Quantum walk on graphs and perfect state transfer

Graphs that exhibit perfect state transfer (PST), by which a quantum walk beginning at one vertex is guaranteed to end at another after a nonzero displacement in time, are of great interest to researchers in the field of quantum computing. The conditions imposed by PST are strong, and many of the graphs that are known to exhibit PST are highly symmetric, which provides motivation to relax the restrictions imposed by perfect state transfer. This project introduces the property of group state transfer (GST), by which quantum walks are guaranteed to transfer states between nontrivial vertex sets. We define GST, examine its implications for finite simple graphs, and provide examples of graphs that exhibit GST.Graphs that exhibit perfect state transfer (PST), by which a quantum walk beginning at one vertex is guaranteed to end at another after a nonzero displacement in time, are of great interest to researchers in the field of quantum computing. The conditions imposed by PST are strong, and many of the graphs that are known to exhibit PST are highly symmetric, which provides motivation to relax the restrictions imposed by perfect state transfer. This project introduces the property of group state transfer (GST), by which quantum walks are guaranteed to transfer states between nontrivial vertex sets. We define GST, examine its implications for finite simple graphs, and provide examples of graphs that exhibit GST.

Apr 24, 20190 citations

Perfect state transfer using Markovian quantum walk

Supriyo Dutta

arXiv:2212.11699Annals of PhysicsFeb 20, 20260 citations

Perfect State Transfer in Weighted Cubelike Graphs

Jaideep Mulherkar, Rishikant Rajdeepak, V. Sunitha

A continuous-time quantum random walk describes the motion of a quantum mechanical particle on an underlying graph. The graph itself is associated with a Hilbert space of dimension equal to the number of vertices. The dynamics of the walk is governed by the unitary operator $\mathcal{U}(t) = e^{iAt}$ , where $A$ is the adjacency matrix of the graph. An important notion in the quantum random walk is the transfer of a quantum state from one vertex to another. If the fidelity of the transfer is unity, we call it a perfect state transfer. Many graph families have been shown to admit PST or periodicity, including cubelike graphs. These graphs are unweighted. In this paper, we generalize the PST or periodicity of cubelike graphs to that of weighted cubelike graphs. We characterize the weights for which they admit PST or show periodicity, both at time $t=\fracπ{2}$ .A continuous-time quantum random walk describes the motion of a quantum mechanical particle on an underlying graph. The graph itself is associated with a Hilbert space of dimension equal to the number of vertices. The dynamics of the walk is governed by the unitary operator $\mathcal{U}(t) = e^{iAt}$, where $A$ is the adjacency matrix of the graph. An important notion in the quantum random walk is the transfer of a quantum state from one vertex to another. If the fidelity of the transfer is unity, we call it a perfect state transfer. Many graph families have been shown to admit PST or periodicity, including cubelike graphs. These graphs are unweighted. In this paper, we generalize the PST or periodicity of cubelike graphs to that of weighted cubelike graphs. We characterize the weights for which they admit PST or show periodicity, both at time $t=\fracπ{2}$.

arXiv:2109.12607Sep 26, 20210 citations

PERFECT STATE TRANSFER IN UNITARY CAYLEY GRAPHS OVER LOCAL RINGS

Yotsanan Meemark, Songpon Sriwongsa, Communicated Dariush Kiani

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.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.

Jan 1, 20143 citations

\mathcal{U}(t) = e^{iAt}

, where

A

is the adjacency matrix of the graph. An important notion in the quantum random walk is the transfer of a quantum state from one vertex to another. If the fidelity of the transfer is unity, we call it a perfect state transfer. Many graph families have been shown to admit PST or periodicity, including cubelike graphs. These graphs are unweighted. In this paper, we generalize the PST or periodicity of cubelike graphs to that of weighted cubelike graphs. We characterize the weights for which they admit PST or show periodicity, both at time

t=\fracπ{2}