  Quantum percolation describes the problem of a quantum particle moving
through a disordered system. While certain similarities to classical
percolation exist, the quantum case has additional complexity due to the
possibility of Anderson localisation. Here, we consider a directed
discrete-time quantum walk as a model to study quantum percolation of a
two-state particle on a two-dimensional lattice. Using numerical analysis we
determine the fraction of connected edges required (transition point) in the
lattice for the two-state particle to percolate with finite (non-zero)
probability for three fundamental lattice geometries, finite square lattice,
honeycomb lattice, and nanotube structure and show that it tends towards unity
for increasing lattice sizes. To support the numerical results we also use a
continuum approximation to analytically derive the expression for the
percolation probability for the case of the square lattice and show that it
agrees with the numerically obtained results for the discrete case. Beyond the
fundamental interest to understand the dynamics of a two-state particle on a
lattice (network) with disconnected vertices, our study has the potential to
shed light on the transport dynamics in various quantum condensed matter
systems and the construction of quantum information processing and
communication protocols.
