Perfect state transfer and Cayley presentations

• URL: http://arxiv.org/abs/2608.20476v1
• Date: Thu, 20 Aug 2026 18:08:45 GMT
• Title: Perfect state transfer and Cayley presentations
• Abstract summary: 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.<n>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.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

Related papers

• Neurosymbolic Discovery of Algebraic Graph Constructions [22.962110170304594]
  We ask whether one can automatically discover a short algebraic description if only this raw data is provided.<n>We propose an agent that runs on a general-purpose large language model with no fine-tuning or per-target training.<n>We test the approach on a benchmark of 100 highly symmetric graphs.
  arXiv  Detail & Related papers  (2026-08-08T13:07:51Z)
• Perfect state transfer in Grover walks on normal Cayley graphs [0.0]
  We investigate perfect state transfer in Grover walks on normal Cayley graphs.<n>We prove that exactly four graphs in the class of unitary Cayley graphs exhibit perfect state transfer.
  arXiv  Detail & Related papers  (2026-07-21T17:23:56Z)
• Quantum state isomorphism problems for groups [41.49556721904551]
  We study the computational complexity of quantum state isomorphism problems under group actions.<n>For pure-state version, we show that the problem is BQP-hard for all nontrivial groups.<n>For the mixed-state version, for nontrivial, finite and efficiently representable groups, the problem is QSZK-complete.
  arXiv  Detail & Related papers  (2026-05-12T18:05:27Z)
• Perfect state transfer in Grover walks on dihedral Cayley graphs [5.8879697428274875]
  The paper investigates perfect state transfer in Grover walks on Cayley graphs over the dihedral group $D_n$.<n>A Cayley graph $operatornameCay(,S)$ is called normal if $S$ is the union of some conjugacy classes of the group $$; otherwise, it is called non-normal.
  arXiv  Detail & Related papers  (2026-05-04T05:58:30Z)
• Bridging conformal field theory and parton approaches to SU(n)_k chiral spin liquids [21.876059213677966]
  We employ the $mathrmSU(n)_k$ Wess-Zumino-Witten (WZW) model in conformal field theory to construct lattice wave functions in both one and two dimensions.<n>The spins on all lattice sites are chosen to transform under the $mathrmSU(n)$ irreducible representation with a single row and $k$ boxes in the Young tableau.
  arXiv  Detail & Related papers  (2025-01-16T14:42:00Z)
• Quantum fractional revival governed by adjacency matrix Hamiltonian in unitary Cayley graphs [0.0]
  We prove existence of quantum fractional revival in unitary Cayley graph utilizing adjacency matrix Hamiltonian. Quantum fractional revival is analogous to quantum entanglement.
  arXiv  Detail & Related papers  (2024-10-04T10:47:43Z)
• CKGConv: General Graph Convolution with Continuous Kernels [24.58050212186722]
  We propose a novel and general graph convolution framework by parameterizing the kernels as continuous functions of pseudo-coordinates derived via graph positional encoding. We name this Continuous Kernel Graph Convolution (CKGConv) We show that CKGConv-based Networks outperform existing graph convolutional networks and perform comparably to the best graph transformers across a variety of graph datasets.
  arXiv  Detail & Related papers  (2024-04-21T10:26:13Z)
• Fractional revival on abelian Cayley graphs [23.909933791900322]
  Fractional revival is essential for entanglement generation in quantum spin networks. Two general constructions of abelian Cayley graphs having fractional revival are presented. We establish several new families of abelian Cayley graphs admitting fractional revival.
  arXiv  Detail & Related papers  (2022-08-10T02:01:44Z)
• On state transfer in Cayley graphs for abelian groups [0.0]
  We characterize perfect state transfer in Cayley graphs for abelian groups that have a cyclic Sylow-2-subgroup. This generalizes a result of Bavsi'c from 2013 where he provides a similar characterization for Cayley graphs of cyclic groups.
  arXiv  Detail & Related papers  (2022-04-20T22:14:09Z)
• Learning on heterogeneous graphs using high-order relations [37.64632406923687]
  We propose an approach for learning on heterogeneous graphs without using meta-paths. We decompose a heterogeneous graph into different homogeneous relation-type graphs, which are then combined to create higher-order relation-type representations.
  arXiv  Detail & Related papers  (2021-03-29T12:02:47Z)
• Hamiltonian systems, Toda lattices, Solitons, Lax Pairs on weighted Z-graded graphs [62.997667081978825]
  We identify conditions which allow one to lift one dimensional solutions to solutions on graphs. We show that even for a simple example of a topologically interesting graph the corresponding non-trivial Lax pairs and associated unitary transformations do not lift to a Lax pair on the Z-graded graph.
  arXiv  Detail & Related papers  (2020-08-11T17:58:13Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.