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Limit theorem for continuous-time quantum walk on the line.

¹Department of Applied Mathematics, Yokohama National University, 79-5 Tokiwadai, Yokohama, 240-8501, Japan. norio@mathlab.sci.ynu.ac.jp

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|October 4, 2005

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This study reveals new weak limit theorems for continuous-time quantum walks, contrasting with classical random walks. It also explores the relationship between discrete and continuous-time quantum walks using matrix representations.

Area of Science:

Quantum Mechanics
Probability Theory
Mathematical Physics

Background:

Discrete-time quantum walks with symmetric distributions exhibit specific weak limit theorems.
Classical random walks follow a different limit behavior described by the central limit theorem.

Purpose of the Study:

To establish a weak limit theorem for continuous-time quantum walks on a line.
To compare the limit distributions of discrete- and continuous-time quantum walks.
To investigate the relationship between discrete- and continuous-time quantum walks.

Main Methods:

Analysis of continuous-time quantum walks using matrix representations.
Derivation of weak limit theorems for quantum walk distributions.
Exhaustive comparison of limit distributions between discrete and continuous cases.

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Main Results:

A novel weak limit theorem is established for continuous-time quantum walks on the line: X(t)(c)/t --> dx/pi * sqrt(1-x^2).
The limit distribution for continuous-time quantum walks differs significantly from the central limit theorem of classical random walks.
The study provides a comparative analysis of discrete- and continuous-time quantum walks within a unified formalism.

Conclusions:

Continuous-time quantum walks exhibit distinct statistical properties compared to their classical counterparts.
The findings contribute to understanding the fundamental differences and connections between discrete and continuous quantum walks.
The matrix representation formalism offers a robust framework for analyzing quantum walk dynamics.