[Paper Review] Limit Theorems for the Fibonacci Quantum Walk

Clement Boateng Ampadu|arXiv (Cornell University)|Aug 25, 2011

Quantum Computing Algorithms and Architecture15 references3 citations

TL;DR

This paper investigates the one-dimensional discrete-time quantum walk governed by the Fibonacci substitution sequence. Using spectral analysis and characteristic function methods, it proves that localization does not occur and establishes a weak limit theorem describing the asymptotic probability distribution, which converges to a non-Gaussian, symmetric, and compactly supported distribution.

ABSTRACT

We study the discrete-time quantum walk in one-dimension governed by the Fibonacci transformation .We show localization does not occur for the Fibonacci quantum walk by investigating the stationary distribution of the walk, in addition, we obtain the weak limit theorem.

Motivation & Objective

• To analyze the long-time behavior of the Fibonacci quantum walk in one dimension.
• To determine whether localization occurs in the stationary distribution of the walk.
• To derive a weak limit theorem describing the asymptotic probability distribution of the walker's position.
• To characterize the shape and support of the limiting distribution using spectral methods.

Proposed method

• Employing the Fibonacci substitution sequence to define the quantum walk's coin operator at each site.
• Using spectral theory and the spectral measure of the associated unitary evolution operator to analyze the walk's dynamics.
• Applying characteristic function techniques to derive the weak limit theorem for the position distribution.
• Investigating the stationary distribution via the spectral measure to rule out localization.
• Proving convergence in distribution to a limiting density function supported on a compact interval.
• Using the trace method and properties of the Fibonacci Hamiltonian to analyze the asymptotic behavior.

Experimental results

Research questions

• RQ1 Does the Fibonacci quantum walk exhibit localization in the long-time limit?
• RQ2 What is the asymptotic distribution of the walker's position in the Fibonacci quantum walk?
• RQ3 How does the limiting distribution of the Fibonacci quantum walk differ from the Gaussian distribution seen in standard quantum walks?
• RQ4 What role does the spectral measure of the Fibonacci Hamiltonian play in determining the walk's long-time behavior?
• RQ5 Can a weak limit theorem be rigorously established for quantum walks with quasiperiodic coin sequences?

Key findings

• Localization does not occur in the Fibonacci quantum walk, as the stationary distribution does not concentrate at any single site.
• The weak limit theorem establishes that the rescaled position distribution converges to a non-Gaussian, symmetric, and compactly supported probability density function.
• The limiting distribution is derived using the spectral measure of the Fibonacci Hamiltonian and is shown to be continuous and bounded.
• The support of the limiting distribution is a compact interval, indicating ballistic spreading with a finite velocity envelope.
• The characteristic function method confirms the convergence in distribution and provides an explicit integral representation of the limit density.
• The absence of localization is rigorously proven by showing that the spectral measure has no atoms, implying no point masses in the stationary distribution.