Modelling the quantum walk search problem

Classically the first time a particle reaches a target, either via a diffusive mechanism or deterministically, controls many processes in science. In the absence of a well defined path the first arrival time to a target state of a quantum particle can be treated in several ways. Such problems arise in the excitation transfer to a reaction centre in light harvesting systems, and more recently in the context of quantum search algorithms. We will review the challenges of search, starting with the quantum renewal equation, dark states and their relation to symmetry, and topological aspects of the first return problem. This is done for a protocol with unitary dynamics pierces by repeated strong measurements aimed to detect a quantum walker on a node of a graph. We will then show how to construct tight binding Hamiltonians that speed up state-transfer both in the presence and the absence of repeated measurements. These are related to a mass-less Dirac quasi particle and a large degenracy of the eigenvalues of the underlying non-Hermitian survival operator.