Abstract This paper constructs a new model for the lackadaisical quantum walk and analyzes its features. Traditionally, the particle state is represented by a three-dimensional complex vector, with the Grover operator as the evolution unitary. Analytical methods calculate position probability distributions, but computational complexity and error grow exponentially with steps. To address this, the paper introduces an evolution unitary operator composed of three operators for left movement, right movement, and staying. The staying state corresponds to multiple self-loops at position vertices. A path analysis method is employed to calculate position probabilities and characteristics in polynomial time. Lackadaisical quantum walks differ from classical walks, increasing self-loops enhances diffusion, following a Gaussian mixture model. Numerical experiments reveal lackadaisical characteristics such as slower diffusion, localization, and ballistic diffusion.
Abstract This paper constructs a new model for the lackadaisical quantum walk and analyzes its features. Traditionally, the particle state is represented by a three-dimensional complex vector, with the Grover operator as the evolution unitary. Analytical methods calculate position probability distributions, but computational complexity and error grow exponentially with steps. To address this, the paper introduces an evolution unitary operator composed of three operators for left movement, right movement, and staying. The staying state corresponds to multiple self-loops at position vertices. A path analysis method is employed to calculate position probabilities and characteristics in polynomial time. Lackadaisical quantum walks differ from classical walks, increasing self-loops enhances diffusion, following a Gaussian mixture model. Numerical experiments reveal lackadaisical characteristics such as slower diffusion, localization, and ballistic diffusion.
Abstract In this paper, we study the properties of lackadaisical quantum walks (LQWs) on a line. This model was first proposed in Wong (2015 J. Phys. A: Math. Theor . 48 435304) as a quantum analogue of lazy random walks where each vertex is attached to τ self-loops. We derive an analytic expression for the localization probability of the walker at the origin after infinite steps, and obtain the peak velocities of the walker. We also calculate the wave function of the walker starting from the origin and obtain a long-time approximation for the entire probability density function. As an application of the density function, we prove that lackadaisical quantum walks spread ballistically for arbitrary τ , and give an analytic solution for the variance of the walker’s probability distribution.
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 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 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 lackadaisical quantum walk, a quantum analog of the lazy random walk, is obtained by adding a weighted self-loop transition to each state. Impacts of the self-loop weight l on the final success probability in finding a solution make it a key parameter for the search process. The number of self-loops can also be critical for search tasks. This article proposes the quantum search algorithm Multi-self-loop Lackadaisical Quantum Walk with Partial Phase Inversion, which can be defined as a lackadaisical quantum walk with multiple self-loops, where the target state phase is partially inverted. In the proposed algorithm, each vertex has m self-loops, with weights l′=l/m, where l is a real parameter. The phase inversion is based on Grover's algorithm and acts partially, modifying the phase of a given quantity s<m of self-loops. On a hypercube structure, we analyzed the situation where 1⩽m⩽30. We also propose two new weight values based on two ideal weights l used in the literature. We investigated the effects of partial phase inversion in the search for 1 to 12 marked vertices. As a result, this proposal improved the maximum success probabilities to values close to 1 in O((n+m)⋅N), where n is the hypercube degree. This article contributes with a new perspective on the use of quantum interferences in constructing new quantum search algorithms.
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.