  We consider to what extent quantum walks can constitute models of
thermalization, analogously to how classical random walks can be models for
classical thermalization. In a quantum walk over a graph, a walker moves in a
superposition of node positions via a unitary time evolution. We show a quantum
walk can be interpreted as an equilibration of a kind investigated in the
literature on thermalization in unitarily evolving quantum systems. This
connection implies that recent results concerning the equilibration of
observables can be applied to analyse the node position statistics of quantum
walks. We illustrate this in the case of a family of graphs known as
fullerenes. We find that a bound from Short et al., implying that certain
expectation values will at most times be close to their time-averaged value,
applies tightly to the node position probabilities. Nevertheless, the node
position statistics do not thermalize in the standard sense. In particular,
quantum walks over fullerene graphs constitute a counter-example to the
hypothesis that subsystems equilibrate to the Gibbs state. We also exploit the
bridge created to show how quantum walks can be used to probe the universality
of the eigenstate thermalisation hypothesis (ETH) relation. We find that whilst
in C60 with a single walker, the ETH relation does not hold for node position
projectors, it does hold for the average position, enforced by a symmetry of
the Hamiltonian. The findings suggest a unified study of quantum walks and
quantum self-thermalizations is natural and feasible.
