Fast and slow dynamics for classical and quantum walks on mean-field small world networks

Area of Science:

• Physics
• Network Science
• Quantum Computing

Background:

• Classical and quantum random walks exhibit distinct dynamical properties on various network structures.
• Mean-field small-world (MFSW) networks offer unique topological characteristics for studying walk dynamics.
• Understanding these dynamics is crucial for applications in quantum information and network analysis.

Purpose of the Study:

• To investigate and compare the dynamical properties of classical and quantum random walks on MFSW networks.
• To derive exact mathematical expressions for transition probabilities.
• To analyze the impact of disorder on the spreading behavior of both classical and quantum walks.

Main Methods:

• Utilizing the exact mathematical properties of adjacency and Laplacian matrices for MFSW networks.
• Deriving exact expressions for transition probabilities using Bessel functions.
• Comparing analytical results with numerical simulations of the model's Hamiltonian.

Main Results:

• Exact expressions for transition probabilities derived in terms of Bessel functions.
• Classical walks exhibit exponential decay to equilibrium with minimal disorder, unlike homogeneous polynomial decay.
• Quantum walks show oscillatory evolution with local maxima, indicating polynomial decay to equilibrium regardless of disorder.
• A key finding is that classical random walks spread faster than quantum random walks on MFSW networks.

Conclusions:

• Classical random walks demonstrate superior spreading speed compared to quantum random walks on MFSW networks.
• This finding contrasts with the established diffusive (classical) and ballistic (quantum) spreading on linear chains.
• The study provides insights into the non-intuitive behavior of quantum dynamics on complex network topologies.