Simulating Non Commutative Geometry with Quantum Walks

TL;DR

This paper demonstrates that quantum walks can simulate non-commutative geometry phenomena both near the continuum limit and in purely discrete regimes, revealing new insights into quantum simulation of geometric structures.

Contribution

It introduces a new marker for non-commutative geometry and shows how quantum walks can exhibit NCG in different regimes, expanding the scope of quantum simulation.

Findings

Quantum walks exhibit NCG near the continuum limit.

Some discrete quantum walks also show NCG properties.

Simplest walks may not exhibit NCG, unlike more complex ones.

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.