Abstract
We formalize a notion of discrete Lorentz transforms for quantum walks (QW) and quantum cellular automata (QCA), in -dimensional discrete spacetime. The theory admits a diagrammatic representation in terms of a few local, circuit equivalence rules. Within this framework, we show the first-order-only covariance of the Dirac QW. We then introduce the clock QW and the clock QCA, and prove that they are exactly discrete Lorentz covariant. The theory also allows for non-homogeneous Lorentz transforms, between non-inertial frames.
Highlights
The Dirac Quantum Walk (QW) would be understood as describing an infinitesimal time evolution, but in the same formalism as that of discrete time evolutions, i.e. in an alternative language to the Hamiltonian formalism
In the context of QW and Quantum Cellular Automata (QCA), we have formalized a notion of discrete Lorentz transform of parameters α, β, which consists in replacing each spacetime point with a lightlike α × β rectangular spacetime patch, Cm E, where E is an isometric encoding, and Cm is the repeated application of the unitary interaction Cm throughout the patch
First we considered the Dirac QW, a natural candidate, given that it has the Dirac equation as continuum limit, which is covariant
Summary
Instead of deforming the translation operator algebra, one could look at dropping translational invariance of the QW evolution Along these lines, models have been constructed for QWs in external fields, including specific cases of gravitational fields [17, 18]. Models have been constructed for QWs in external fields, including specific cases of gravitational fields [17, 18] Another non-statistical, early approach is to restrict the class of allowed Lorentz transforms, to a subgroup of the Lorentz group whose matrices are over the integers numbers tt x x.
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