Recurrence of biased quantum walks on a line

Abstract

The Polya number of a classical random walk on a regular lattice is known to depend solely on the dimension of the lattice. For one and two dimensions it equals one, meaning unit probability to return to the origin. This result is extremely sensitive to the directional symmetry, any deviation from the equal probability to travel in each direction results in a change of the character of the walk from recurrent to transient. Applying our definition of the Polya number to quantum walks on a line we show that the recurrence character of quantum walks is more stable against bias. We determine the range of parameters for which biased quantum walks remain recurrent. We find that there exist genuine biased quantum walks which are recurrent.

Journal reference added, minor corrections in the text

References in corpus (16)

• A Quantum Random Walk Search Algorithm
• Controlling discrete quantum walks: coins and intitial states
• Weak limits for quantum random walks
• One-Dimensional Three-State Quantum Walk
• Quantum walk on the line as an interference phenomenon
• Hitting time for quantum walks on the hypercube
• Recurrence and Pólya number of quantum walks
• Quantum walks with infinite hitting times
• Recurrence properties of unbiased coined quantum walks on infinite $d$-dimensional lattices
• Localization of Multi-State Quantum Walk in One Dimension
• Three routes to the exact asymptotics for the one-dimensional quantum walk
• Exact Operator Quantization of a Model of Two-Dimensional Dilaton Gravity
• Wigner formula of rotation matrices and quantum walks
• Symmetry-noise interplay in quantum walk on an n-cycle
• Scattering quantum random-walk search with errors
• Large Qudit Limit of One-dimensional Quantum Walks

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