Discrete Geometry from Quantum Walks

Fabrice Debbasch
2019 Condensed Matter  
A particular family of Discrete Time Quantum Walks (DTQWs) simulating fermion propagation in 2D 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
more » ... he 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.
doi:10.3390/condmat4020040 fatcat:cbli5ltdtzdozfelw6y6bcm4vq