From classical to quantum walks with stochastic resetting on networks

Area of Science:

• Stochastic processes
• Quantum mechanics
• Network science

Background:

• Random walks are foundational models for stochastic processes across science and technology.
• Stochastic resetting introduces a mechanism to interrupt and restart random walks, impacting their dynamics.
• Understanding classical and quantum walks on networks is crucial for diverse applications.

Purpose of the Study:

• To develop a unified framework for classical and quantum random walks with stochastic resetting on arbitrary networks.
• To analyze the influence of quantum effects on the stationary and long-time average probability distributions.
• To compare the effects of resetting on sampling properties between classical and quantum walks.

Main Methods:

• Utilizing the mathematical formalism of quantum stochastic walks.
• Employing graph Laplacians to define the evolution of both classical and quantum walks.
• Interpolating between classical and quantum regimes to study the transition of effects.
• Conducting numerical simulations on various network structures.

Main Results:

• A framework for studying classical and quantum walks with stochastic resetting on networks was established.
• Quantum effects were shown to modify the stationary and long-time average probability distributions.
• Significant differences were observed in how stochastic resetting affects the sampling properties of classical versus quantum walks.
• Analytical predictions were validated through numerical simulations.

Conclusions:

• Stochastic resetting has distinct impacts on classical and quantum random walks.
• Quantum mechanics introduces unique behaviors in random walk dynamics, particularly under resetting conditions.
• The study provides insights into the interplay between quantum effects, network structure, and resetting in stochastic processes.