[Paper Review] Continuous-time open quantum walks

Chaobin Liu, Radhakrishnan Balu|arXiv (Cornell University)|Apr 19, 2016

Quantum Computing Algorithms and Architecture3 citations

TL;DR

This paper introduces continuous-time open quantum walks (CTOQW) as quantum Markov semigroups on graphs, modeling open quantum systems with trace-preserving, completely positive maps. It proves that CTOQW always converge to a steady state on connected graphs, and to the maximally mixed state on connected, regular graphs, distinguishing its long-time behavior from other continuous-time quantum processes.

ABSTRACT

Continuous-time open quantum walks (CTOQW) are introduced as the formulation of quantum dynamical semigroups of trace-preserving and completely positive linear maps (or quantum Markov semigroups) on graphs. We show that a CTOQW always converges to a steady state regardless of the initial state when a graph is connected. When the graph is both connected and regular, it is shown that the steady state is the maximally mixed state. The difference of long-time behaviors between CTOQW and other two continuous-time processes on graphs is exemplified.

Motivation & Objective

• To formalize continuous-time open quantum walks (CTOQW) as quantum Markov semigroups on graphs.
• To analyze the long-time behavior of CTOQW under various graph topologies.
• To establish conditions under which CTOQW converge to the maximally mixed state.
• To contrast CTOQW dynamics with other continuous-time quantum processes on graphs.

Proposed method

• Formulate CTOQW as quantum dynamical semigroups using trace-preserving and completely positive linear maps on graph-structured state spaces.
• Model the generator of the semigroup via a graph Laplacian-like structure encoding transition rates between nodes.
• Use the theory of quantum Markov semigroups to analyze convergence to steady states.
• Apply spectral graph theory to characterize the steady state under connectivity and regularity conditions.
• Compare asymptotic dynamics of CTOQW with unitary quantum walks and classical continuous-time Markov chains on the same graphs.

Experimental results

Research questions

• RQ1 Under what conditions does a continuous-time open quantum walk converge to a steady state on a graph?
• RQ2 What is the structure of the steady state when the underlying graph is connected and regular?
• RQ3 How does the long-time behavior of CTOQW differ from that of unitary quantum walks or classical continuous-time Markov processes?
• RQ4 What role does the graph topology play in determining the asymptotic state of CTOQW?

Key findings

• A CTOQW always converges to a steady state for any connected graph, regardless of the initial state.
• When the graph is both connected and regular, the steady state is the maximally mixed state, i.e., the uniform density matrix.
• The convergence is guaranteed by the properties of quantum Markov semigroups and the irreducibility induced by graph connectivity.
• The long-time dynamics of CTOQW differ fundamentally from those of unitary quantum walks and classical continuous-time Markov chains, particularly in the nature of the steady state.