2013/11/30 by Stefan Boettcher, S. Boettcher, Stefan Falkner +3 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Density matrix renormalization group #Geometry #Mathematical physics #Mathematics #Physics #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum algorithm #Quantum and electron transport phenomena #Quantum mechanics #Quantum walk #Renormalization #Scaling #Statistical physics #cond-mat.stat-mech #quant-ph
paper · pdf · doi:10.1103/physreva.90.032324
published as Physical Review A 90, 032324 (2014) · 10 pages, revtex4, for more information, see http://www.physics.emory.edu/faculty/boettcher/
openalex publication_date 2014/09/23 · arxiv created 2014/09/28 · arxiv updated 2014/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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. For 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^\ensuremath'=2L, characteristic times rescale as t^\ensuremath'=2^dwt, with the exact walk exponent dw=log2√(5)=1.1609... Despite the lack of translational invariance, this value is very close to the ballistic spreading, dw=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 dw 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 decreases with a power law from the initial site, in contrast to a classical random walk, which would pass all sites with certainty.