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.
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.
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.
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 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.
The lackadaisical quantum walk is a quantum analogue of the lazy random walk obtained by adding a self-loop to each vertex in the graph. We analytically prove that lackadaisical quantum walks can find a unique marked vertex on any regular locally arc-transitive graph with constant success probability quadratically faster than the hitting time. This result proves several speculations and numerical findings in previous work, including the conjectures that the lackadaisical quantum walk finds a unique marked vertex with constant success probability on the torus, cycle, Johnson graphs, and other classes of vertex-transitive graphs. Our proof establishes and uses a relationship between lackadaisical quantum walks and quantum interpolated walks for any regular locally arc-transitive graph.
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.
Lackadaisical quantum walk (LQW) has been an efficient technique in searching for a target state in a database which is distributed in a two-dimensional lattice. We numerically study the quantum search algorithm based on the lackadaisical quantum walk in one and two dimensions. It is observed that specific values of the self-loop weight at each vertex of the graph is responsible for such a speedup of the algorithm. Searching for a target state in one-dimensional lattice with periodic boundary conditions is possible using lackadaisical quantum walk, which can find a target state with [Formula: see text] success probability after [Formula: see text] time steps. In two dimensions, our numerical simulation up to [Formula: see text] for specific sets of target states suggests that the lackadaisical quantum walk can search one of the [Formula: see text] target states in [Formula: see text] time steps.