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Ergodicity in discrete-time quantum walks

Kiran Kumar, Mostafa Sabri

Abstract

We undertake a detailed analysis of ergodicity for homogeneous discrete-time quantum walks on the integer lattice. The most significant result of our paper holds in dimension one, and gives a complete equivalence between the absolutely continuous spectrum of the unitary operator encoding the walk, and the equidistribution of its dynamics in position space, which appears for the first time in the context of large-volume quantum ergodicity. In higher dimensions, we give a criterion for full and partial ergodicity in terms of a finer property of the spectrum which we dub ``No Repeating Graphs'', and distinguish how strongly the equidistribution is taking place (weak convergence vs total variation). The paper includes a wealth of examples where we apply our criteria, with certain families of walks fully characterized.

Ergodicity in discrete-time quantum walks

Abstract

We undertake a detailed analysis of ergodicity for homogeneous discrete-time quantum walks on the integer lattice. The most significant result of our paper holds in dimension one, and gives a complete equivalence between the absolutely continuous spectrum of the unitary operator encoding the walk, and the equidistribution of its dynamics in position space, which appears for the first time in the context of large-volume quantum ergodicity. In higher dimensions, we give a criterion for full and partial ergodicity in terms of a finer property of the spectrum which we dub ``No Repeating Graphs'', and distinguish how strongly the equidistribution is taking place (weak convergence vs total variation). The paper includes a wealth of examples where we apply our criteria, with certain families of walks fully characterized.
Paper Structure (39 sections, 36 theorems, 195 equations, 3 figures, 1 table)

This paper contains 39 sections, 36 theorems, 195 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model and definitions
  3. Main results
  4. Open questions
  5. Organization of the paper
  6. Proof of the general criterion
  7. Basic Fourier Analysis.
  8. Step 1. From the Heisenberg picture to a phase space operator.
  9. Step 2. The time limit in a finite spacial box.
  10. Step 3. The zero order symbol yields a weighted average.
  11. Step 4. The higher order modes vanish asymptotically.
  12. From FQE to PQE
  13. From QE to weak convergence
  14. Convergence in total variation
  15. Compact support of eigenvectors
  16. ...and 24 more sections

Key Result

Theorem 1.3

Let $U$ be a quantum walk e:ugen-eqn:homogen_unitary satisfying e:flo. Let $\psi = \sum_{\mathbf{k}\in \Lambda} \psi(\mathbf{k})\delta_{\mathbf{k}}$ be an initial state of compact support, i.e. $\Lambda\subset \mathbb{Z}^d$ is finite and independent of $N$, and assume $\|\psi\|=1$. Then for any regu where $\langle \phi \rangle_{T,\psi} = \sum_{\mathbf{r}\in \mathbb{L}_N^d} \phi(\mathbf{r}) \mu_{T,

Figures (3)

  • Figure 1: Summary of results for walks on $\mathbb{Z}$.
  • Figure 2: Time evolution of the $1d$ quantum walk with coin matrix $0110$ and step sizes $\alpha=2$ and $\beta=5$, and initial state $\psi=\delta_0 \otimes 10$.
  • Figure 3: Illustration of the possible positions that a quantum walk without entanglement on 2-dimensional integer lattice reaches, with dotted arrows depicting the four vectors $\pm e_1 \pm e_2$. The blue vertices are the ones that can be reached from $(0,0)$ in one step and the red vertices are the ones that can be reached in two steps.

Theorems & Definitions (79)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Lemma 1.6
  • Theorem 1.7
  • Proposition 1.8
  • Theorem 1.9: One-dimensional walks
  • Proposition 1.10: Convergence in total variation
  • ...and 69 more