One-dimensional lazy quantum walks and occupancy rate

Abstract

State Key Laboratory of N etworking and Switching T echnology, Beijing U niversity of P osts and T elecommunications, Beijing, 100876, China a) T he De Brun Centre f or Computational Algebra, School of M athematics, Statistics and Applied M athematics, N ational U niversity of Ireland, Galway b) (Received X XX XXXX; revised manuscript received X XX XXXX) Lazy quantum walks were presented by Andrew M. Childs to prove that the continuous-time quantum walk is a limit of the discrete-time quantum walk [Commun.Math.Phys.294,581-603(2010)].In this paper, we discuss properties of lazy quantum walks. Our analysis shows that lazy quantum walks have O(t n ) order of the n-th moment of the corresponding probability distribution, which is the same as that for normal quantum walks. Also, the lazy quantum walk with DFT (Discrete Fourier Transform) coin operator has a similar probability distribution concentrated interval to that of the normal Hadamard quantum walk. Most importantly, we introduce the concepts of occupancy number and occupancy rate to measure the extent to which the walk has a (relatively) high probability at every position in its range. We conclude that lazy quantum walks have a higher occupancy rate than other walks such as normal quantum walks, classical walks and lazy classical walks.

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].…”

Photonic quantum walks with four-dimensional coins

“…Already on the one-dimensional (1D) line DTQWs with higher-dimensional coins have been shown to exhibit unique features not possessed by two-dimensional coins, among the most striking the so-called trapping [47][48][49]. While due to the simplicity of the 1D structure these may be regarded as toy systems, they can be efficiently used to demonstrate several fundamental differences between classical and quantum walks.…”

Localization for a one-dimensional split-step quantum walk with bound states robust against perturbations

“…From (2.3), this walk is interpreted as a lazy quantum walk. We emphasize that our walk is defined as a two-state quantum walk on ℓ 2 (Z; C 2 ), whereas standard lazy quantum walks [16], [24] are defined as a three-state quantum walk on ℓ 2 (Z; C 3 ). Our evolution U partially covers several examples of one-dimensional two-sate quantum walks as seen below.…”