  Lazy quantum walks were presented by Andrew M. Childs to prove that the
continuous-time quantum walk is a limit of the discrete-time quantum walk
[Commun.Math.Phys.294,581-603(2010)]. In this paper, we discuss properties of
lazy quantum walks. Our analysis shows that lazy quantum walks have $O(t^n)$
order of the n-th moment of the corresponding probability distribution, which
is the same as that for normal quantum walks. Also, the lazy quantum walk with
DFT (Discrete Fourier Transform) coin operator has a similar probability
distribution concentrated interval to that of the normal Hadamard quantum walk.
Most importantly, we introduce the concepts of occupancy number and occupancy
rate to measure the extent to which the walk has a (relatively) high
probability at every position in its range. We conclude that lazy quantum walks
have a higher occupancy rate than other walks such as normal quantum walks,
classical walks and lazy classical walks.
