Quantum Walks and discrete Gauge Theories

Abstract

A particular example is produced to prove that quantum walks can be used to simulate full-fledged discrete gauge theories. A new family of $2D$ walks is introduced and its continuous limit is shown to coincide with the dynamics of a Dirac fermion coupled to arbitrary electromagnetic fields. The electromagnetic interpretation is extended beyond the continuous limit by proving that these DTQWs exhibit an exact discrete local $U(1)$ gauge invariance and possess a discrete gauge-invariant conserved current. A discrete gauge-invariant electromagnetic field is also constructed and that field is coupled to the conserved current by a discrete generalization of Maxwell equations. The dynamics of the DTQWs under crossed electric and magnetic fields is finally explored outside the continuous limit by numerical simulations. Bloch oscillations and the so-called $\mathbf{E}\times \mathbf{B}$ drift are recovered in the weak-field limit. Localization is observed for some values of the gauge fields.

7 pages, 7 figures

References in corpus (7)

• Universal computation by quantum walk
• Quantum Walk in Position Space with Single Optically Trapped Atoms
• Realization of quantum walks with negligible decoherence in waveguide lattices
• Quantum Communication through Spin Chain Dynamics: an Introductory Overview
• Mimicking the probability distribution of a two-dimensional Grover walk with a single-qubit coin
• Quantum Lattice Boltzmann is a quantum walk
• Two-dimensional quantum walk under artificial magnetic field

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