  Augmenting the unitary transformation which generates a quantum walk by a
generalized phase gate G is a symmetry for both noisy and noiseless quantum
walk on a line, in the sense that it leaves the position probability
distribution invariant. However, this symmetry breaks down in the case of a
quantum walk on an n-cycle, and hence can be regarded as a probe of the walk
topology. Noise, modelled here as phase flip and generalized amplitude damping
channels, tends to restore the symmetry because it classicalizes the walk.
However, symmetry restoration happens even in the regime where the walker is
not entirely classical, because noise also has the effect of desensitizing the
operation G to the walk topology. We discuss methods for physical
implementation, and talk about the wider implications to condensed matter
systems.
