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Simulating Non Commutative Geometry with Quantum Walks

Fabrice Debbasch

Abstract

Non Commutative Geometry (NCG) is considered in the context of a charged particle moving in a uniform magnetic field. The classical and quantum mechanical treatments are revisited and a new marker of NCG is introduced. This marker is then used to investigate NCG in magnetic Quantum Walks. It is proven that these walks exhibit NCG at and near the continuum limit. For the purely discrete regime, two illustrative walks of different complexities are studied in full detail. The most complex walk does exhibit NCG but the simplest, most degenerate one does not. Thus, NCG can be simulated by QWs, not only in the continuum limit, but also in the purely discrete regime.

Simulating Non Commutative Geometry with Quantum Walks

Abstract

Non Commutative Geometry (NCG) is considered in the context of a charged particle moving in a uniform magnetic field. The classical and quantum mechanical treatments are revisited and a new marker of NCG is introduced. This marker is then used to investigate NCG in magnetic Quantum Walks. It is proven that these walks exhibit NCG at and near the continuum limit. For the purely discrete regime, two illustrative walks of different complexities are studied in full detail. The most complex walk does exhibit NCG but the simplest, most degenerate one does not. Thus, NCG can be simulated by QWs, not only in the continuum limit, but also in the purely discrete regime.

Paper Structure

This paper contains 11 sections, 24 equations, 2 figures.

Table of Contents

  1. I. Introduction
  2. II. Spinless particle
  3. II.1. Classical mechanics
  4. 2.2 Quantum mechanics
  5. III. Quantum Walks
  6. III.1 Magnetic Quantum Walks
  7. III.2 NCG at and near the continuous limit
  8. III.3 Purely discrete case
  9. III.3.1 Example 1
  10. III.3.2 Example 2
  11. IV. Discussion

Figures (2)

  • Figure 1: Example 1. Angle $\theta_S$ as a function of the reduced momentum variable $\bar{p}$.
  • Figure 2: Example 1. Average position $< q >$ as a function of the reduced momentum variable $\bar{p}$.