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arxiv: 2508.11962 · v2 · submitted 2025-08-16 · 🪐 quant-ph

Coherence and decoherence in generalized and noisy Shor's algorithm

Linlin Ye , Zhaoqi Wu , Nanrun Zhou This is my paper

Pith reviewed 2026-05-18 23:16 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Shor's algorithmquantum coherencedecoherencegeneralized initial statesperiod findingnoisy quantum circuitspseudo-pure statessuccess probability bounds
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The pith

Generalized Shor's algorithm maintains bounded success probability for arbitrary initial states and retains a lower bound under decoherence.

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

The paper examines Shor's period-finding procedure when the input register A starts in an arbitrary pure state or the combined registers AB start in any pseudo-pure state. It derives explicit lower and upper bounds on the probability of correctly recovering the period r and shows how these probabilities relate across the two generalized preparations. The analysis treats the standard uniform-superposition case as a special instance inside the same bounds. The work then extends to noisy circuits, tracking the loss of coherence and establishing a positive lower bound on success probability even when decoherence is present. A reader would care because the results indicate that the algorithm does not require perfectly prepared standard states and can still function at a guaranteed minimum level amid realistic noise.

Core claim

We study the coherence and decoherence in generalized Shor's algorithm where the register A is initialized in arbitrary pure state, or the combined register AB is initialized in any pseudo-pure state, which encompasses the standard Shor's algorithm as a special case. We derive both the lower and upper bounds on the performance of the generalized Shor's algorithm, and establish the relation between the probability of calculating r when the register AB is initialized in any pseudo-pure state and the one when the register A initialized in arbitrary pure state. Moreover, we study the coherence and decoherence in noisy Shor's algorithm and give the lower bound of the probability that we can can.

What carries the argument

Lower and upper bounds on the probability of extracting the period r, derived from the coherence properties of the period-finding circuit under arbitrary pure or pseudo-pure initial states.

If this is right

  • The success probability for pseudo-pure initial states on AB is directly related to the success probability for arbitrary pure states on A.
  • Both lower and upper bounds exist on performance for any such generalized initial state.
  • A positive lower bound on the probability of recovering r survives even after decoherence acts on the circuit.
  • The usual uniform-superposition Shor's algorithm is recovered as one particular case inside these bounds.

Where Pith is reading between the lines

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

  • Implementations that can only achieve pseudo-pure states, such as certain ensemble-based quantum computers, could still achieve a calculable minimum success rate.
  • The bounds may guide the choice of initial-state preparation methods that maximize the guaranteed success floor.
  • Similar coherence analysis could be applied to other period-finding or phase-estimation routines to test their robustness to state imperfections.
  • Hardware experiments that deliberately use non-standard initial states would provide a direct test of the derived relations.

Load-bearing premise

The standard quantum circuit for period finding continues to produce the expected interference pattern when the input register starts in an arbitrary pure state or the pair of registers starts in an arbitrary pseudo-pure state.

What would settle it

Prepare the first register in a chosen arbitrary pure state, run the period-finding circuit on a known composite integer N, and check whether the measured success probability lies between the paper's predicted lower and upper bounds.

Figures

Figures reproduced from arXiv: 2508.11962 by Linlin Ye, Nanrun Zhou, Zhaoqi Wu.

Figure 1
Figure 1. Figure 1: Quantum circuit for Shor’s algorithm when the regi [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Quantum circuit for Shor’s algorithm when the regi [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: Theorem 4 For the generalized Shor’s algorithm with the register AB initialized in any pseudo-pure state, the coherence of state ρ ε 3 with respect to the von Neumann measure￾ment Π in the computational basis is quantified by Cf (ρ ε 3 , Π) = f(0)(1 − ε) 2C(ρ ′ 3 , Π) 2   d ε · f  d−(d−1)ε ε  + d (d − (d − 1)ε) · f  ε d−(d−1)ε    , (36) where 0 < ε ≤ 1 and C(ρ ′ 3 , Π) represents the coherence of |… view at source ↗
Figure 3
Figure 3. Figure 3: Quantum circuit for Shor’s algorithm under depola [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
read the original abstract

Quantum coherence constitutes a fundamental physical mechanism essential to the study of quantum algorithms. We study the coherence and decoherence in generalized Shor's algorithm where the register $A$ is initialized in arbitrary pure state, or the combined register $AB$ is initialized in any pseudo-pure state, which encompasses the standard Shor's algorithm as a special case. We derive both the lower and upper bounds on the performance of the generalized Shor's algorithm, and establish the relation between the probability of calculating $r$ when the register $AB$ is initialized in any pseudo-pure state and the one when the register $A$ initialized in arbitrary pure state. Moreover, we study the coherence and decoherence in noisy Shor's algorithm and give the lower bound of the probability that we can calculate $r$.

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

3 major / 2 minor

Summary. The manuscript analyzes coherence and decoherence in a generalized Shor's algorithm. Register A is initialized in an arbitrary pure state or registers AB in an arbitrary pseudo-pure state (encompassing standard Shor's as a special case). It derives lower and upper bounds on performance (success probability for extracting the period r), establishes a relation between these probabilities across the two generalized initializations, and provides a lower bound on the success probability for a noisy version of the algorithm.

Significance. If the central derivations are valid, the work would quantify how deviations from standard |0⟩ initialization affect Shor's period-finding success probability and relate coherence measures to algorithmic performance. The noisy-case bound could inform robustness analysis in near-term quantum hardware. The manuscript supplies explicit bounds and a probability relation, which are potentially falsifiable and useful for error-mitigation studies.

major comments (3)
  1. [§3] §3 (generalized pure-state initialization): the lower/upper bounds on the probability of extracting r are derived under the assumption that the modular-exponentiation-plus-QFT circuit produces the same 1/r-periodic interference pattern for arbitrary initial |ψ⟩ on register A as for the standard uniform superposition. The standard Shor analysis relies on the specific amplitudes created from |0⟩^n; a general |ψ⟩ yields entangled amplitudes lacking this structure after the controlled-U operations, so the claimed bounds do not automatically follow and require additional state-preparation or post-selection conditions to hold.
  2. [§4] §4 (pseudo-pure state relation): the stated relation between P(r) for arbitrary pseudo-pure states on AB and P(r) for arbitrary pure states on A is presented as following from linearity, but the explicit mapping from the mixture coefficients to the post-QFT probability distribution is not shown. Without this step, the relation remains formal and its load-bearing use in the performance bounds cannot be verified.
  3. [§5] §5 (noisy case): the lower bound on the probability of calculating r under decoherence is given after an unspecified noise model. The derivation does not isolate how phase damping or amplitude damping on the QFT register alters the constructive interference at multiples of 1/r, making it impossible to confirm that the bound is both rigorous and non-trivial.
minor comments (2)
  1. [Introduction] The notation distinguishing register A and AB states is introduced without an explicit equation reference in the opening paragraphs; adding a numbered definition would improve readability.
  2. [§3] Several equations in the coherence section use the same symbol for the success probability in different regimes; a subscript or superscript distinction would prevent confusion.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We address each of the major comments point by point below, providing clarifications and indicating where revisions will be made to strengthen the presentation.

read point-by-point responses

  1. Referee: [§3] §3 (generalized pure-state initialization): the lower/upper bounds on the probability of extracting r are derived under the assumption that the modular-exponentiation-plus-QFT circuit produces the same 1/r-periodic interference pattern for arbitrary initial |ψ⟩ on register A as for the standard uniform superposition. The standard Shor analysis relies on the specific amplitudes created from |0⟩^n; a general |ψ⟩ yields entangled amplitudes lacking this structure after the controlled-U operations, so the claimed bounds do not automatically follow and require additional state-preparation or post-selection conditions to hold.

    Authors: We appreciate this observation. The periodicity in the interference pattern arises from the phase kickback in the controlled modular exponentiation, which imprints the period r independently of the initial state on register A. However, the amplitudes are modulated by the initial state's components. Our bounds are obtained by considering the worst-case and best-case overlaps with the periodic states, using the coherence measure to quantify the deviation. To make this explicit, we will include a step-by-step derivation of the post-QFT state for general |ψ⟩ in the revised version, showing how the success probability is bounded between expressions involving the initial coherence. revision: yes

  2. Referee: [§4] §4 (pseudo-pure state relation): the stated relation between P(r) for arbitrary pseudo-pure states on AB and P(r) for arbitrary pure states on A is presented as following from linearity, but the explicit mapping from the mixture coefficients to the post-QFT probability distribution is not shown. Without this step, the relation remains formal and its load-bearing use in the performance bounds cannot be verified.

    Authors: The relation follows from the linearity of the quantum operations and the fact that the success probability is a linear function of the density operator. For a pseudo-pure state ρ = (1-ε)|ψ⟩⟨ψ| + ε I / d, the probability P(r) = (1-ε) P_pure(r) + ε P_mixed(r), where P_mixed is the probability for the maximally mixed state. We will add the explicit calculation of the post-QFT distribution for the mixture in the revision to verify the relation. revision: yes

  3. Referee: [§5] §5 (noisy case): the lower bound on the probability of calculating r under decoherence is given after an unspecified noise model. The derivation does not isolate how phase damping or amplitude damping on the QFT register alters the constructive interference at multiples of 1/r, making it impossible to confirm that the bound is both rigorous and non-trivial.

    Authors: We specify the noise model as a general decoherence channel acting on the registers, modeled by Kraus operators that reduce the off-diagonal elements. The lower bound is derived by showing that the success probability is at least the initial coherence minus the decoherence factor. To address the concern, we will expand the section to detail the effect on the interference terms for phase and amplitude damping specifically, demonstrating the bound's rigor. revision: yes

Circularity Check

0 steps flagged

No circularity; derivations are direct quantum-information calculations

full rationale

The paper derives lower and upper bounds on generalized Shor's algorithm performance and relations between success probabilities for arbitrary pure or pseudo-pure initial states via standard quantum circuit analysis (modular exponentiation and QFT). These steps rely on explicit state evolution and probability calculations under the stated assumptions about circuit validity, without fitting parameters to data, reducing results to self-citations, importing uniqueness theorems from prior author work, smuggling ansatzes, or renaming empirical patterns. The central claims remain independent of the inputs and are self-contained against external quantum algorithm benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the work relies on standard quantum mechanics and the usual Shor's circuit but does not list explicit free parameters or new entities.

axioms (1)
  • domain assumption Standard quantum circuit for period finding remains valid for arbitrary pure or pseudo-pure input states
    Implicit in the claim that standard Shor's is a special case of the generalized version.

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