Abstract We define a discrete-time, coined quantum walk on weighted graphs that is inspired by Szegedy’s quantum walk. Using this, we prove that many lackadaisical quantum walks, where each vertex has l integer self-loops, can be generalized to a quantum walk where each vertex has a single self-loop of real-valued weight l . We apply this real-valued lackadaisical quantum walk to two problems. First, we analyze it on the line or one-dimensional lattice, showing that it is exactly equivalent to a continuous deformation of the three-state Grover walk with faster ballistic dispersion. Second, we generalize Grover’s algorithm, or search on the complete graph, to have a weighted self-loop at each vertex, yielding an improved success probability when l < 3 + 2 2 ≈ 5.828 .
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.
Abstract The evolution operator of the lackadaisical quantum walk on a weighted cycle graph with self-loop weight
w>0 w > 0 is examined. With Grover oracle, the effect of different values of
w w on the success probability of the search under various hyperparameter combinations is analyzed. The success probability function of the first few steps is provided algebraically, which in turn highlights the role of
w w in governing the evolutionary behavior of the quantum walk search. These studies subsequently allow the numerical results to be obtained for various hyperparameter combinations, showing that the highest success probability for the weighted cycle graph with
N N vertices is achieved using a flip-flop shift operator, a weighted coin superposition initial state, and
w=N1.26 w = 1.26 N . A comparison is also made with other known oracle, demonstrating that the proposed configuration provides a better trade-off between success probability and runtime. An extension of the study to the SKW scheme is also included, and it demonstrates that the scheme provides more variability and potential for quantum walk search.
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 lackadaisical quantum walk is a lazy version of a discrete-time, coined quantum walk, where each vertex has a weighted self-loop that permits the walker to stay put. They have been used to speed up spatial search on a variety of graphs, including periodic lattices, strongly regular graphs, Johnson graphs, and the hypercube. In these prior works, the weights of the self-loops preserved the symmetries of the graphs. In this paper, we show that the self-loops can break all the symmetries of vertex-transitive graphs while providing the same computational speedups. Only the weight of the self-loop at the marked vertex matters, and the remaining self-loop weights can be chosen randomly, as long as they are small compared to the degree of the graph.
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.
Lackadaisical quantum walks are coined quantum walk models in which each vertex is associated with a weighted self-loop, allowing the walker to remain at its current position. They have been shown to effectively enhance spatial search on various graphs under different oracle models, with the cyclic graph being the most fundamental instance. In this paper, lackadaisical quantum walk search is studied on Cayley graphs of dicyclic groups, a non-abelian extension of cyclic groups, under three different oracle models. A preliminary analysis of the search performance is first conducted on dicyclic Cayley graphs with respect to the minimal generating set of the underlying dicyclic group. By expanding the generating sets to enable symmetric exploration and applying an appropriate oracle, the quantum walk search achieves a near-one success probability. Under our proposed configurations, the expected runtime to search on dicyclic Cayley graphs outperforms both quantum walk search on the cycle graph and classical unstructured search.