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arxiv: 2604.11620 · v1 · submitted 2026-04-13 · 🪐 quant-ph · math-ph· math.MP

Quantum state transfer on a scalable network under unital and non-unital noise

Monika Rani , Subhashish Banerjee , Nikhil Swami , Supriyo Dutta This is my paper

Pith reviewed 2026-05-10 15:35 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords quantum state transferbutterfly graphsdiscrete-time quantum walknon-Markovian noiseunital noiseamplitude dampingquantum networksscalability
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The pith

Butterfly graphs enable perfect quantum state transfer in discrete-time quantum walks, with robustness under unital and non-unital noise.

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

The paper establishes that a class of bipartite graphs called butterfly graphs supports high-fidelity quantum state transfer when the system evolves under discrete-time quantum walk rules. This holds both in the ideal noiseless setting and when the walker interacts with the environment through non-Markovian noise channels. The authors examine several concrete constructions of these graphs and several noise types, including unital random telegraph and modified Ornstein-Uhlenbeck processes as well as non-unital amplitude damping. A sympathetic reader would see this as evidence that certain network topologies can be used to build scalable sender-receiver links that remain effective even when realistic decoherence is present.

Core claim

The authors show that quantum state transfer occurs across different butterfly graphs within the discrete-time quantum walk framework, thereby extending the known families of networks that support high-fidelity transfer. They further demonstrate that this transfer remains robust when the evolution is subject to non-Markovian environmental noise consisting of unital random telegraph noise and modified Ornstein-Uhlenbeck noise together with non-unital amplitude damping noise.

What carries the argument

Discrete-time quantum walks on butterfly graphs that produce perfect state transfer between designated sender and receiver vertices in the noiseless limit.

If this is right

  • Scalable quantum networks can be assembled from multiple butterfly-graph building blocks for sender-receiver communication.
  • High-fidelity transfer survives the memory effects captured by the chosen unital and non-unital noise models.
  • The same graph family supports transport under both unital and non-unital environmental interactions.
  • Butterfly graphs enlarge the catalog of topologies known to permit reliable quantum state transfer.

Where Pith is reading between the lines

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

  • Device designers could adopt butterfly-graph layouts as a concrete template for noise-tolerant quantum links.
  • The same noise-analysis approach could be applied to other graph families to identify additional resilient transfer architectures.
  • Explicit control of non-Markovian memory time scales might be used to further optimize fidelity in real hardware.

Load-bearing premise

The chosen butterfly graph constructions together with the discrete-time quantum walk evolution rules produce perfect state transfer when noise is absent.

What would settle it

A direct calculation of the noiseless walk on any of the examined butterfly graphs that yields state-transfer fidelity significantly below unity at the predicted arrival time.

Figures

Figures reproduced from arXiv: 2604.11620 by Monika Rani, Nikhil Swami, Subhashish Banerjee, Supriyo Dutta.

Figure 1
Figure 1. Figure 1: A path graph with n vertices. [30, 32], as well as a non-unital non-Markovian Amplitude Damping Noise (ADN) [33]. Although these noise models were originally introduced for qubits, they have been generalised to higher-dimensional systems using Weyl operators [34]. Among the various ways to incorporate noise into quantum walks, we adopt an approach in which the quantum noise acts on the walker’s state after… view at source ↗
Figure 2
Figure 2. Figure 2: Different butterfly graphs generated by the seed graph [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: State transfer on P2. 4 Discrete-time quantum walk on the butterfly graphs We consider that there is a sender and a receiver placed at two distinct vertices of a butterfly graph. We mark these vertices with s and r, respectively. It classifies the vertex set V (Bk) into the marked and unmarked vertices. We consider different cases based on the locations of the marked vertices. 4.1 State transfer on the but… view at source ↗
Figure 4
Figure 4. Figure 4: State transfer on the butterfly graph B1. (6) we have |ψ(s)⟩ = 1 √ 2 h | −−−→ (0, 1)⟩ + | −−−→ (0, 2)⟩ i ; and |ψ(r)⟩ = 1 √ 2 h | −−−→ (1, 0)⟩ + | −−−→ (1, 3)⟩ i . (14) The evolution operator UB1 = SB1CB1 where CB1 is a block diagonal matrix represented by CB1 = diag{−X, −X, X, X}, in this case. We observe that the fidelity becomes 0.25 at t = 1, 3, 5, . . . . The fidelity plot is shown in Figure 4b. When … view at source ↗
Figure 5
Figure 5. Figure 5: State transfer on the butterfly graph B2. 2. The sender and receivers are placed on the wings of the butterfly with the maximum distance, and they are in different partite set. Now we assume that s = 5 and r = 6. Similarly, |ψ(s)⟩ = 1 √ 2 h | −−−→ (5, 1)⟩ + | −−−→ (5, 4)⟩ i ; and |ψ(r)⟩ = 1 √ 2 h | −−−→ (6, 0)⟩ + | −−−→ (6, 7)⟩ i . (33) The coin operator in this case is represented by C = C0 MC1 MC2 MC3 MC… view at source ↗
Figure 6
Figure 6. Figure 6: State transfer on the butterfly graph B3 generated by the path graph P2. The coin operator is given by C = −C0 MC1 M−C2 MC3 MC4 MC5 MC6 MC7. (38) In this case, fidelity is less than 0.5 always, as depicted in Figure 6e. The maximum fidelity of state transfer is 0.4183, which is at the time step t = 175. Now we calculate the average fidelity for different locations of the sender and the receiver on the butt… view at source ↗
Figure 7
Figure 7. Figure 7: State transfer on the butterfly graph B3 formed by seed graph P3. Following equation (3), the coin operator is given by C = C0 MC1 MC2 MC3 MC4 M−C5 M−C6 MC7 MC8. (40) From Figure 7a, we notice that d(0) = d(2) = 4, d(1) = 5. Also, the degree of all other vertices is 2. The coin operator for nodes of degree four is given by equation (32). The coin operators for the vertices with degree 5 is, C1 =      … view at source ↗
Figure 8
Figure 8. Figure 8: State-transfer on the butterfly graph B3 generated by path graph P2, in the absence of noise and under non-Markovian RTN (8a), modified non-Markovian OUN (8b) and NMAD noise (8c). The RTN parameters are set to a = 0.1, γ = 0.01, while the OUN parameters are λ = 1, γ = 0.05. The channel parameters for NMAD noise are γ = 5 and g = .001. (a) (b) (c) [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Coherence of a quantum walker for a butterfly graph [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: State transfer on the the butterfly graph [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
read the original abstract

We investigate quantum state transfer on a class of bipartite graphs, namely the butterfly graphs, within the framework of discrete-time quantum walks. These graphs facilitate the construction of scalable quantum networks that enable communication between a sender and a receiver via perfect state transfer. Our analysis demonstrates that state transfer occurs across different butterfly graphs, thereby extending the known families of networks that support high-fidelity quantum state transfer. In addition to the ideal noiseless dynamics, we further investigate the robustness of quantum state transfer in the presence of non-Markovian environmental noise, specifically, random telegraph noise, modified Ornstein-Uhlenbeck noise, which are examples of unital noise and non-Markovian amplitude damping noise, non-unital noise. These noise models capture different types of system-environment interactions and memory effects that influence the coherence of the quantum walk. These findings contribute to the theoretical understanding of how butterfly graph constructions influence quantum transport phenomena.

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 quantum state transfer via discrete-time quantum walks on butterfly graphs. It claims that these graphs support perfect state transfer in the noiseless limit, extending known families of networks for high-fidelity QST. It also examines the robustness of this transfer under unital non-Markovian noises (random telegraph and modified Ornstein-Uhlenbeck) and non-unital noise (amplitude damping).

Significance. If the noiseless PST baseline is rigorously verified, this would extend the catalog of graphs known to support high-fidelity quantum state transfer, which is relevant for scalable quantum network design. The distinction between unital and non-unital noise models and their memory effects adds useful insight into decoherence in quantum walks. The work receives credit for addressing both ideal dynamics and realistic noise, but its significance is currently constrained by the unverified central assumption.

major comments (2)
  1. Abstract and noiseless dynamics section: The assertion that perfect state transfer occurs on the butterfly graphs (with unit fidelity at a finite time) requires explicit verification that the coin-position evolution operator satisfies the necessary spectral conditions (eigenvalue phases aligning to map the sender state exactly onto the receiver vertex). No derivation, eigenvalue analysis, or fidelity check is referenced, rendering the noiseless baseline unestablished and the subsequent noise-robustness results dependent on an unproven premise.
  2. Noise robustness analysis section: All quantitative claims about fidelity decay under the listed noise channels presuppose that the ideal (noiseless) transfer reaches probability exactly 1. Without confirmation of this prerequisite via the standard conditions for PST in discrete-time walks, the reported robustness metrics cannot be interpreted as deviations from a verified perfect baseline.
minor comments (2)
  1. The abstract refers to 'different butterfly graphs' without specifying the graph sizes, parameters, or explicit constructions used in the analysis, hindering reproducibility.
  2. The implementation details for the noise models (e.g., how the random telegraph or modified Ornstein-Uhlenbeck processes are coupled to the walk evolution, including all parameter values and discretization) should be expanded for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The two major comments both center on the need for explicit verification of perfect state transfer (PST) in the noiseless limit. We agree that this verification is essential to establish the baseline before analyzing noise robustness, and we will add the required derivation in the revised manuscript.

read point-by-point responses

  1. Referee: Abstract and noiseless dynamics section: The assertion that perfect state transfer occurs on the butterfly graphs (with unit fidelity at a finite time) requires explicit verification that the coin-position evolution operator satisfies the necessary spectral conditions (eigenvalue phases aligning to map the sender state exactly onto the receiver vertex). No derivation, eigenvalue analysis, or fidelity check is referenced, rendering the noiseless baseline unestablished and the subsequent noise-robustness results dependent on an unproven premise.

    Authors: We acknowledge that the current manuscript does not contain an explicit spectral analysis or derivation confirming that the coin-position evolution operator on the butterfly graphs satisfies the PST conditions. In the revised version we will insert a new subsection in the noiseless-dynamics part that (i) recalls the standard PST criterion for discrete-time quantum walks, (ii) computes the eigenvalues and eigenvectors of the unitary operator restricted to the relevant subspace, and (iii) verifies that the phase alignment maps the sender state exactly onto the receiver vertex at the predicted time, yielding fidelity 1. This will make the baseline rigorous and independent of numerical checks. revision: yes

  2. Referee: Noise robustness analysis section: All quantitative claims about fidelity decay under the listed noise channels presuppose that the ideal (noiseless) transfer reaches probability exactly 1. Without confirmation of this prerequisite via the standard conditions for PST in discrete-time walks, the reported robustness metrics cannot be interpreted as deviations from a verified perfect baseline.

    Authors: We agree that the noise-robustness figures are only interpretable once the noiseless fidelity is rigorously shown to equal unity. The explicit spectral verification described in our response to the first comment will be placed before the noise sections, so that all subsequent fidelity-decay curves are clearly presented as deviations from a proven perfect baseline. We will also add a short statement in the noise-analysis section reminding the reader of this verified starting point. revision: yes

Circularity Check

0 steps flagged

No circularity: standard quantum walk formalism applied to butterfly graphs with explicit noise analysis

full rationale

The derivation relies on the established discrete-time quantum walk operator on bipartite graphs and standard unital/non-unital noise channels (random telegraph, Ornstein-Uhlenbeck, amplitude damping). The claim of perfect state transfer in the noiseless limit is presented as a consequence of the graph construction and evolution rules rather than a self-referential definition or fitted parameter. No load-bearing step reduces to a prior self-citation, ansatz smuggled via citation, or renaming of known results; the noise robustness analysis uses independent models. The work is self-contained against external benchmarks for the chosen graphs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard discrete-time quantum walk formalism on graphs and established noise models (random telegraph, Ornstein-Uhlenbeck, amplitude damping) without introducing new free parameters, axioms beyond standard quantum mechanics, or invented entities.

axioms (2)
  • standard math Discrete-time quantum walk evolution on bipartite graphs follows the standard coin and shift operator construction.
    Invoked implicitly when stating that state transfer occurs on butterfly graphs.
  • domain assumption The listed noise models (unital and non-unital) accurately represent relevant environmental effects on the quantum walk.
    Used to investigate robustness without deriving the noise from first principles.

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