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On the Optimal Rate of Convergence for Translation-Invariant 1D Quantum Walks

Benjamin Hinrichs, Pascal Mittenbühler

Abstract

We study the convergence rate of translation-invariant discrete-time quantum dynamics on a one-dimensional lattice. We prove that the cumulative distributions function of the ballistically scaled position $\mathbb X(n)/{n}$ after $n$ steps converges at a rate of $n^{-1/3}$ in the Lévy metric as $n\to\infty$. In the special case of step-coin quantum walks with two-dimensional coin space, we recover the same convergence rate for the supremum distance and prove optimality.

On the Optimal Rate of Convergence for Translation-Invariant 1D Quantum Walks

Abstract

We study the convergence rate of translation-invariant discrete-time quantum dynamics on a one-dimensional lattice. We prove that the cumulative distributions function of the ballistically scaled position after steps converges at a rate of in the Lévy metric as . In the special case of step-coin quantum walks with two-dimensional coin space, we recover the same convergence rate for the supremum distance and prove optimality.

Paper Structure

This paper contains 17 sections, 23 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Model and Results
  3. Lattice Dynamics
  4. General Setup
  5. Translation-Invariance and Weak Limit Theorems
  6. Lévy Metric and General Upper Bound
  7. Kolmogorov Metric and Optimality for Step-Coin Walks
  8. Esseen--Zolotarev Type Estimates
  9. A Generalized Zolotarev Inequality
  10. Estimates on The Characteristic Functions
  11. Berry-Esseen Bounds for Coin-Step Walks
  12. The Asymptotic Distribution
  13. Smooth Region $|x|<|a|$
  14. Wavefront Region $|x|\approx |a|$
  15. Proof of Theorem 2.2
  16. ...and 2 more sections

Key Result

Theorem 2.1

In eq:Wp additionally assume that $\omega_k\in C^2({\mathds T};{\mathbb R})$ and $\Pi_k\in C^1({\mathds T};{\mathcal{B}}({\mathcal{H}}))$ such that Then, for any density matrix $\rho$ satisfying $\operatorname{tr}(\lvert {\mathds X}\rvert \rho)<\infty$, there exists a constant $C>0$ such that

Theorems & Definitions (47)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of \ref{['lem:ZolotarevSim']}
  • ...and 37 more