Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dynamical Localization for General Scattering Quantum Walks

Alain Joye, Andreas Schaefer, Simone Warzel

Abstract

We consider quantum walks defined on arbitrary infinite graphs, parameterized by a family of scattering matrices attached to the vertices. Multiplying each scattering matrix by an i.i.d. random phase, we obtain a random scattering quantum walk. We prove dynamical localization for random scattering walks in a large-disorder regime. The result is based on a relation between fractional moment estimates and eigenfunction correlators of independent interest, which we establish for general random unitary operators.

Dynamical Localization for General Scattering Quantum Walks

Abstract

We consider quantum walks defined on arbitrary infinite graphs, parameterized by a family of scattering matrices attached to the vertices. Multiplying each scattering matrix by an i.i.d. random phase, we obtain a random scattering quantum walk. We prove dynamical localization for random scattering walks in a large-disorder regime. The result is based on a relation between fractional moment estimates and eigenfunction correlators of independent interest, which we establish for general random unitary operators.
Paper Structure (23 sections, 10 theorems, 157 equations, 3 figures)

This paper contains 23 sections, 10 theorems, 157 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Eigenfunction correlators for random unitaries
  3. Definitions and basic properties
  4. Consistent families of random unitaries on subgraphs
  5. Relation to fractional moments
  6. Application to scattering quantum walks
  7. Scattering quantum walks
  8. Random scattering quantum walks
  9. Fractional moment bounds
  10. Spectral averaging
  11. Localization proof
  12. Subspaces associated to balls
  13. Spectral gaps of the fully localized random operator
  14. Finite-volume resolvent estimate
  15. The iterative step
  16. ...and 8 more sections

Key Result

Proposition 1

Given a sequence of finite subsets $E_L \in \mathcal{F}$ in a consistent family $\mathcal{F}$ for random unitaries on the Hilbert space over a graph $(V,E)$ such that the boundary operators converge strongly, i.e., for all $e \in E$ and all $\omega \in \mathbb{R}^{E}$ Then for all $e,f \in E$ and all $\omega \in \mathbb{R}^{E}$:

Figures (3)

  • Figure 1: A ball $B$ centered at $f$ of radius $d_E(e,f)/2$.
  • Figure 2: A graph $(V,E)$ and its digraph $(V,D)$
  • Figure 3: Action of $U_S$

Theorems & Definitions (24)

  • Definition 1
  • Remark 1
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Remark 2
  • Theorem 1
  • proof : Proof of Theorem \ref{['thm:fmec']}
  • Theorem 2
  • ...and 14 more