discussion (0)

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