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

• URL: http://arxiv.org/abs/2402.17341v2
• Date: Thu, 07 Nov 2024 12:44:49 GMT
• Title: Circulant graphs with valency up to 4 that admit perfect state transfer in Grover walks
• Authors: Sho Kubota, Kiyoto Yoshino,
• Abstract summary: We characterize circulant graphs with valency up to $4$ that admit perfect state transfer. On the other hand, circulant graphs with valency $4$ admit perfect state transfer only in two infinite families: one discovered by Zhan and another new family, while no others do.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We completely characterize circulant graphs with valency up to $4$ that admit perfect state transfer. Those of valency $3$ do not admit it. On the other hand, circulant graphs with valency $4$ admit perfect state transfer only in two infinite families: one discovered by Zhan and another new family, while no others do. The main tools for deriving these results are symmetry of graphs and eigenvalues. We describe necessary conditions for perfect state transfer to occur based on symmetry of graphs, which mathematically refers to automorphisms of graphs. As for eigenvalues, if perfect state transfer occurs, then certain eigenvalues of the corresponding isotropic random walks must be the halves of algebraic integers. Taking this into account, we utilize known results on the rings of integers of cyclotomic fields.

Related papers

• Strongly regular and strongly walk-regular graphs that admit perfect state transfer [0.0]
  We study perfect state transfer in Grover walks on two important classes of graphs: strongly regular graphs and strongly walk-regular graphs.<n>We first give a complete classification of strongly regular graphs that admit perfect state transfer. The only such graphs are the complete bipartite graph $K_2,2 and the complete graph $K_2,2,2.
  arXiv  Detail & Related papers  (2025-06-03T07:10:06Z)
• Real state transfer on edge perturbed graphs with generalized clusters [0.0]
  We study the existence of real state transfer in edge-perturbed graphs containing generalized clusters.<n>A central observation is that the evolution of certain quantum states depends solely on the local structure of the underlying graph.
  arXiv  Detail & Related papers  (2025-05-12T18:26:59Z)
• Perfect state transfer between real pure states [0.0]
  Pure states correspond to one-dimensional subspaces of $mathbbCn$ represented by unit vectors. We develop the theory of perfect state transfer (PST) between real pure states with emphasis on the adjacency and Laplacian matrices as Hamiltonians of a graph representing a quantum spin network.
  arXiv  Detail & Related papers  (2025-02-12T04:02:15Z)
• A generalization of quantum pair state transfer [0.0]
  An $s$-pair state in a graph is a quantum state of the form $mathbfe_u+smathbfe_v$. We develop the theory of perfect $s$-pair state transfer in continuous quantum walks.
  arXiv  Detail & Related papers  (2024-04-25T14:45:49Z)
• 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)
• The quantum commuting model (Ia): The CHSH game and other examples: Uniqueness of optimal states [91.3755431537592]
  We use the universal description of quantum commuting correlations as state space on the universal algebra for two player games. We find that the CHSH game leaves a single optimal state on this common algebra.
  arXiv  Detail & Related papers  (2022-10-07T17:38:31Z)
• Exact solution of a family of staggered Heisenberg chains with conclusive pretty good quantum state transfer [68.8204255655161]
  We work out the exact solutions in the one-excitation subspace. We present numerical evidence that pretty good transmission is achieved by chains whose length is not a power of two.
  arXiv  Detail & Related papers  (2022-06-28T18:31:09Z)
• Quantum state transfer between twins in weighted graphs [0.0]
  We explore the role of twin vertices in quantum state transfer. We provide characterizations of periodicity, perfect state transfer, and pretty good state transfer. As an application, we provide characterizations of all simple unweighted double cones on regular graphs that exhibit periodicity, perfect state transfer, and pretty good state transfer.
  arXiv  Detail & Related papers  (2022-01-08T01:15:24Z)
• Perfect State Transfer in Weighted Cubelike Graphs [0.0]
  A continuous-time quantum random walk describes the motion of a quantum mechanical particle on a graph. We generalize the PST or periodicity of cubelike graphs to that of weighted cubelike graphs.
  arXiv  Detail & Related papers  (2021-09-26T13:44:44Z)
• Perfect state transfer in Grover walks between states associated to vertices of a graph [0.0]
  We study perfect state transfer in Grover walks, which are typical discrete-time quantum walk models. We call such states type states. We derive a necessary condition on eigenvalues of a graph for perfect state transfer between type states to occur.
  arXiv  Detail & Related papers  (2021-09-14T03:59:47Z)
• A Thorough View of Exact Inference in Graphs from the Degree-4 Sum-of-Squares Hierarchy [37.34153902687548]
  We tackle the problem of exactly recovering an unknown ground-truth binary labeling of the nodes from a single corrupted observation of each edge. We apply a hierarchy of relaxations known as the sum-of-squares hierarchy, to the problem. We show that the solution of the dual of the relaxed problem is related to finding edge weights of the Johnson and Kneser graphs.
  arXiv  Detail & Related papers  (2021-02-16T08:36:19Z)
• 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)
• Continuous-time quantum walks in the presence of a quadratic perturbation [55.41644538483948]
  We address the properties of continuous-time quantum walks with Hamiltonians of the form $mathcalH= L + lambda L2$. We consider cycle, complete, and star graphs because paradigmatic models with low/high connectivity and/or symmetry.
  arXiv  Detail & Related papers  (2020-05-13T14:53:36Z)
• Perfect State Transfer on Oriented Graphs [0.0]
  We study the phenomena, unique to oriented graphs, of multiple state transfer. We give a characterization of multiple state transfer, and a new example of a graph where it occurs.
  arXiv  Detail & Related papers  (2020-02-11T20:34:54Z)

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.