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Quantum walk on a random comb

François David, Thordur Jonsson

Abstract

We study continuous time quantum walk on a random comb graph with infinite teeth. Due to localization effects along the spine, the walk cannot go to infinity in the spine direction, while it can escape to infinity along the teeth of the comb. Starting from an initial vertex the walk has a nonzero probability to stay trapped in a finite region. These results are obtained by studying the spectrum and eigenstates of the random Hamiltonian for the graph and analysing its properties. We use both analytic and numerical methods many of which come from the theory of Anderson localization in one dimension.

Quantum walk on a random comb

Abstract

We study continuous time quantum walk on a random comb graph with infinite teeth. Due to localization effects along the spine, the walk cannot go to infinity in the spine direction, while it can escape to infinity along the teeth of the comb. Starting from an initial vertex the walk has a nonzero probability to stay trapped in a finite region. These results are obtained by studying the spectrum and eigenstates of the random Hamiltonian for the graph and analysing its properties. We use both analytic and numerical methods many of which come from the theory of Anderson localization in one dimension.

Paper Structure

This paper contains 60 sections, 3 theorems, 242 equations, 19 figures.

Table of Contents

  1. Introduction
  2. The model
  3. The finite random comb
  4. The geometry
  5. The eigenvalue equation
  6. $E<4$ states and the $S$-matrix
  7. The $S$-matrix
  8. The full scalar product
  9. The $S$-matrix eigenvector basis
  10. The $S$-matrix column vector basis
  11. The $E>4$ states
  12. Mapping on the Anderson model
  13. The spectrum of the binary chain
  14. The spectral flow
  15. Symmetries of the spectrum
  16. ...and 45 more sections

Key Result

Lemma 1

Consider a chain $\mathcal{C}$ and its energy spectrum $\{ E_\alpha(V)\}$. The following statements are equivalent: $\blacktriangleleft$$\blacktriangleleft$

Figures (19)

  • Figure 1: A random comb ($p=.33$) vs. a regular comb ($p=0$)
  • Figure 2: In and Out states
  • Figure 3: The ellipses describe the location of the $|E|<2$ eigenstates in the $E,V$ plane for a fixed value of the phase shift $\delta$ and the red curve corresponds to the $|E|>2$ states, cf. (\ref{['A1']}) and (\ref{['A2']})
  • Figure 4: The flow of eigenvalues $E$ as a function of the disorder $V$ for a random comb $\mathcal{C}$ of length $L=20$ (here with $N_t=9$ teeth and $N_h=11$ holes) and with periodic boundary conditions.
  • Figure 5: Enlarged view of the spectral flow of Fig. \ref{['fEflow1']}, with the 4 segments considered in the argument.
  • ...and 14 more figures

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • Lemma 3