Discrete-Time Quantum Walks with Lazy and Chaotic Coins

Abstract

We review and study quantum walks and their properties. We highlight the contrasts between quantum and classical walks by comparing the walker’s probability distribution and hitting times on different types of graphs. We focus on discrete-time quantum walks. We see that the quantum walks do not always give an advantage over the classical random walks. As in the case of Lackadaisical quantum walks for a specific range of parameter values, the classical walker has lower hitting times as compared to the quantum walker. We study the lackadaisical quantum walk search algorithm and its limitations in finding adjacent marked nodes on a graph. We develop a modified algorithm to find any type of marked node at the cost of increased algorithm running time. We carry out the search on a binary tree and a hypercube and build a quantum circuit to search for a marked node on the 3−dimensional hypercube. Finally, we study the quantum-chaotic walks by coupling a quantum kicked rotor as a coin to the quantum walker. The kicked rotor affects the walker’s probability distribution, and it seems that the quantum walker gets localized with high probability for a short time interval. This behavior occurs for low ℏ and kick strengths, K ≲2 (for the kicking time period T = 1), accompanied by the rapid growth of von Neumann entropy which is used to measure coin-walker entanglement.