Mapping a Hadamard Quantum Walk to a Unique Case of a Birth and Death Process

TL;DR

This paper introduces a novel model that maps a Hadamard quantum walk to a birth and death process, preserving quantum state distributions and Markovian properties through a higher-dimensional representation.

Contribution

It presents a new approach linking quantum walks with classical birth and death processes via a 2D Markov chain, enabling analysis of quantum state distributions in a Markovian framework.

Findings

Maps quantum walks to birth and death processes

Preserves probability distributions of quantum states

Reveals quantum distributions through a Markovian model

Abstract

A new model maps a quantum random walk described by a Hadamard operator to a particular case of a birth and death process. The model is represented by a 2D Markov chain with a stochastic matrix, i.e., all the transition rates are positive, although the Hadamard operator contains negative entries (this is possible by increasing the dimensionality of the system). The probability distribution of the walker population is preserved using the Markovian property. By applying a proper transformation to the population distribution of the random walk, the probability distributions of the quantum states |0>, 1> are revealed. Thus, the new model has two unique properties: it reveals the probability distribution of the quantum states as a unitary system and preserves the population distribution of the random walker as a Markovian system.