Properties of Quantum Walks within Various One Dimensional Media Öffentlichkeit

Abstract

Recent work has shown that the canonical interpretation of Renormalization Group (RG) analysis on the quantum ultra walk produces incorrect results for determining the walk dimension, which describes the long term scaling of the system. Motivated by this inconsistency, we develop two numerical methods for approximating the walk dimension for 1D quantum walks. First, we approximate the walk dimension using the Nth moment of position in the large N limit. Then, we reproduce the walk dimension through envelope collapse.

These methods are used to compare the quantum ultra walk to its classical analog, the persistent random walk. These methodologies are then extended to the quantum random walk, as well as various walks with three chiral states, including the Grover walk, the cyclic Grover Walk, and the hierarchical Grover walk, all of which provide insight into the efficacy of our numerical methods as well as impart some understanding of how lattice geometry affects the scaling of the walk.

Through our analysis, it is found that both methods are widely applicable, yet become less accurate for walks with large coin variability relative to system size. Finally, we discuss how these methods can be made more robust, and conclude that, from the results of our numerical methods, the canonical interpretation of the RG flow for the quantum ultra walk is indeed misguided.

Table of Contents

Table of Contents

I. Introduction......................................................................1

II. Structure of the One Dimensional Quantum Walk..................2

A. Walk Dimension........................................................2

B. Two State Walker, n = 2............................................2

1. Delta Function Barriers, n = 2............................2

2. Random Quantum Walk, n = 2...........................3

3. Ultra Walk, n=2................................................3

4. Persistent Classical Walk, n = 2..........................3

C. Three State Walker, n = 3..........................................4

1. Constant Grover Walk, n = 3..............................4

2. Constant Cyclical Grover Walk, n = 3..................4

3. Hierarchical Cyclical Grover Walk, n = 3..............4

III. Modelling a Quantum Walk on a Classical Computer.............4

IV. First Passage Time within Delta Function Barriers.................5

V. Methods for Determining dw...............................................6

A. Limiting Moments.....................................................6

B. Determining dw through Scaling.................................6

VI. Results of dw Finding Methods on Non-Ultra Walks...............6

VII. Conclusion.....................................................................8

About this Honors Thesis

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School	Emory College
Department	Physics
Degree	BS
Submission	Honors Thesis
Language	English
Research Field	Physics, General
Stichwort	Renormalization Group Numerical Methods Ultra Walk Grover Walk Quantum Random Walk Quantum Walk Scaling Walk Dimension
Committee Chair / Thesis Advisor	Boettcher, Stefan, Emory University
Committee Members	Brody, Jed, Emory University Ettinger, Bree, Emory University

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