Combinatorics Seminar – Oct 15, 2021

'Continuous-Time Quantum Walks and Twin Vertices'

15 October 2021, 2:30 pm, Zoom

Speaker: Hermie Monterde (Manitoba, Math)
Title: Continuous-Time Quantum Walks and Twin Vertices

Abstract:Undirected graphs are used to model quantum spin networks, with the vertices and edges representing the qubits and their interactions, respectively. Each of these qubits has an associated quantum state that contains information, and in order to construct an operational quantum computer, two tasks must be performed: the accurate transmission of quantum states from one location in the quantum computer to another, and the generation of entanglements between quantum states.

Let G be an undirected graph, and M be a Hermitian matrix associated with G. By axioms of quantum mechanics, the matrix U(t)=e^{itM} governs the evolution of the quantum system represented by G at any time t and determines a continuous-time quantum walk on X. The entries of U(t) give information about the probability of state transfer between any two vertices of G at time t. In particular, if |U(\tau)_{j,k}|^2=1, then we say that perfect state transfer occurs from vertex j to vertex k at time \tau, while if |U(\tau)_{j,k}|^2 can be made arbitrarily close to 1 by appropriate choices of \tau, then we say that pretty good state transfer occurs from j to k. On the other hand, if entries on the jth and kth columns of U(t) are zero at time \tau except for those indexed by j and k, then we say that fractional revival occurs between vertices j and k at time \tau.

In this talk, we look at the properties perfect state transfer, pretty good state transfer and fractional revival, as well as determine which of these quantum phenomena occur between twin vertices in an undirected graph.

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Oct 14, 2021