Quantum Hitting Time on the Complete Graph

TL;DR

This paper derives analytical formulas for quantum hitting time and success probability of quantum walks on the complete graph, advancing understanding of quantum algorithms based on quantum Markov chains.

Contribution

It provides the first explicit analytical expressions for quantum hitting time and success probability on the complete graph using Szegedy's framework.

Findings

Analytical expressions for quantum hitting time derived

Success probability formulas established

Enhanced understanding of quantum walk dynamics on complete graphs

Abstract

Quantum walks play an important role in the area of quantum algorithms. Many interesting problems can be reduced to searching marked states in a quantum Markov chain. In this context, the notion of quantum hitting time is very important, because it quantifies the running time of the algorithms. Markov chain-based algorithms are probabilistic, therefore the calculation of the success probability is also required in the analysis of the computational complexity. Using Szegedy's definition of quantum hitting time, which is a natural extension of the definition of the classical hitting time, we present analytical expressions for the hitting time and success probability of the quantum walk on the complete graph.