Universal Behavior of Quantum Walks with Long-Range Steps

Abstract

Quantum walks with long-range steps $R^{-\gamma }$ ($R$ being the distance between sites) on a discrete line behave in similar ways for all $\gamma \ge 2$. This is in contrast to classical random walks, which for $\gamma >3$ belong to a different universality class than for $\gamma \le 3$. We show that the average probabilities to be at the initial site after time $t$ as well as the mean square displacements are of the same functional form for quantum walks with $\gamma =2$, 4, and with nearest neighbor steps. We interpolate this result to arbitrary $\gamma \ge 2$.

4 pages, 3 figures

References in corpus (10)

• Quantum Communication Through an Unmodulated Spin Chain
• Quantum random walks - an introductory overview
• Quantum Computation and Decision Trees
• Perfect state transfer in quantum spin networks
• Quantum random walks in optical lattices
• Quantum quincunx in cavity quantum electrodynamics
• Survival Probabilities in Coherent Exciton Transfer with Trapping
• Efficiency of quantum and classical transport on graphs
• Slow transport by continuous time quantum walks
• Diffusive-ballistic crossover in 1D quantum walks

Cited by in corpus (12)

• Continuous-Time Quantum Walks: Models for Coherent Transport on Complex Networks
• Quantum walks with sequential aperiodic jumps
• Pólya number of continuous-time quantum walks
• Coherent exciton transport and trapping on long-range interacting cycles
• Multiple transitions between normal and hyperballistic diffusion in quantum walks with time-dependent jumps
• The quantum Levy walk
• Discrete-time quantum walks on one-dimensional lattices
• Scaling and crossover behaviour in a truncated long range quantum walk
• Enhancing entanglement with the generalized elephant quantum walk from localized and delocalized states
• Slow Excitation Trapping in Quantum Transport with Long-Range Interactions
• The effect of large-decoherence on mixing-time in Continuous-time quantum walks on long-range interacting cycles
• Quantum centipedes: collective dynamics of interacting quantum walkers