Pair state transfer
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Pair state transfer Qiuting Chen1
· Chris Godsil1
Received: 11 February 2020 / Accepted: 18 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Let L denote the Laplacian matrix of a graph G. We study continuous quantum walks on G defined by the transition matrix U (t) = exp (it L). The initial state is of the pair state form, ea − eb with a, b being any two vertices of G. We provide two ways to construct infinite families of graphs that have perfect pair state transfer. We study a “transitivity” phenomenon which cannot occur in vertex state transfer. We characterize perfect pair state transfer on paths and cycles. We also study the case when quantum walks are generated by the unsigned Laplacians of underlying graphs and the initial states are of the plus state form, ea + eb . When the underlying graphs are bipartite, plus state transfer is equivalent to pair state transfer. Keywords Quantum state transfer · Continuous quantum walk · Graph spectra · Perfect state transfer
1 Introduction The concept of a quantum walk was introduced by Farhi and Gutmann [10] as a quantum mechanical analogue of a classical random walk on decision trees. Exploiting the interference effects of quantum mechanics and quantum walks outperform classical random walks for some computational tasks [4]. In the field of quantum information processing, Christandl et al. [6] brought our attention to the topic of perfect state transfer. Using the tool of quantum scattering theory, Childs [3] proved that continuous time quantum walks can be regarded as a universal computational primitive and any desired quantum computation can be encoded in some underlying graph of the quantum walk. Quantum walks have become powerful tools to improve existing quantum algorithms and develop new quantum
Research supported by Natural Sciences and Engineering Council of Canada, Grant No. RGPIN-9439.
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Qiuting Chen [email protected] Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON, Canada 0123456789().: V,-vol
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algorithms. In this paper, we use graphs to represent networks of interacting qubits and study quantum state transfer during quantum communication over the network. Let G be a graph. The evolution of a continuous quantum walk on G is given by the matrices U (t) = exp(it H ),
(t ∈ R).
Here, H is a matrix, called the Hamiltonian of the walk, and is usually either the adjacency matrix, the Laplacian or the signless Laplacian of G. In any case, H is Hermitian and its rows and columns are indexed by the vertices of G. If n = |V (G)|, then the walk represents a quantum system with an n-dimensional state space. We identify the states of the system by density matrices, i.e. positive semidefinite matrices with trace 1. The physically meaningful questions are the form: “Given that the quantum system is initially in state represented by a density matrix D0 , what is the probability that, at time t, its state is D1 ?”. Let e1 , e2 , . . . , en de
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