Percolated quantum walks with a general shift operator: From trapping to transport

Abstract

We present a generalized definition of discrete-time quantum walks convenient for capturing a rather broad spectrum of walker's behavior on arbitrary graphs. It includes and covers both: the geometry of possible walker's positions with interconnecting links and the prescribed rule in which directions the walker will move at each vertex. While the former allows for the analysis of inhomogeneous quantum walks on graphs with vertices of varying degree, the latter offers us to choose, investigate, and compare quantum walks with different shift operators. The synthesis of both key ingredients constitutes a well-suited playground for analyzing percolated quantum walks on a quite general class of graphs. Analytical treatment of the asymptotic behavior of percolated quantum walks is presented and worked out in details for the Grover walk on graphs with maximal degree 3. We find, that for these walks with cyclic shift operators the existence of an edge-3-coloring of the graph allows for non-stationary asymptotic behavior of the walk. For different shift operators the general structure of localized attractors is investigated, which determines the overall efficiency of a source-to-sink quantum transport across a dynamically changing medium. As a simple nontrivial example of the theory we treat a single excitation transport on a percolated cube.

arXiv comment23 pages, 18 figures

References in corpus (33)

• Quantum walks: a comprehensive review
• Environment-Assisted Quantum Transport
• Quantum Walk in Position Space with Single Optically Trapped Atoms
• Photons Walking the Line: A quantum walk with adjustable coin operations
• From quantum cellular automata to quantum lattice gases
• Realization of a quantum walk with one and two trapped ions
• Quantum walk of a trapped ion in phase space
• Quantum walks in higher dimensions
• Decoherence in quantum walks - a review
• One-Dimensional Three-State Quantum Walk
• Quantum random walks with decoherent coins
• Quantum Walks on a Random Environment
• Quantum walks on general graphs
• Limit distributions of two-dimensional quantum walks
• Quantum walks based on an interferometric analogy
• Disordered Quantum Walks in one lattice dimension
• Quantum searches on highly symmetric graphs
• Localization of Multi-State Quantum Walk in One Dimension
• Asymptotic dynamics of coined quantum walks on percolation graphs
• Coined quantum walks on percolation graphs
• Quantum Walks
• Weak Limit of the 3-State Quantum Walk on the Line
• Inhomogeneous Quantum Walks
• Mixing Times in Quantum Walks on the Hypercube
• Wigner formula of rotation matrices and quantum walks
• Asymptotic Evolution of Random Unitary Operations
• Equivalence between discrete quantum walk models in arbitrary topologies
• Localization in Quantum Walks on a Honeycomb Network
• Localization of the Grover walks on spidernets and free Meixner laws
• Quantum walk transport properties on graphene structures
• Percolation induced effects in 2D coined quantum walks: analytic asymptotic solutions
• Percolation assisted excitation transport in discrete-time quantum walks
• Quantum Walks on Sierpinski Gaskets

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• Quantum Ultra-Walks: Walks on a Line with Hierarchical Spatial Heterogeneity
• Quantum walks driven by quantum coins with two multiple eigenvalues
• A comfortable graph structure for Grover walk
• Relation between quantum walks with tails and quantum walks with sinks on finite graphs
• Two-particle Hadamard walk on dynamically percolated line and circle
• Key graph properties affecting transport efficiency of flip-flop Grover percolated quantum walks