  We address the scattering of a quantum particle by a one-dimensional barrier
potential over a set of discrete positions. We formalize the problem as a
continuous-time quantum walk on a lattice with an impurity, and use the quantum
Fisher information as a mean to quantify the maximal possible accuracy in the
estimation of the height of the barrier. We introduce suitable initial states
of the walker and derive the reflection and transmission probabilities of the
scattered state. We show that while the quantum Fisher information is affected
by the width and central momentum of the initial wave packet, this dependency
is weaker for the quantum signal-to-noise ratio. We also show that a dichotomic
position measurement provides a nearly optimal detection scheme.
