Grover search with lackadaisical quantum walks

Abstract

Abstract. 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 Shenvi, Kempe, and Whaley (2003) 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.

Search by lackadaisical quantum walks with nonhomogeneous weights

“…Such quantum walks have been used to design a variety of quantum algorithms, such as for searching [4], solving element distinctness [5], and evaluating boolean formulas [6]. Since then, generalizations have been introduced that allow a randomly walking quantum particle to stay put, including lazy quantum walks [7,8] and lackadaisical quantum walks [9,10]. Here, we focus on the lackadaisical quantum walk, where a single, weighted self-loop is added to each vertex [11].…”

“…Here, we focus on the lackadaisical quantum walk, where a single, weighted self-loop is added to each vertex [11]. This can cause faster dispersion on the line [11][12][13], and it also speeds up quantum search on the complete graph [10,11] and two-dimensional periodic square lattice (or discrete torus) [14,15].…”

Isolated vertices in continuous-time quantum walks on dynamic graphs

“…3, and CNOT gates. Another approach is using lackadaisical quantum walks [15], where vertices have self-loops of various weights to indicate how "lazy" the walk is at each vertex [16].…”

Perfect state transfer by means of discrete-time quantum walk search algorithms on highly symmetric graphs

“…The reason we consider the additional self-loops is that the discrete-time quantum walk search algorithm on the complete graph does not find the marked vertex with unit probability. Nevertheless, it was shown [8,25] that adding self-loops makes two steps of the discretetime quantum walk equivalent to the Grover search algorithm and increases probability of finding the marked vertex to one. In the following we show explicitly that the algorithm achieves state transfer with unit fidelity independent of the size of the graph.…”