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Condens. Matter 2019, 4(2), 40;

Discrete Geometry from Quantum Walks

Sorbonne Université, Observatoire de Paris, Université PSL, CNRS, LERMA, F-75005 Paris, France
Received: 28 February 2019 / Revised: 7 April 2019 / Accepted: 9 April 2019 / Published: 11 April 2019
(This article belongs to the Special Issue Quantum Cellular Automata and Quantum Walks)
PDF [273 KB, uploaded 11 April 2019]


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 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.
Keywords: discrete time quantum walks; discrete geometry; discrete Riemann curvature; discrete metric discrete time quantum walks; discrete geometry; discrete Riemann curvature; discrete metric
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Debbasch, F. Discrete Geometry from Quantum Walks. Condens. Matter 2019, 4, 40.

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