The concept of lackadaisical quantum walk -- quantum walk with self loops -- was first introduced for discrete-time quantum walk on one-dimensional line. Later it was successfully applied to improve the running time of the spacial search on two-dimensional grid. In this paper we study search by lackadaisical quantum walk on the two-dimensional grid with multiple marked vertices. First, we show that the lackadaisical quantum walk, similarly to the regular (non-lackadaisical) quantum walk, has exceptional configuration, i.e. placements of marked vertices for which the walk has no speed-up over the classical exhaustive search. Next, we demonstrate that the weight of the self-loop suggested in the previous papers is not optimal for multiple marked vertices. And, last, we show how to adjust the weight of the self-loop to overcome the aforementioned problem.
The concept of lackadaisical quantum walk -- quantum walk with self loops -- was first introduced for discrete-time quantum walk on one-dimensional line. Later it was successfully applied to improve the running time of the spacial search on two-dimensional grid. In this paper we study search by lackadaisical quantum walk on the two-dimensional grid with multiple marked vertices. First, we show that the lackadaisical quantum walk, similarly to the regular (non-lackadaisical) quantum walk, has exceptional configuration, i.e. placements of marked vertices for which the walk has no speed-up over the classical exhaustive search. Next, we demonstrate that the weight of the self-loop suggested in the previous papers is not optimal for multiple marked vertices. And, last, we show how to adjust the weight of the self-loop to overcome the aforementioned problem.
Abstract Lackadaisical quantum walk (LQW) is a quantum analog of a classical lazy walk, where each vertex has a self-loop of weight l . For a regular N × N 2D grid LQW can find a single marked vertex with O (1) probability in O ( N log N ) steps using l = d / N , where d is the degree of the vertices of the grid [11]. For multiple marked vertices, however, l = d / N is not optimal as the success probability decreases with the increase of the number of marked vertices [12]. In this paper, we numerically study search by LQW for different types of 2D grids—triangular, rectangular and honeycomb—with multiple marked vertices. We show that in all cases the weight l = m ⋅ d / N , where m is the number of marked vertices, still leads to O (1) success probability.
Nature of quantum walk in presence of multiple marked state has been studied by Nahimovs and Rivosh \cite{10.1007/978-3-662-49192-8_31}. They have shown that if the marked states are arranged in a k×k cluster in a N×N grid, then to find a single marked state among the multiple ones, quantum walk requires Ω(N−k) time. In this paper, we show that using lackadaisical quantum walk with the weight of the self-loop as N(k+⌊2k⌋)4, where k is odd, the probability of finding a marked state increases by ∼0.2. Furthermore, we show that instead of applying the quantum walk O(k) times to find all the marked states, classical search in the vicinity of the marked state found after the first implementation of the quantum walk can find all the marked states in O(k) time on average.
The lackadaisical quantum walk, which is a quantum walk with a weighted self-loop at each vertex, has been shown to speed up dispersion on the line and improve spatial search on the complete graph and periodic square lattice. In these investigations, each self-loop had the same weight, owing to each graph's vertex transitivity. In this paper, we propose lackadaisical quantum walks where the self-loops have different weights. We investigate spatial search on the complete bipartite graph, which can be irregular with N1 and N2 vertices in each partite set, and this naturally leads to self-loops in each partite set having different weights l1 and l2, respectively. We analytically prove that for large N1 and N2, if the k marked vertices are confined to, say, the first partite set, then with the typical initial uniform state over the vertices, the success probability is improved from its nonlackadaisical value when l1=kN2/2N1 and ${N}_{2}>(3\ensuremath{-}2\sqrt{2}){N}_{1}$, regardless of l2. When the initial state is stationary under the quantum walk, however, then the success probability is improved when l1=kN2/2N1, now without a constraint on the ratio of N1 and N2, and again independent of l2. Next, when marked vertices lie in both partite sets, then for either initial state there are many configurations for which the self-loops yield no improvement in quantum search, no matter what weights they take.
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.
In the typical model, a discrete-time coined quantum walk search has the same running time of O(NlogN) for 2D rectangular, triangular and honeycomb grids. It is known that for 2D rectangular grid the running time can be improved to O(NlogN) using several different techniques. One of such techniques is adding a self-loop of weight 4/N to each vertex (i.e. making the walk lackadaisical). In this paper we apply lackadaisical approach to quantum walk search on triangular and honeycomb 2D grids. We show that for both types of grids adding a self-loop of weight 6/N and 3/N for triangular and honeycomb grids, respectively, results in O(NlogN) running time.
The running time of a quantum walk search algorithm depends on both the structure of the search space (graph) and the configuration (the placement and the number) of marked locations. While the first dependence has been studied in a number of papers, the second dependence remains mostly unstudied. We study search by quantum walks on the two-dimensional grid using the algorithm of Ambainis, Kempe and Rivosh [3]. The original paper analyses one and two marked locations only. We move beyond two marked locations and study the behaviour of the algorithm for several configurations of multiple marked locations. In this paper, we prove two results showing the importance of how the marked locations are arranged. First, we present two placements of [Formula: see text] marked locations for which the number of steps of the algorithm differs by a factor of [Formula: see text]. Second, we present two configurations of [Formula: see text] and [Formula: see text] marked locations having the same number of steps and probability of finding a marked location.