"I think I can safely say that nobody understands quantum mechanics."
--- Richard Feynman
Given a graph G=(V,E) with adjacency matrix A, a quantum walk on G is defined by the unitary matrix U(t) = exp(-itA). The probability of finding the quantum particle on vertex b at time t when the walk started at vertex a is given by the squared absolute value of the (a,b) entry of U(t). Quantum walk on graphs has proved useful for designing quantum algorithms and for implementing information transfer in quantum networks. We study quantum walks mathematically using techniques from algebraic graph theory.
A graph G=(V,E) has perfect state transfer (PST) between vertices a and b at time t if the absolute value of the (a,b) entry of U(t) is one. The graph is periodic at time t if it has perfect state transfer between a and itself. The goal is to understand spectral and combinatorial conditions which allow graphs to have perfect state transfer.
Potsdam-Clarkson Algebraic Graph Theory REU group (time reversed: present - 2000).
Relevant books: