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Quantum percolation in quasicrystals using continuous-time quantum walk

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Pith's one-line read Quantum walkers on quasicrystal lattices are trapped at 32–40% edge disconnection, versus 50% on a square lattice.

desk verdict Useful numerical data on CTQW percolation on quasicrystal tilings, but the central claim is confounded by the degree-dependent diagonal term in the Laplacian Hamiltonian. read the letter →

arxiv 1908.03051 v2 pith:4RXQ3SYM submitted 2019-08-08 quant-ph cond-mat.dis-nncond-mat.mes-hall

Prateek Chawla , C. V. Ambarish , C. M. Chandrashekar This is my paper · ORCID
classification quant-phcond-mat.dis-nncond-mat.mes-hall
keywords continuous-timequantumwalkpercolationquasicrystalPenrosetilingAmmann-BeenkerAndersonlocalizationedgedisconnectiondisordergraphLaplacian
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that a quasicrystal lattice is not just a backdrop for quantum motion but a structure that actively traps a quantum particle. Using a continuous-time quantum walk on Penrose and Ammann-Beenker tilings, the authors find that a significant part of the wave function remains localized near the starting point even with no disorder, whereas on a square lattice it spreads quickly. When edges are randomly removed to simulate percolation disorder, the probability of escaping a 40-hop zone falls much faster on both quasicrystals, vanishing at roughly 32% disconnection for Ammann-Beenker and 40% for Penrose, versus 50% for the square lattice. The authors conclude that quasicrystal lattices slow quantum percolation and could store a quantum state for longer.

What carries the argument

The engine of the calculation is the graph-Laplacian Hamiltonian $H=\gamma(D-A)$ for the continuous-time quantum walk, where $A$ is the adjacency matrix and $D$ is the diagonal matrix of vertex degrees; the walk evolves as $e^{-iHt}$. On a quasicrystal the diagonal term is not uniform, so the model carries an effective on-site potential that varies from vertex to vertex. The lattices are a fivefold-symmetric Penrose tiling built from fat and thin rhombi and an octagonal Ammann-Beenker tiling built from squares and rhombi. Disorder is introduced by randomly deleting edges, and percolation is scored by whether at least 2% of the probability has left a zone of 40 hopping steps; curves are averaged over 50 random configurations.

What would settle it

Rerun the identical percolation protocol with the graph Laplacian replaced by the adjacency Hamiltonian $H=-\gamma A$, so that no degree-dependent on-site term exists, on the same Penrose and Ammann-Beenker tilings. If the localization signatures and the 32%/40% thresholds persist, the tiling's aperiodic order is doing the work; if they move toward the square-lattice threshold, the degree term is the cause. A second decisive control is a square lattice whose vertices are assigned the same degree sequence as each quasicrystal, which should mimic the effect if degrees alone matter.

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Extended reading notes

Core claim

The central claim is that the aperiodic order of quasicrystals, not merely the presence of disorder, suppresses the spread of a continuous-time quantum walk. On Penrose and Ammann-Beenker tilings the walker retains a notable fraction of probability around its initial vertex even when all edges are present, a signature the square lattice does not show. Random edge removal suppresses percolation faster on both quasicrystals than on the square lattice; percolation probability reaches zero after about 32% of edges are removed on Ammann-Beenker lattices, about 40% on Penrose lattices, and only at 50% on square lattices. The authors also report that the choice of starting vertex matters on aperiodic tilings, and that Ammann-Beenker traps the walker slightly more than Penrose. From this they conclude that quasicrystal lattices can hold a quantum state longer than rectangular lattices, which they propose as a route to longer-lived quantum storage.

Load-bearing premise

Everything about the quasicrystal effect rests on comparing the chosen lattices, but the Hamiltonian's diagonal term varies with vertex degree on aperiodic tilings, so the slowdown may come from that built-in quasiperiodic potential rather than from the tiling's long-range order; the paper does not test a periodic lattice with matched degree variations.

Editorial extensions

If this is right

  • Quasicrystal lattices can serve as effective traps for a quantum state, with percolation stopping at roughly 32% broken edges on Ammann-Beenker and 40% on Penrose tilings.
  • On square lattices the same walker keeps percolating until about half the edges are removed, so quasicrystal geometry, rather than disorder alone, is what accelerates localization.
  • The starting vertex matters on aperiodic tilings, so experiments and applications must specify where the walker enters; some points are more trapping than others.
  • Among quasicrystals, the Ammann-Beenker tiling localizes the walker more strongly than the Penrose tiling, giving a first ordering of candidate tiling geometries for quantum storage.
  • The localization effect persists at nonzero disorder, supporting the idea that quasicrystal lattices could extend the time a quantum state stays available for readout.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An alternative reading of the same numbers is that the degree-dependent diagonal of the graph Laplacian acts as a quasiperiodic on-site potential; repeating the simulations with $H=-\gamma A$ would show how much of the reported 32%/40% thresholds is due to that potential rather than to the tiling itself.
  • Because the two quasicrystals differ mainly in their vertex degree statistics, the slightly stronger trapping on Ammann-Beenker could be a degree-distribution effect; comparing with a random lattice that has the same degree distribution would test this without any quasicrystalline order.
  • A natural next question is whether the localization survives interactions between walkers or a continuously monitored walk; if it does, quasicrystal arrays become a concrete platform for disorder-resilient quantum memory, though the paper itself does not quantify decoherence times.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper studies continuous-time quantum walks (CTQW) on two-dimensional quasicrystal tilings (Penrose and Ammann-Beenker) and compares their spreading behavior with that on a square lattice. The dynamics is generated by the graph-Laplacian Hamiltonian H = γ(D − A) of Eq. (4), and disorder is modeled by randomly removing edges. Percolation is defined as the probability that at least 2% of the wavefunction lies outside a hopping zone of length 40. The paper reports that, for a 100×100 square lattice, the percolation probability vanishes after about 50% edge disconnection fraction, while for Ammann-Beenker and Penrose lattices it vanishes after about 32% and 40%, respectively (Figs. 11–13). The authors conclude that the aperiodic nature of quasicrystals slows quantum percolation and that such lattices could be used to store quantum states for longer times.

Significance. If the central conclusion held, the work would be a useful numerical contribution on quantum walks in aperiodic media, potentially relevant for designing lattices with suppressed spreading and extended coherence times. The paper is transparent about its numerical setup, explicitly states the Hamiltonian and the percolation criterion, considers multiple initial positions on each quasicrystal lattice, and compares two different quasicrystal tilings against a square-lattice baseline. These are genuine strengths. However, the central attribution of the observed suppression to aperiodicity is not yet established, because the chosen Hamiltonian introduces a degree-dependent on-site potential that is different on quasicrystals and square lattices, and because the quantitative threshold claims rest on arbitrary definitions and unquantified statistical error. The manuscript would need additional control simulations before the main conclusion can be accepted.

major comments (4)
  1. [Section II, Eq. (4)] The Hamiltonian H = γ(D − A) is the graph Laplacian. On a square lattice D is a constant multiple of the identity, so H is equivalent to the adjacency Hamiltonian up to a global phase. On the Penrose and Ammann-Beenker tilings the vertex degree takes several distinct values, so the diagonal term γ d_i acts as a deterministic quasiperiodic on-site potential. Such a potential can by itself slow spreading or localize the wavefunction, independently of the aperiodicity of the hopping graph. The paper never compares with the adjacency Hamiltonian H = −A, nor with a Hamiltonian having a constant on-site term, nor with a regular lattice that has the same degree distribution. Consequently, the conclusion in Section IV that the reduced percolation is due to the aperiodic nature of the tilings is not yet supported; a control calculation with the adjacency Hamiltonian is needed before the central claim can be assigned to quasicrystalline geometry.
  2. [Section II] The percolation criterion is defined as having at least 2% of the total probability outside a hopping zone of length 40. These two numbers are presented without justification, and the quantitative threshold fractions reported in Figs. 11–13 (about 50%, 32%, and 40% for the square, Ammann-Beenker, and Penrose lattices) are tied to these choices. The paper should either show robustness of the thresholds when the 2% probability threshold and the hopping-zone length are varied, or provide a physical argument that fixes these scales; otherwise the specific numeric threshold claims are not robust enough to be the main quantitative result.
  3. [Section III C, Figs. 12–13] All disorder-averaged results are stated to be averaged over 50 runs, but no error bars, standard deviations, or confidence intervals are provided. It is therefore unclear whether the differences between the three lattices, and the apparent vanishing of percolation at 32% and 40% edge disconnection, are statistically significant rather than consequences of the particular finite sample. The authors should report a measure of spread over runs and over initial positions, and they should define how 'percolation probability vanishes' is determined (for example, the value falling below a specified cutoff) instead of relying only on visual inspection of the plotted curves.
  4. [Section III] No finite-size analysis is included. The text says the actual calculations used a 41-iteration Ammann-Beenker tiling and a 7-iteration Penrose tiling, whereas the probability-distribution figures (Figs. 4 and 5) are captioned as three- and four-iteration tilings; the square lattice is 100×100. If the percolation thresholds in Figs. 11–13 are compared across the three systems, the linear size and boundary conditions of the three lattices need to be matched, or the dependence of the thresholds on lattice size must be checked. Without this, the differences in threshold fractions could reflect the finite sizes of the tilings rather than the intrinsic lattice geometry.
minor comments (6)
  1. [Section III C] There is a typo in the text: 'Amman-Beeker' should be 'Ammann-Beenker'.
  2. [Fig. 13 caption] The caption contains a duplicated phrase: 'edge edge disconnection fraction' should be 'edge disconnection fraction'.
  3. [Fig. 1 caption] The caption refers to a '15 hopping length percolation test zone', which conflicts with the hopping-zone length of 40 used throughout the text and other figures.
  4. [Section III, Figs. 4–5] The iteration numbers of the Ammann-Beenker and Penrose tilings in the figure captions (three and four iterations) are inconsistent with the statement in the text that the actual calculations used a 41-iteration Ammann-Beenker tiling and a 7-iteration Penrose tiling; this inconsistency should be resolved.
  5. [Section III B] The phrase 'the positon of dislocations may be different' should be 'the positions of the disconnections may be different', since the disorder is implemented as removed edges, not as lattice dislocations.
  6. [References] Reference [4] appears to have a corrupted author name ('B. Bollobs' should be 'B. Bollobás') and reference [52] has a typo in the publisher name ('Freemann' should be 'Freeman').

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: percolation thresholds are genuine outputs of the stated CTQW Hamiltonian and operational definitions, not fitted inputs or self-citation-derived claims.

full rationale

The paper's central comparison is a direct numerical simulation: CTQW dynamics are defined by Eqs. (3)-(4) with H = γ(D−A) and γ=1, and the percolation probability is an operational count of runs in which at least 2% of the probability lies outside a 40-hop zone (Section II). Every parameter is stated up front; none is fitted to the percolation probabilities later reported. The vanishing-disconnection fractions (50%, about 32%, and about 40% for square, Ammann-Beenker, and Penrose lattices) are outputs of that stated model and criterion, not reconstructions of values already used to calibrate the model. The self-citations (e.g., Refs. [14], [15], [35]) are background context on quantum walks and disorder and are not load-bearing for the present results; no uniqueness theorem or ansatz is imported from the authors' prior work. A genuine scientific caveat exists but is not circularity: because H uses the graph Laplacian, the quasicrystal lattices carry a degree-dependent diagonal term that the square lattice lacks, so the attribution of slower percolation to 'aperiodic nature' conflates aperiodic hopping with an on-site coordination potential. The paper also states in Section III B that the square-lattice result 'can be derived analytically' without giving the derivation, but that standard claim is not used to justify the quasicrystal conclusion. These are modeling and presentation concerns, not identities between prediction and input; the reported quantities follow from the stated Hamiltonian rather than by construction. No circular step meeting the quoted-reduction standard was found.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central result depends on the arbitrary percolation criterion (2% outside a 40-hop zone), on the choice of the graph Laplacian as Hamiltonian (which adds a degree-dependent potential on quasicrystals), and on finite-size patches that are not tested for scaling. No new entities are introduced.

free parameters (4)
  • percolation probability threshold = 2%
    The particle is considered percolated if at least 2% of probability lies outside the hopping zone (Section II). This cutoff is chosen by hand and directly determines the reported percolation curves and threshold fractions.
  • hopping zone length = 40
    The test zone radius is 40 hop lengths (Section II). Results for square lattice use multiple zone lengths (20-50), but quasicrystal comparisons use only 40, which may not be directly comparable.
  • number of disorder runs = 50
    Averaging over 50 random edge-disconnection realizations (Section III B); no error bars are reported, so statistical uncertainty is unknown.
  • evolution time = 200
    Percolation probabilities as a function of disconnection fraction are reported at time t=200 (Figs. 11-13). Different times could change thresholds.
assumptions (4)
  • domain assumption The graph Laplacian H = γ(D - A) is the appropriate CTQW Hamiltonian for these lattices.
    Invoked in Eq. (4). On quasicrystals the diagonal degree term creates a quasiperiodic on-site potential, which is not controlled for.
  • domain assumption Finite-size quasicrystal patches are representative of infinite quasicrystals.
    No finite-size scaling is presented; the cited iteration counts are inconsistent between text and figures.
  • standard math The Schrödinger equation governs the evolution and the norm is conserved.
    Section II, Eq. (3). This is standard and not in question.
  • domain assumption Averaging over 50 independent edge-disconnection realizations with fixed fraction gives a converged estimate of percolation probability.
    Section III B. No error bars or convergence checks are provided.

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Pith. "Pith review of Quantum percolation in quasicrystals using continuous-time quantum walk." pith.science (2026). https://pith.science/paper/4RXQ3SYM

@misc{pith2026190803051,
  author       = {Pith},
  title        = {Pith review of: Quantum percolation in quasicrystals using continuous-time quantum walk},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4RXQ3SYM}},
  note         = {Machine review of arXiv:1908.03051}
}
read the original abstract

We study the percolation of a quantum particle on quasicrystal lattices and compare it with the square lattice. For our study, we have considered quasicrystal lattices modelled on the pentagonally symmetric Penrose tiling and the octagonally symmetric Ammann-Beenker tiling. The dynamics of the quantum particle is modelled using continuous-time quantum walk (CTQW) formalism. We present a comparison of the behaviour of the CTQW on the two aperiodic quasicrystal lattices and the square lattice when all the vertices are connected and when disorder is introduced in the form of disconnections between the vertices. Unlike on a square lattice, we see a significant fraction of quantum state localised around the origin in quasicrystal lattice. With increase in disorder, the percolation probability of a particle on a quasicrystal lattice decreases significantly faster when compared to the square lattice. This study sheds light on the minimum fraction of disconnections allowed to see percolation of quantum particle on these quasicrystal lattices.

Figures

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Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
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Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
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Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p004_6.png]
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Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]
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Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p005_8.png]
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Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p005_9.png]
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Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p006_11.png]
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Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p006_12.png]
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Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p006_13.png]

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