Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

PERFECT STATE TRANSFER IN UNITARY CAYLEY GRAPHS OVER LOCAL RINGS

Yotsanan Meemark, Songpon Sriwongsa, Communicated Dariush Kiani

Published Jan 1, 2014• 3 citations
OpenAlex

Abstract

Abstract. In this work, using eigenvalues and eigenvectors of unitary Cayley graphs over finite local rings and elementary linear algebra, we characterize which local rings allow a PST occurring in its unitary Cayley graph. Moreover, we have some developments when R is a product of local rings. 1. Perfect State Transfer and Unitary Cayley Graphs Let G be an undirected graph whose vertex set V (G) = {v1,..., vn}. The adjacency matrix of G, written AG, is the n×n matrix in which entry ajk is the number of edges in G with endpoint {vj, vk}. Define the matrix-valued function H(t) = exp(itAG) for all t ≥ 0. We say there is a perfect state transfer (PST) from vertex vj to vertex vk if there is a time t such that |H(t)jk | = 1. We note that our matrix H(t) determines what is known in graph theory as a continuous quantum walk. For background on quantum walks, we refer the reader to [9] and [10]. A perfect state transfer in continuous-time quantum walk on graphs has received considerable attention in quantum information and computations in Physics (e.g., [2, 4]). An excellent survey of perfect state transfer graphs and related questions are given by Godsil [8]. Observe that H(t) has the following properties: (i) H(t) is symmetric, (ii) H(t) = H(t)−1, where ¯ is the complex conjugate, (iii) H(t) is unitary, i.e., (H(t))T = H(t)−1. Thus, we have the next proposition.

Related papers

State Transfer in Complex Quantum Walks

Antonio Acuaviva, Ada Chan, Eldridge, Summer +5 more

Given a graph with Hermitian adjacency matrix $H$, perfect state transfer occurs from vertex $a$ to vertex $b$ if the $(b,a)$-entry of the unitary matrix $\exp(-iHt)$ has unit magnitude for some time $t$. This phenomenon is relevant for information transmission in quantum spin networks and is known to be monogamous under real symmetric matrices. We prove the following results: 1. For oriented graphs (whose nonzero weights are $\pm i$), the oriented $3$-cycle and the oriented edge are the only graphs where perfect state transfer occurs between every pair of vertices. This settles a conjecture of Cameron et al. On the other hand, we construct an infinite family of oriented graphs with perfect state transfer between any pair of vertices on a subset of size four. 2. There are infinite families of Hermitian graphs with one-way perfect state transfer, where perfect state transfer occurs without periodicity. In contrast, perfect state transfer implies periodicity whenever the adjacency matrix has algebraic entries (as shown by Godsil). 3. There are infinite families with non-monogamous pretty good state transfer in rooted graph products. In particular, we generalize known results on double stars (due to Fan and Godsil) and on paths with loops (due to Kempton, Lippner and Yau). The latter extends the experimental observation of quantum transport (made by Zimborás et al.) and shows non-monogamous pretty good state transfer can occur amongst distant vertices.

arXiv:2301.01473•Jan 4, 2023•0 citations

Quantum State Transfer in Graphs

Gabriel Coutinho

Let X be a graph, A its adjacency matrix, and t a non-negative real number. The matrix exp(i t A) determines the evolution in time of a certain quantum system defined on the graph. It represents a continuous-time quantum walk in X. We say that X admits perfect state transfer from a vertex u to a vertex v if there is a time t such that \n| exp(i t A)_(u,v) | = 1. \n \nThe main problem we study in this thesis is that of determining which simple graphs admit perfect state transfer. For some classes of graphs the problem is solved. For example, a path on n vertices admits perfect state transfer if and only if n=2 or n=3. However, the general problem of determining all graphs that admit perfect state transfer is substantially hard. \n \nIn this thesis, we focus on some special cases. We provide necessary and sufficient conditions for a distance-regular graph to admit perfect state transfer. In particular, we provide a detailed account of which distance-regular graphs of diameter three do so. \n \nA graph is said to be spectrally extremal if the number of distinct eigenvalues is equal to the diameter plus one. Distance-regular graphs are examples of such graphs. We study perfect state transfer in spectrally extremal graphs and explore rich connections to the topic of orthogonal polynomials. We characterize perfect state transfer in such graphs. \n \nWe also provide a general framework in which perfect state transfer in graph products can be studied. We use this to determine when direct products and double covers of graphs admit perfect state transfer. As a consequence, we provide many new examples of simple graphs admitting perfect state transfer. We also provide some advances in the understanding of perfect state transfer in Cayley graphs for the groups (Z_2)^d and Z_n. \n \nFinally, we consider the problem of determining which trees admit perfect state transfer. We show more generally that, except for the path on two vertices, if a connected bipartite graph contains a unique perfect matching, then it cannot admit perfect state transfer. We also consider this problem in the context of another model of quantum walks determined by the matrix exp(i t L), where L is the Laplacian matrix of the graph. In particular, we show that no tree on an odd number of vertices admits perfect state transfer according to this model.

•Aug 13, 2014•34 citations

Perfect state transfer in Grover walks on normal Cayley graphs

Koushik Bhakta, Bikash Bhattacharjya

A Cayley graph $\operatorname{Cay}(Γ,S)$ over a finite group $Γ$ is said to be normal if its connection set $S$ is a union of some conjugacy classes of $Γ$. 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.

quant-pharXiv:2607.19309•Jul 22, 2026•0 citations

Grover walks on unitary Cayley graphs and integral regular graphs

Koushik Bhakta, Bikash Bhattacharjya

The unitary Cayley graph has vertex set $\{0,1, \hdots ,n-1\}$, where two vertices $u$ and $v$ are adjacent if $\gcd(u - v, n) = 1$. In this paper, we study periodicity and perfect state transfer of Grover walks on the unitary Cayley graphs. We characterize all periodic unitary Cayley graphs. We prove that periodicity is a necessary condition for occurrence of perfect state transfer on a vertex-transitive graph. Also, we provide a necessary and sufficient condition for the occurrence of perfect state transfer on circulant graphs. Using these, we prove that only four graphs in the class of unitary Cayley graphs exhibit perfect state transfer. Also, we provide a spectral characterization of the periodicity of Grover walks on integral regular graphs.

arXiv:2405.01020•May 2, 2024•0 citations

Perfect state transfer on Cayley graphs over dihedral groups: A complete and practical characterization

Shixin Wang

Perfect state transfer on graphs has attracted extensive attention due to its application in quantum information and quantum computation. Explicit characterizations of connection sets admitting perfect state transfer in Cayley graphs are rare and, so far, are known only for a few abelian Cayley graphs. In this paper, we characterize the conjugation-closed connection sets of connected Cayley graphs over dihedral groups that admit perfect state transfer. By applying Ramanujan sums, Möbius inversion, and arguments based on the $p$-adic exponential valuation of rational numbers, we convert the eigenvalue constraints imposed by perfect state transfer into explicit structural conditions on the connection set. This yields a complete and practical characterization, which gives an effective criterion for recognizing and constructing such Cayley graphs and also determines the exact minimum perfect state transfer time.

arXiv:2608.30347•Aug 31, 2026•0 citations

Perfect state transfer and Cayley presentations

Arnbjörg Soffía Árnadóttir, Krystal Guo

We study perfect state transfer on Cayley graphs from the point of view that state transfer is a property of a graph and not of a group. This paper is a bridge between the classical question about isomorphic Cayley graphs of non-isomorphic groups and quantum walks on graphs. We show that a Cayley graph of a group with an abelian subgroup of index two is a Cayley graph of an abelian group under any one of three hypotheses, two drawn from the theory of isomorphic Cayley graphs. A statement of the same kind holds for extraspecial groups: every Cayley graph of an extraspecial $p$-group of order $p^{2n+1}$ with a conjugacy-closed connection set is a Cayley graph of $Z_p^{2n+1}$. From these results we deduce that every explicit construction of perfect state transfer in the six papers we survey, on dihedral, dicyclic, generalized dihedral, $V_{8n}$ and extraspecial $2$-groups, is a non-abelian presentation of an abelian Cayley graph. Moreover, we show that a non-abelian group with an abelian subgroup of index two admits a connected Cayley graph with perfect state transfer if and only if its order is divisible by four. Genuinely non-abelian examples do exist. We prove that, for every odd prime power $q\ge 5$, the $SL(2,q)$ graph of Pantangi and Sin, which they showed to admit perfect state transfer, is a Cayley graph of no abelian group; to our knowledge, this is the first infinite family of Cayley graphs with perfect state transfer provably admitting no abelian Cayley presentation. We also construct an infinite family of Cayley graphs with peak state transfer and determine all regular subgroups of the automorphism group of every member. An appendix records a census of the connected vertex-transitive graphs with perfect state transfer on at most $30$ vertices.

math.COCross-listedarXiv:2608.20476•Aug 24, 2026•0 citations
Abstract. In this work, using eigenvalues and eigenvectors of unitary Cayley graphs over finite local rings and elementary linear algebra, we characterize which local rings allow a PST occurring in its unitary Cayley graph. Moreover, we have some developments when R is a product of local rings. 1. Perfect State Transfer and Unitary Cayley Graphs Let G be an undirected graph whose vertex set V (G) = {v1,..., vn}. The adjacency matrix of G, written AG, is the n×n matrix in which entry ajk is the number of edges in G with endpoint {vj, vk}. Define the matrix-valued function H(t) = exp(itAG) for all t ≥ 0. We say there is a perfect state transfer (PST) from vertex vj to vertex vk if there is a time t such that |H(t)jk | = 1. We note that our matrix H(t) determines what is known in graph theory as a continuous quantum walk. For background on quantum walks, we refer the reader to [9] and [10]. A perfect state transfer in continuous-time quantum walk on graphs has received considerable attention in quantum information and computations in Physics (e.g., [2, 4]). An excellent survey of perfect state transfer graphs and related questions are given by Godsil [8]. Observe that H(t) has the following properties: (i) H(t) is symmetric, (ii) H(t) = H(t)−1, where ¯ is the complex conjugate, (iii) H(t) is unitary, i.e., (H(t))T = H(t)−1. Thus, we have the next proposition.
Given a graph with Hermitian adjacency matrix HHH, perfect state transfer occurs from vertex aaa to vertex bbb if the (b,a)(b,a)(b,a)-entry of the unitary matrix exp⁡(−iHt)\exp(-iHt)exp(−iHt) has unit magnitude for some time ttt. This phenomenon is relevant for information transmission in quantum spin networks and is known to be monogamous under real symmetric matrices. We prove the following results: 1. For oriented graphs (whose nonzero weights are ±i\pm i±i), the oriented 333-cycle and the oriented edge are the only graphs where perfect state transfer occurs between every pair of vertices. This settles a conjecture of Cameron et al. On the other hand, we construct an infinite family of oriented graphs with perfect state transfer between any pair of vertices on a subset of size four. 2. There are infinite families of Hermitian graphs with one-way perfect state transfer, where perfect state transfer occurs without periodicity. In contrast, perfect state transfer implies periodicity whenever the adjacency matrix has algebraic entries (as shown by Godsil). 3. There are infinite families with non-monogamous pretty good state transfer in rooted graph products. In particular, we generalize known results on double stars (due to Fan and Godsil) and on paths with loops (due to Kempton, Lippner and Yau). The latter extends the experimental observation of quantum transport (made by Zimborás et al.) and shows non-monogamous pretty good state transfer can occur amongst distant vertices.
Let X be a graph, A its adjacency matrix, and t a non-negative real number. The matrix exp(i t A) determines the evolution in time of a certain quantum system defined on the graph. It represents a continuous-time quantum walk in X. We say that X admits perfect state transfer from a vertex u to a vertex v if there is a time t such that \n| exp(i t A)_(u,v) | = 1. \n \nThe main problem we study in this thesis is that of determining which simple graphs admit perfect state transfer. For some classes of graphs the problem is solved. For example, a path on n vertices admits perfect state transfer if and only if n=2 or n=3. However, the general problem of determining all graphs that admit perfect state transfer is substantially hard. \n \nIn this thesis, we focus on some special cases. We provide necessary and sufficient conditions for a distance-regular graph to admit perfect state transfer. In particular, we provide a detailed account of which distance-regular graphs of diameter three do so. \n \nA graph is said to be spectrally extremal if the number of distinct eigenvalues is equal to the diameter plus one. Distance-regular graphs are examples of such graphs. We study perfect state transfer in spectrally extremal graphs and explore rich connections to the topic of orthogonal polynomials. We characterize perfect state transfer in such graphs. \n \nWe also provide a general framework in which perfect state transfer in graph products can be studied. We use this to determine when direct products and double covers of graphs admit perfect state transfer. As a consequence, we provide many new examples of simple graphs admitting perfect state transfer. We also provide some advances in the understanding of perfect state transfer in Cayley graphs for the groups (Z_2)^d and Z_n. \n \nFinally, we consider the problem of determining which trees admit perfect state transfer. We show more generally that, except for the path on two vertices, if a connected bipartite graph contains a unique perfect matching, then it cannot admit perfect state transfer. We also consider this problem in the context of another model of quantum walks determined by the matrix exp(i t L), where L is the Laplacian matrix of the graph. In particular, we show that no tree on an odd number of vertices admits perfect state transfer according to this model.
A Cayley graph Cay⁡(Γ,S)\operatorname{Cay}(Γ,S)Cay(Γ,S) over a finite group ΓΓΓ is said to be normal if its connection set SSS is a union of some conjugacy classes of ΓΓΓ. 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.
The unitary Cayley graph has vertex set $\{0,1, \hdots ,n-1\}$, where two vertices uuu and vvv are adjacent if gcd⁡(u−v,n)=1\gcd(u - v, n) = 1gcd(u−v,n)=1. In this paper, we study periodicity and perfect state transfer of Grover walks on the unitary Cayley graphs. We characterize all periodic unitary Cayley graphs. We prove that periodicity is a necessary condition for occurrence of perfect state transfer on a vertex-transitive graph. Also, we provide a necessary and sufficient condition for the occurrence of perfect state transfer on circulant graphs. Using these, we prove that only four graphs in the class of unitary Cayley graphs exhibit perfect state transfer. Also, we provide a spectral characterization of the periodicity of Grover walks on integral regular graphs.
Perfect state transfer on graphs has attracted extensive attention due to its application in quantum information and quantum computation. Explicit characterizations of connection sets admitting perfect state transfer in Cayley graphs are rare and, so far, are known only for a few abelian Cayley graphs. In this paper, we characterize the conjugation-closed connection sets of connected Cayley graphs over dihedral groups that admit perfect state transfer. By applying Ramanujan sums, Möbius inversion, and arguments based on the ppp-adic exponential valuation of rational numbers, we convert the eigenvalue constraints imposed by perfect state transfer into explicit structural conditions on the connection set. This yields a complete and practical characterization, which gives an effective criterion for recognizing and constructing such Cayley graphs and also determines the exact minimum perfect state transfer time.
We study perfect state transfer on Cayley graphs from the point of view that state transfer is a property of a graph and not of a group. This paper is a bridge between the classical question about isomorphic Cayley graphs of non-isomorphic groups and quantum walks on graphs. We show that a Cayley graph of a group with an abelian subgroup of index two is a Cayley graph of an abelian group under any one of three hypotheses, two drawn from the theory of isomorphic Cayley graphs. A statement of the same kind holds for extraspecial groups: every Cayley graph of an extraspecial ppp-group of order p2n+1p^{2n+1}p2n+1 with a conjugacy-closed connection set is a Cayley graph of Zp2n+1Z_p^{2n+1}Zp2n+1​. From these results we deduce that every explicit construction of perfect state transfer in the six papers we survey, on dihedral, dicyclic, generalized dihedral, V8nV_{8n}V8n​ and extraspecial 222-groups, is a non-abelian presentation of an abelian Cayley graph. Moreover, we show that a non-abelian group with an abelian subgroup of index two admits a connected Cayley graph with perfect state transfer if and only if its order is divisible by four. Genuinely non-abelian examples do exist. We prove that, for every odd prime power q≥5q\ge 5q≥5, the SL(2,q)SL(2,q)SL(2,q) graph of Pantangi and Sin, which they showed to admit perfect state transfer, is a Cayley graph of no abelian group; to our knowledge, this is the first infinite family of Cayley graphs with perfect state transfer provably admitting no abelian Cayley presentation. We also construct an infinite family of Cayley graphs with peak state transfer and determine all regular subgroups of the automorphism group of every member. An appendix records a census of the connected vertex-transitive graphs with perfect state transfer on at most 303030 vertices.