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We formalize a notion of discrete Lorentz transforms for Quantum Walks (QW)and Quantum Cellular Automata (QCA), in (1 + 1)-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-onlycovariance 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 alsoallows for non-homogeneous Lorentz transforms, between non-inertial frames.

Discrete Lorentz covariance for Quantum Walks and Quantum Cellular Automata