Anyonic Quantum Walks

Abstract

The one dimensional quantum walk of anyonic systems is presented. The anyonic walker performs braiding operations with stationary anyons of the same type ordered canonically on the line of the walk. Abelian as well as non-Abelian anyons are studied and it is shown that they have very different properties. Abelian anyonic walks demonstrate the expected quadratic quantum speedup. Non-Abelian anyonic walks are much more subtle. The exponential increase of the system's Hilbert space and the particular statistical evolution of non-Abelian anyons give a variety of new behaviors. The position distribution of the walker is related to Jones polynomials, topological invariants of the links created by the anyonic world-lines during the walk. Several examples such as the SU(2) level k and the quantum double models are considered that provide insight to the rich diffusion properties of anyons.

17 pages, 10 figures

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• Quantum walks: a comprehensive review
• Bringing order through disorder: Localization of errors in topological quantum memories
• Review on Quantum Walk Computing: Theory, Implementation, and Application
• Thermalization, Error-Correction, and Memory Lifetime for Ising Anyon Systems
• Statistical dynamics of a non-Abelian anyonic quantum walk
• Transport properties of anyons in random topological environments
• Environment-induced mixing processes in quantum walks
• Quantum walk on a square lattice with identical particles