On Hitting Times for General Quantum Markov Processes

TL;DR

This paper introduces a quantum Markov chain model using density matrices, extending classical random walk tools like hitting times to quantum settings, and applies it to algorithms such as Grover's search.

Contribution

It generalizes classical hitting time analysis to quantum Markov processes using the density-matrix formalism, bridging classical and quantum random walk theories.

Findings

Hitting times can be computed similarly in quantum and classical models.

The framework applies to known quantum algorithms like Grover's algorithm.

Provides a foundation for analyzing quantum random walks with classical tools.

Abstract

Random walks (or Markov chains) are models extensively used in theoretical computer science. Several tools, including analysis of quantities such as hitting and mixing times, are helpful for devising randomized algorithms. A notable example is Sch\"oning's algorithm for the satisfiability (SAT) problem. In this work, we use the density-matrix formalism to define a quantum Markov chain model which directly generalizes classical walks, and we show that a common tools such as hitting times can be computed with a similar formula as the one found in the classical theory, which we then apply to known quantum settings such as Grover's algorithm.