Perfect state transfer in Grover walks on normal Cayley graphs

Bikash Bhattacharjya; Koushik Bhakta

Perfect state transfer in Grover walks on normal Cayley graphs

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Pith's one-line read The paper proves that perfect state transfer in Grover walks on normal Cayley graphs occurs exactly when the vertex displacement is a central involution and the transfer-time Chebyshev polynomial takes the corresponding character signs, wit

desk verdict The main characterization is new and correct; the unitary Cayley classification holds; but Theorem 4.1's periods are wrong for n=2,3,6 and need a fix. read the letter →

arxiv 2607.19309 v1 pith:2Z2PEYPV submitted 2026-07-21 quant-ph math.CO

Koushik Bhakta , Bikash Bhattacharjya This is my paper

classification quant-phmath.CO MSC 05C5005C2581Q99

keywords GroverwalkperfectstatetransfernormalCayleygraphChebyshevpolynomialdiscriminantmatrixcentralinvolutionunitarydicyclicgroup

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

The paper gives a complete answer to when a Grover walk on a normal Cayley graph transfers a vertex-localized state to another vertex with certainty. The answer is group-theoretic: the two vertices must differ by an order-2 element in the center of the group, and the transfer-time Chebyshev polynomial must evaluate to 1 or −1 at each discriminant eigenvalue according to the sign of the corresponding irreducible character at that element. This replaces a dynamical search over walk times with a finite check on the character table and the small spectrum of the graph. The payoff is concrete: among unitary Cayley graphs, exactly the four graphs of orders 2, 4, 6, and 12 exhibit perfect state transfer, while infinite families over abelian, dicyclic, and dihedral groups are constructed. A reader should care because it makes one of the standard discrete-time quantum walk models completely tractable for an important class of highly symmetric graphs.

What carries the argument

The engine is the discriminant matrix P (for a k-regular graph, P=(1/k)A) together with the spectral mapping of the Grover walk: the eigenvalues of the time-evolution matrix are exp(±i arccos μ) for eigenvalues μ of P. A previously established lemma converts perfect state transfer at time τ into the single matrix equation T_τ(P)e_u = e_v, so only Chebyshev polynomials of P matter. On a normal Cayley graph, P's eigenspaces are indexed by irreducible representations, and the trace formula T_τ(P)_{u,v} = (1/|Γ|) Σ_ρ d_ρ T_τ(μ_ρ) χ_ρ(uv^{-1}) carries the whole argument: it forces the displacement to be a central involution and dictates the required sign pattern. The same machinery yields the per

What would settle it

Numerically diagonalize the Grover transition matrix of the unitary Cayley graph on 8 vertices and test every power acting on each vertex state. The theorem predicts no perfect state transfer to any vertex at any time; observing U^τ Φ_u = γΦ_v for any τ and unit γ would refute the classification.

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Extended reading notes

Core claim

The central claim is Theorem 3.4: for a normal Cayley graph Cay(Γ,S), perfect state transfer from u to v at time τ occurs if and only if z = uv^{-1} is an order-2 element in the center of Γ, and for every irreducible representation ρ of Γ, the Chebyshev polynomial T_τ satisfies T_τ(μ_ρ) = 1 when χ_ρ(z) = d_ρ and T_τ(μ_ρ) = −1 when χ_ρ(z) = −d_ρ, where μ_ρ are the eigenvalues of the discriminant matrix P and d_ρ is the degree of ρ. The proof shows that PST forces T_τ(P) to be the permutation matrix of right multiplication by z, and the trace formula decomposes T_τ(P) over the irreducible representations, making the signs above necessary and sufficient. Applied to unitary Cayley graphs, the cr

Load-bearing premise

The load-bearing premise is the previously established equivalence that perfect state transfer in a Grover walk occurs exactly when T_τ(P)e_u = e_v with phase exactly 1; if an unavoidable phase factor can survive in that equivalence, the sufficiency direction of the criterion collapses.

Editorial extensions

If this is right

Perfect state transfer on normal Cayley graphs is decidable from the character table and the discriminant spectrum; no search over arbitrarily long walk times is required.
Groups without a central involution (for example, odd-order dihedral groups) can never support perfect state transfer in Grover walks on their normal Cayley graphs.
The criterion constructs infinite families of perfect-state-transfer graphs over abelian, dicyclic, and dihedral groups, with minimum transfer times often given by lcm(4,m0)/2.
For unitary Cayley graphs, the only perfect-state-transfer graphs are those of orders 2, 4, 6, and 12; every other unitary Cayley graph fails the criterion.
Whenever perfect state transfer occurs at its earliest time, the graph is 2τ-periodic, tying the transfer time to the graph's period.

Reading between the lines

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

Editorial inference: the rigidity of the two conditions suggests that among random large Cayley graphs, perfect state transfer is rare; testing small groups with random connection sets should show almost no examples unless a central involution and spectral coincidences are engineered.
Editorial inference: the same trace formula could be pushed toward 'pretty good state transfer' by replacing the exact ±1 requirements with asymptotically approaching values; the paper does not address this.
Editorial inference: the unitary-Cayley classification might extend to other circulant or quadratic unitary families, and the character-table criterion would be the natural tool for generating analogous finite lists.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the central result is real and the paper is mostly in good shape, but there is a concrete false statement in Theorem 4.1 that needs correcting. The main theorem, Theorem 3.4, gives a necessary and sufficient condition for PST in Grover walks on normal Cayley graphs, and it is genuinely new: it covers all normal Cayley graphs, recovers the earlier circulant/dihedral cases, and yields the first complete PST classification of unitary Cayley graphs (n = 2, 4, 6, 12). I checked the proof carefully. Lemma 3.1's reduction to a permutation matrix is sound, the spectral expansion (3.4) is correct, and the sign-counting in the necessity direction works. The dependence on imported lemmas — including the authors' own periodicity criterion — is normal building-block use, not circularity.

The soft spot is Theorem 4.1. The period values stated there are wrong for n = 2, 3, and 6. G_Z2 is K2 and has period 2, not 4; G_Z3 is K3 and has period 3, not 12; G_Z6 is C6 and has period 6, not 12. The 'if and only if' part of the theorem — periodicity occurs exactly for n = 2^α 3^β — is still correct; only the computation of the actual period in those three small cases is off. And this error does not infect Theorem 4.4: the PST classification handles n = 2, 4, 6 via the cycle/complete-graph lemmas and n = 12 directly, so the wrong periods are never loaded. Still, a false theorem in the paper needs a correction before publication. Also, the abstract calls Corollaries 3.13 and 3.17 'characterizations,' but they are only necessary conditions; and in both corollaries the two items are numbered (i) and (i).

This is a paper worth taking seriously. The proof method is clean, the applications are worked out in real detail, and the unitary Cayley classification is a satisfying capstone. The right outcome is minor revision: fix the periods, fix the labels, and tone down 'characterizations.' I'd cite Theorem 3.4 and the classification, and I'd bring it to reading group to discuss both the method and the small-case period subtlety.

Referee Report

0 major / 6 minor

Summary. The paper studies perfect state transfer (PST) in discrete-time Grover walks on normal Cayley graphs. The main result, Theorem 3.4, gives an if-and-only-if condition: PST from u to v at time tau occurs exactly when z = u v^{-1} is a central element of order 2 and, for every irreducible representation rho, the Chebyshev evaluation T_tau(mu_rho) equals +1 or -1 according as chi_rho(z)=d_rho or chi_rho(z)=-d_rho. The proof combines the spectral mapping theorem for Grover walks with the representation-theoretic spectral decomposition of the discriminant. The authors apply this to abelian, dicyclic, and dihedral Cayley graphs, giving infinite families with explicit transfer times, and they prove a complete classification of unitary Cayley graphs: PST occurs exactly for n in {2,4,6,12}. They also characterize periodicity of unitary Cayley graphs as n = 2^alpha 3^beta.

Significance. If the main theorem is correct, it reduces a dynamical question to a finite character-table computation, which is a genuine conceptual advance for this class of graphs. The resulting unitary Cayley classification is crisp, falsifiable, and parameter-free, and the paper recovers previously known dihedral results as special cases. I checked the sign-counting argument in Theorem 3.4, the applications to dicyclic and dihedral groups, and the unitary Cayley classification; the central derivations appear sound. The proofs are self-contained given standard imported lemmas. I found one localized mathematical error in the period formula of Theorem 4.1 for n=2,3,6, plus several wording and presentation issues that do not affect the main PST classification.

minor comments (6)

[Section 4, Theorem 4.1] The period formula is incorrect for n=2,3,6. For n=2 the Grover walk on K2 is a transposition, so the period is 2, not 4. For n=3 the time-evolution eigenvalues are {1,omega,omega^2}, so the period is 3, not 12. For n=6 the unitary Cayley graph is C6, whose period is 6, not 12. The proof itself treats n=2, n=3, and n=6 in separate initial cases, but the final displayed formula omits these exceptions. The error is local: Theorem 4.4 handles n=2,3,6 directly and invokes the period formula only for n=12, where it is correct.
[Abstract; Corollaries 3.13 and 3.17] The abstract and introduction describe Corollaries 3.13 and 3.17 as 'simple combinatorial characterizations of the existence of perfect state transfer.' As stated, these corollaries are only necessary conditions ('If G exhibits PST, then ...'), and the authors explicitly note that no example satisfying the first alternative is currently known. Please reword to 'necessary combinatorial conditions' or add a true sufficiency statement.
[Corollaries 3.13 and 3.17] In both corollary statements the second alternative is labelled '(i)' again; it should be labelled '(ii)'.
[Example 3.16] The variable n is overloaded and the proof contains apparent typos. The text 'First, assume that n is odd' cannot be right because n=2m is even by construction; it should refer to m. In that case the Chebyshev index should be T_{2m} (which equals T_n with the group parameter n=2m), not T_{2n}=T_{4m}; otherwise the required value T(0)=-1 is not obtained. The subsequent 'n is even' case likewise refers to m.
[After Corollary 3.17] The sentence 'Example 3.12 provides such an example exhibiting perfect state transfer' in the dihedral subsection should refer to Example 3.16, since Example 3.12 is in the dicyclic subsection.
[Section 4, proof of Theorem 4.1] The proof refers to 'Corollary 2.5' when determining the period; the cited statement is Lemma 2.5, not a corollary.

Circularity Check

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No significant circularity: the central PST characterization and unitary-Cayley classification are derived from standard imported lemmas and group representation theory, not from fitted inputs or definitional equivalences.

full rationale

The paper's main result, Theorem 3.4, is derived from external lemmas (Kubota–Yoshino [23], Guo–Schmeits [13], Higuchi et al. [14]) plus the standard representation-theoretic eigenspace decomposition of normal Cayley graphs. No parameter is fitted and no 'prediction' is just a renamed input. The proof reduces PST to verifying T_tau(P)e_u = e_v, then shows for normal Cayley graphs this forces T_tau(P) to be a right multiplication by a central involution, and the character sum computation is a genuine condition rather than a restatement of the definition. The self-citation in Lemma 2.6 (periodicity criterion from the authors' [4]) is used as a supporting necessary condition to restrict unitary Cayley graphs, but it is a general theorem with stated assumptions and is not equivalent to the target PST result; it therefore provides independent support rather than circularity. The recovery of the dihedral result from [5] via Theorem 3.15 is presented as a consequence, not as the basis of the argument. The localized period-formula error in Theorem 4.1 for n=2,3,6 is a correctness issue, not a circularity, and does not affect the central 'if and only if' PST classifications. Therefore the derivation chain is self-contained and no circular step can be exhibited by quote and reduction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted; the paper is a pure derivation. The self-cited Lemma 2.6 ([4]) is a background theorem, not an input whose choice forces the result. No new entities are introduced.

assumptions (5)

standard math Spectral mapping theorem for Grover walks (Lemma 2.2)
Imported from Higuchi et al. [14, Prop 1]; used to go from discriminant eigenvalues to time-evolution eigenvalues in Corollary 3.2 and in period calculations.
standard math PST iff T_τ(P)e_u=e_v (Lemma 2.9)
Imported from Kubota–Yoshino [23] and Guo–Schmeits [13]; the foundation for Lemmas 3.1 and 3.4.
standard math Periodicity criterion for regular graphs (Lemma 2.6)
Imported from the authors' prior paper [4, Thm 4.4]; essential for Theorem 4.1 (unitary Cayley periodicity).
standard math Eigenvalue/eigenvector decomposition of normal Cayley graphs (Theorem 2.11)
Imported from Steinberg [29]; yields equation (3.4) and all eigenvalue computations.
standard math Characters: Schur's lemma and faithfulness of the regular representation
Used in Lemma 3.3 to characterize central involutions.

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Pith. "Pith review of Perfect state transfer in Grover walks on normal Cayley graphs." pith.science (2026). https://pith.science/paper/2Z2PEYPV

@misc{pith2026260719309,
  author       = {Pith},
  title        = {Pith review of: Perfect state transfer in Grover walks on normal Cayley graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2Z2PEYPV}},
  note         = {Machine review of arXiv:2607.19309}
}

abstract

A Cayley graph $\operatorname{Cay}(\Gamma,S)$ over a finite group $\Gamma$ is said to be normal if its connection set $S$ is a union of some conjugacy classes of $\Gamma$. This paper investigates perfect state transfer in Grover walks on normal Cayley graphs. The Grover walk is a widely studied discrete-time quantum walk. We establish a necessary and sufficient condition for the occurrence of perfect state transfer on normal Cayley graphs. As applications, we derive explicit spectral criteria for perfect state transfer on Cayley graphs over abelian groups, dicyclic groups, and dihedral groups. These results yield several infinite families of Cayley graphs exhibiting perfect state transfer. We further obtain simple combinatorial characterizations of the existence of perfect state transfer on Cayley graphs over dihedral and dicyclic groups. Our general characterization also recovers a number of previously known results as special cases. As a further consequence, we obtain a complete characterization of perfect state transfer on unitary Cayley graphs. In particular, we prove that exactly four graphs in the class of unitary Cayley graphs exhibit perfect state transfer.

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Works this paper leans on

32 extracted references · 3 linked inside Pith

[1]

Non-uniform mixing of quantum walks on the symmetric group.Linear Algebra Appl., 742:37–65, 2026

Avah Banerjee. Non-uniform mixing of quantum walks on the symmetric group.Linear Algebra Appl., 742:37–65, 2026

2026
[2]

Periodicity and perfect state transfer in quantum walks on variants of cycles.Quantum Inf

Katharine Barr, Timothy Proctor, Daniel Allen, and Viv Kendon. Periodicity and perfect state transfer in quantum walks on variants of cycles.Quantum Inf. Comput., 14:417–438, 2012

2012
[3]

Perfect state transfer in Grover walks on association schemes and distance-regular graphs.arXiv.2506.07439, 2025

Koushik Bhakta and Bikash Bhattacharjya. Perfect state transfer in Grover walks on association schemes and distance-regular graphs.arXiv.2506.07439, 2025

arXiv 2025
[4]

Periodicity and perfect state transfer of Grover walks on quadratic unitary Cayley graphs.Quantum Inf

Koushik Bhakta and Bikash Bhattacharjya. Periodicity and perfect state transfer of Grover walks on quadratic unitary Cayley graphs.Quantum Inf. Process., 24(8):Paper No. 260, 21, 2025

2025
[5]

Perfect state transfer in Grover walks on dihedral Cayley graphs.arXiv:2605.02254, 2026

Koushik Bhakta, Bikash Bhattacharjya, and Xiwang Cao. Perfect state transfer in Grover walks on dihedral Cayley graphs.arXiv:2605.02254, 2026

arXiv 2026
[6]

Quantum communication through an unmodulated spin chain.Phys

Sougato Bose. Quantum communication through an unmodulated spin chain.Phys. Rev. Lett., 91(20):Paper No. 207901, 4, 2003

2003
[7]

Pretty good state transfer in discrete-time quantum walks.J

Ada Chan and Hanmeng Zhan. Pretty good state transfer in discrete-time quantum walks.J. Phys. A, 56(16):Paper No. 165305, 25, 2023

2023
[8]

Hamiltonians of bipartite walks

Qiuting Chen, Chris Godsil, Mariia Sobchuk, and Hanmeng Zhan. Hamiltonians of bipartite walks. Electron. J. Combin., 31(4):Paper No. 4.10, 24, 2024

2024

Show all 32 references

[9]

Matthias Christandl, Nilanjana Datta, Artur Ekert, and Andrew J. Landahl. Perfect state transfer in quantum spin networks.Phys. Rev. Lett., 92(18):Paper No. 187902, 4, 2004

2004
[10]

Perfect state transfer on distance-regular graphs and association schemes.Linear Algebra Appl., 478:108–130, 2015

Gabriel Coutinho, Chris Godsil, Krystal Guo, and Fr´ ed´ eric Vanhove. Perfect state transfer on distance-regular graphs and association schemes.Linear Algebra Appl., 478:108–130, 2015

2015
[11]

Periodic graphs.Electron

Chris Godsil. Periodic graphs.Electron. J. Combin., 18(1):Paper No. 23, 15, 2011. 24

2011
[12]

Discrete-time quantum walks and graph structures.J

Chris Godsil and Hanmeng Zhan. Discrete-time quantum walks and graph structures.J. Combin. Theory Ser. A, 167:181–212, 2019

2019
[13]

State transfer in discrete-time quantum walks via projected transition matrices.arXiv:2411.05560, 2025

Krystal Guo and Vincent Schmeits. State transfer in discrete-time quantum walks via projected transition matrices.arXiv:2411.05560, 2025

2025 arXiv
[14]

Spectral and asymptotic properties of Grover walks on crystal lattices.J

Yusuke Higuchi, Norio Konno, Iwao Sato, and Etsuo Segawa. Spectral and asymptotic properties of Grover walks on crystal lattices.J. Funct. Anal., 267(11):4197–4235, 2014

2014
[15]

A convergence time of Grover walk on regular graph to stationary state with constant inflow to every vertex.Linear Multilinear Algebra, 72(15):2427– 2438, 2024

Ayaka Ishikawa, Sho Kubota, and Etsuo Segawa. A convergence time of Grover walk on regular graph to stationary state with constant inflow to every vertex.Linear Multilinear Algebra, 72(15):2427– 2438, 2024

2024
[16]

Periodicity of Grover walks on complete graphs with self-loops.Linear Algebra Appl., 599:121–132, 2020

Naoharu Ito, Toyoki Matsuyama, and Tatsuya Tsurii. Periodicity of Grover walks on complete graphs with self-loops.Linear Algebra Appl., 599:121–132, 2020

2020
[17]

Cambridge university press, 2001

Gordon Douglas James and Martin W Liebeck.Representations and characters of groups. Cambridge university press, 2001

2001
[18]

Perfect state transfer in quantum walks on graphs.J

Vivien Kendon and Christino Tamon. Perfect state transfer in quantum walks on graphs.J. Comput. Theor. Nanosci., 8:422–433, 2010

2010
[19]

A generalization of quantum pair state transfer.Quantum Inf

Sooyeong Kim, Hermie Monterde, Bahman Ahmadi, Ada Chan, Stephen Kirkland, and Sarah Plosker. A generalization of quantum pair state transfer.Quantum Inf. Process., 23(11):Paper No. 369, 35, 2024

2024
[20]

Some properties of unitary Cayley graphs.Electron

Walter Klotz and Torsten Sander. Some properties of unitary Cayley graphs.Electron. J. Combin., 14(1):Paper No. 45, 12, 2007

2007
[21]

Perfect state transfer in Grover walks between states associated to vertices of a graph.Linear Algebra Appl., 646:238–251, 2022

Sho Kubota and Etsuo Segawa. Perfect state transfer in Grover walks between states associated to vertices of a graph.Linear Algebra Appl., 646:238–251, 2022

2022
[22]

Periodicity of quantum walks defined by mixed paths and mixed cycles.Linear Algebra Appl., 630:15–38, 2021

Sho Kubota, Hiroto Sekido, and Harunobu Yata. Periodicity of quantum walks defined by mixed paths and mixed cycles.Linear Algebra Appl., 630:15–38, 2021

2021
[23]

Circulant graphs with valency up to 4 that admit perfect state transfer in Grover walks.J

Sho Kubota and Kiyoto Yoshino. Circulant graphs with valency up to 4 that admit perfect state transfer in Grover walks.J. Combin. Theory Ser. A, 216:Paper No. 106064, 31, 2025

2025
[24]

A characterization of state transfer on double subdivided stars.Linear Multilinear Algebra, 73(12):2713–2729, 2025

Sarojini Mohapatra and Hiranmoy Pal. A characterization of state transfer on double subdivided stars.Linear Multilinear Algebra, 73(12):2713–2729, 2025

2025
[25]

On certain trigonometrical sums and their applications in the theory of numbers.Trans

Srinivasa Ramanujan. On certain trigonometrical sums and their applications in the theory of numbers.Trans. Cambridge Philos. Soc., 22(13):259–276, 1918. 25

1918
[26]

Periodicity of lively quantum walks on cycles with generalized Grover coin.Linear Algebra Appl., 604:399–424, 2020

Rohit Sarma Sarkar, Amrita Mandal, and Bibhas Adhikari. Periodicity of lively quantum walks on cycles with generalized Grover coin.Linear Algebra Appl., 604:399–424, 2020

2020
[27]

Volume 42 of Graduate Texts in Math- ematics

Jean-Pierre Serre.Linear Representations of Finite Groups. Volume 42 of Graduate Texts in Math- ematics. Springer New York, 1996

1996
[28]

Perfect state transfer on hyper- cubes and its implementation using superconducting qubits.Phys

Siddhant Singh, Bibhas Adhikari, Supriyo Dutta, and David Zueco. Perfect state transfer on hyper- cubes and its implementation using superconducting qubits.Phys. Rev. A, 102(6):Paper No. 062609, 9, 2020

2020
[29]

Universi- text

Benjamin Steinberg.Representation Theory of Finite Groups: An Introductory Approach. Universi- text. Springer New York, 2011

2011
[30]

An infinite family of circulant graphs with perfect state transfer in discrete quantum walks.Quantum Inf

Hanmeng Zhan. An infinite family of circulant graphs with perfect state transfer in discrete quantum walks.Quantum Inf. Process., 18(12):Paper No. 369, 26, 2019

2019
[31]

Quantum walks on embeddings.J

Hanmeng Zhan. Quantum walks on embeddings.J. Algebraic Combin., 53(4):1187–1213, 2021

2021
[32]

Algebraic Combin., 61(4):Paper No

Hanmeng Zhan.ϵ-uniform mixing in discrete quantum walks.J. Algebraic Combin., 61(4):Paper No. 46, 35, 2025. 26

2025

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Non-uniform mixing of quantum walks on the symmetric group.Linear Algebra Appl., 742:37–65, 2026

Avah Banerjee. Non-uniform mixing of quantum walks on the symmetric group.Linear Algebra Appl., 742:37–65, 2026

2026

[2] [2]

Periodicity and perfect state transfer in quantum walks on variants of cycles.Quantum Inf

Katharine Barr, Timothy Proctor, Daniel Allen, and Viv Kendon. Periodicity and perfect state transfer in quantum walks on variants of cycles.Quantum Inf. Comput., 14:417–438, 2012

2012

[3] [3]

Perfect state transfer in Grover walks on association schemes and distance-regular graphs.arXiv.2506.07439, 2025

Koushik Bhakta and Bikash Bhattacharjya. Perfect state transfer in Grover walks on association schemes and distance-regular graphs.arXiv.2506.07439, 2025

arXiv 2025

[4] [4]

Periodicity and perfect state transfer of Grover walks on quadratic unitary Cayley graphs.Quantum Inf

Koushik Bhakta and Bikash Bhattacharjya. Periodicity and perfect state transfer of Grover walks on quadratic unitary Cayley graphs.Quantum Inf. Process., 24(8):Paper No. 260, 21, 2025

2025

[5] [5]

Perfect state transfer in Grover walks on dihedral Cayley graphs.arXiv:2605.02254, 2026

Koushik Bhakta, Bikash Bhattacharjya, and Xiwang Cao. Perfect state transfer in Grover walks on dihedral Cayley graphs.arXiv:2605.02254, 2026

arXiv 2026

[6] [6]

Quantum communication through an unmodulated spin chain.Phys

Sougato Bose. Quantum communication through an unmodulated spin chain.Phys. Rev. Lett., 91(20):Paper No. 207901, 4, 2003

2003

[7] [7]

Pretty good state transfer in discrete-time quantum walks.J

Ada Chan and Hanmeng Zhan. Pretty good state transfer in discrete-time quantum walks.J. Phys. A, 56(16):Paper No. 165305, 25, 2023

2023

[8] [8]

Hamiltonians of bipartite walks

Qiuting Chen, Chris Godsil, Mariia Sobchuk, and Hanmeng Zhan. Hamiltonians of bipartite walks. Electron. J. Combin., 31(4):Paper No. 4.10, 24, 2024

2024