Quantum random walks on the $N$-cycle subject to decoherence on the coin degree of freedom

Abstract

For a discrete time quantum walk (QW) on the $N$-cycle, allowing for decoherence on the coin, we derive a number of new results, including an explicit formula for the position probability distribution. For a QW of this type, we show that the mixing behavior tends, in the long-run, to a uniform distribution, regardless of the initial state of the system and irrespective of the parity of the number of nodes $N$. These results confirm the findings of previous authors who arrived at similar conclusions through extensive numerical simulations. In particular, we infer that the mixing time $\overline{M(\epsilon )}$ for the time-everaged probability distribution is of order no greater than $O(N^2/\epsilon )$.

References in corpus (6)

• Realization of quantum walks with negligible decoherence in waveguide lattices
• Decoherence in quantum walks - a review
• Implementing the one-dimensional quantum (Hadamard) walk using a Bose-Einstein Condensate
• Almost uniform sampling via quantum walks
• Symmetry-noise interplay in quantum walk on an n-cycle
• Quantum random walks using quantum accelerator modes

Cited by in corpus (12)

• Quantum walks: a comprehensive review
• Quantumness of noisy quantum walks: a comparison between measurement-induced disturbance and quantum discord
• Quantumness in decoherent quantum walk using measurement-induced disturbance
• Extraordinary behaviors in two-dimensional decoherent alternative quantum walk
• Asymptotic evolution of quantum walks on the $N$-cycle subject to decoherence on both the coin and position degrees of freedom
• Measurement induced quantum walks
• Open quantum random walks: ergodicity, hitting times, gambler's ruin and potential theory
• On limiting distributions of quantum Markov chains
• Probing coherence and noise tolerance in discrete-time quantum walks: unveiling self-focusing and breathing dynamics
• Noise-enhanced quantum transport on a closed loop using quantum walks
• Limit Theorems for Decoherent Two Dimensional Quantum Walks
• On the von Neumann entropy of certain quantum walks subject to decoherence