  We build a quantum cellular automaton (QCA) which coincides with 1+1 QED on
its known continuum limits. It consists in a circuit of unitary gates driving
the evolution of particles on a one dimensional lattice, and having them
interact with the gauge field on the links. The particles are massive fermions,
and the evolution is exactly U(1) gauge-invariant. We show that, in the
continuous-time discrete-space limit, the QCA converges to the Kogut-Susskind
staggered version of 1+1 QED. We also show that, in the continuous spacetime
limit and in the free one particle sector, it converges to the Dirac equation,
a strong indication that the model remains accurate in the relativistic regime.
