Abstract
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.
Highlights
Discrete Time Quantum Walks (DTQWs) are unitary quantum automata
No exact discrete gauge invariance has been displayed for DTQWs which converge towards fermions coupled to gravitational fields, and no discrete field strength i.e., Riemann curvature has been defined either
We focus on a certain family of DTQWs in discrete 2D space-time whose continuous limit coincides the dynamics of a Dirac fermion
Summary
Discrete Time Quantum Walks (DTQWs) are unitary quantum automata. They have been first considered by Feynman [1] as tools to discretise path integrals for fermions, and later introduced in a more formal and systematic way in Aharonov [2] and Meyer [3]. DTQWs have been realized experimentally with a wide range of physical objects and setups [4,5,6,7,8,9,10], and are studied in a large variety of contexts, ranging from quantum optics [10] to quantum algorithmics [11,12], condensed matter physics [13,14,15,16,17], hydrodynamics [18] and biophysics [19,20] It has been shown recently [21,22,23,24,25,26,27,28,29,30] that several DTQWs admit as continuous limit the dynamics of Dirac fermions coupled to arbitrary Yang–Mills gauge fields (including electromagnetic fields) and to arbitrary relativistic gravitational fields. No exact discrete gauge invariance has been displayed for DTQWs which converge towards fermions coupled to gravitational fields, and no discrete field strength i.e., Riemann curvature has been defined either
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