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.

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... 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.