Renormalization and Scaling in Quantum Walks

Abstract

We show how to extract the scaling behavior of quantum walks using the renormalization group (RG). We introduce the method by efficiently reproducing well-known results on the one-dimensional lattice. As a nontrivial model, we apply this method to the dual Sierpinski gasket and obtain its exact, closed system of RG-recursions. Numerical iteration suggests that under rescaling the system length, $L^{\prime }=2L$, characteristic times rescale as $t^{\prime }=2^{d_w}t$ with the exact walk exponent $d_w={\log }_2\sqrt{5}=1.1609\dots$. Despite the lack of translational invariance, this is very close to the ballistic spreading, $d_w=1$, found for regular lattices. However, we argue that an extended interpretation of the traditional RG formalism will be needed to obtain scaling exponents analytically. Direct simulations confirm our RG-prediction for $d_w$ and furthermore reveal an immensely rich phenomenology for the spreading of the quantum walk on the gasket. Invariably, quantum interference localizes the walk completely with a site-access probability that declines with a powerlaw from the initial site, in contrast with a classical random walk, which would pass all sites with certainty.

10 pages, revtex4, for more information, see http://www.physics.emory.edu/faculty/boettcher/

References in corpus (8)

• Single-Spin Addressing in an Atomic Mott Insulator
• Realization of quantum walks with negligible decoherence in waveguide lattices
• Anderson localization of entangled photons in an integrated quantum walk
• Quantum Google in a Complex Network
• Weak Limit of the 3-State Quantum Walk on the Line
• Geometry and Dynamics for Hierarchical Regular Networks
• Quantum Walks on Sierpinski Gaskets
• Localization and Fractality in Inhomogeneous Quantum Walks with Self-Duality

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• Renormalization of the Unitary Evolution Equation for Coined Quantum Walks
• Transport and Localization in Quantum Walks on a Random Hierarchy of Barriers
• Quantum Ultra-Walks: Walks on a Line with Hierarchical Spatial Heterogeneity
• Renormalization of Discrete-Time Quantum Walks with non-Grover Coins
• Renormalisation of Quantum Cellular Automata
• Renormalization Group Solution of the Chutes&Ladder Model