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 (QWs). 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.

中文翻译：

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量子行走模拟朗道问题中的非交换几何

非交换几何 (NCG) 是在带电粒子在均匀磁场中运动的背景下考虑的。重新审视经典和量子力学处理，并引入了新的 NCG 标记。然后，该标记用于研究磁性量子行走 (QW) 中的 NCG。事实证明，这些行走在连续体极限处和附近表现出 NCG。对于纯粹的离散机制，详细研究了两个不同复杂性的说明性路径。最复杂的步行确实表现出 NCG，但最简单、最退化的步行则没有。因此，NCG 可以通过 QW 进行模拟，不仅可以在连续体极限中模拟，而且可以在纯离散状态中模拟。