One-dimensional discrete-time quantum walks with general coin

TL;DR

This paper explores the effects of a general coin operator on one-dimensional discrete-time quantum walks, providing insights into how parameter tuning influences probability distributions and including algorithms for various coin types.

Contribution

It introduces a comprehensive analysis of the general coin operator in 1D quantum walks, detailing its impact and offering algorithms for implementation.

Findings

Parameter tuning of the coin affects the walker's probability distribution.

The study includes analysis of Hadamard, Grover, and Fourier coins.

Provides an algorithm for quantum walks with a general coin.

Abstract

Quantum walk (QW) is the quantum analog of the random walk. QW is an integral part of the development of numerous quantum algorithms. Hence, an in-depth understanding of QW helps us to grasp the quantum algorithms. We revisit the one-dimensional discrete-time QW and discuss basic steps in detail by incorporating the most general coin operator. We investigate the impact of each parameter of the general coin operator on the probability distribution of the quantum walker. We show that by tuning the parameters of the general coin, one can regulate the probability distribution of the walker. We provide an algorithm for the one-dimensional quantum walk driven by the general coin operator. The study conducted on general coin operator also includes the popular coins -- Hadamard, Grover, and Fourier coins.