  The lazy random walk, where the walker has some probability of staying put,
is a useful tool in classical algorithms. We propose a quantum analogue, the
lackadaisical quantum walk, where each vertex is given $l$ self-loops, and we
investigate its effects on Grover's algorithm when formulated as search for a
marked vertex on the complete graph of $N$ vertices. For the discrete-time
quantum walk using the phase flip coin, adding a self-loop to each vertex
boosts the success probability from 1/2 to 1. Additional self-loops, however,
decrease the success probability. Using instead the Ambainis, Kempe, and Rivosh
(2005) coin, adding self-loops simply slows down the search. These coins also
differ in that the first is faster than classical when $l$ scales less than
$N$, while the second requires that $l$ scale less than $N^2$. Finally,
continuous-time quantum walks differ from both of these discrete-time
examples---the self-loops make no difference at all. These behaviors generalize
to multiple marked vertices.
