Discrete Geometry from Quantum Walks

Abstract

A particular family of Discrete Time Quantum Walks (DTQWs) simulating fermion propagation in $2$D curved space-time is revisited. Usual continuous covariant derivatives and spin-connections are generalized into discrete covariant derivatives along the lattice coordinates and discrete connections. The concepts of metrics and $2$-beins are also extended to the discrete realm. Two slightly different Riemann curvatures are then defined on the space-time lattice as the curvatures of the discrete spin connection. These two curvatures are closely related and one of them tends at the continuous limit towards the usual, continuous Riemann curvature. A simple example is also worked out in full.

References in corpus (12)

• Quantum Walk in Position Space with Single Optically Trapped Atoms
• Two-particle bosonic-fermionic quantum walk via 3D integrated photonics
• Realization of quantum walks with negligible decoherence in waveguide lattices
• Realization of a quantum walk with one and two trapped ions
• Quantum walk of a trapped ion in phase space
• Quantum quincunx in cavity quantum electrodynamics
• Quantum Walks in artificial electric and gravitational Fields
• Quantum Walks and discrete Gauge Theories
• Quantum walks as massless Dirac Fermions in curved Space-Time
• Quantum walks and gravitational waves
• Electric quantum walks in two dimensions
• Einstein's vierbein field theory of curved space