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We present a quantum cellular automaton model in one space-dimension whichhas the Dirac equation as emergent. This model, a discrete-time and causalunitary evolution of a lattice of quantum systems, is derived from theassumptions of homogeneity, parity and time-reversal invariance. The comparisonbetween the automaton and the Dirac evolutions is rigorously set as adiscrimination problem between unitary channels. We derive an exact lower boundfor the probability of error in the discrimination as an explicit function ofthe mass, the number and the momentum of the particles, and the duration of theevolution. Computing this bound with experimentally achievable values, we seethat in that regime the QCA model cannot be discriminated from the usual Diracevolution. Finally, we show that the evolution of one-particle states withnarrow-band in momentum can be effi- ciently simulated by a dispersivedifferential equation for any regime. This analysis allows for a comparisonwith the dynamics of wave-packets as it is described by the usual Diracequation. This paper is a first step in exploring the idea that quantum fieldtheory could be grounded on a more fundamental quantum cellular automaton modeland that physical dynamics could emerge from quantum information processing. Inthis framework, the discretization is a central ingredient and not only a toolfor performing non-perturbative calculation as in lattice gauge theory. Theautomaton model, endowed with a precise notion of local observables and a fullprobabilistic interpretation, could lead to a coherent unification of anhypothetical discrete Planck scale with the usual Fermi scale of high-energyphysics.

Quantum Field as a quantum cellular automaton: the Dirac free evolution in one dimension