  Classical $(1+1)D$ cellular automata, as for instance Domany-Kinzel cellular
automata, are paradigmatic systems for the study of non-equilibrium phenomena.
Such systems evolve in discrete time-steps, and are thus free of
time-discretisation errors. Moreover, information about critical phenomena can
be obtained by simulating the evolution of an initial seed that, at any finite
time, has support only on a finite light-cone. This allows for essentially
numerically exact simulations, free of finite-size errors or boundary effects.
Here, we show how similar advantages can be gained in the quantum regime: The
many-body critical dynamics occurring in $(1+1)D$ quantum cellular automata
with an absorbing state can be studied directly on an infinite lattice when
starting from seed initial conditions. This can be achieved efficiently by
simulating the dynamics of an associated one-dimensional, non-unitary quantum
cellular automaton using tensor networks. We apply our method to a model
introduced recently and find accurate values for universal exponents,
suggesting that this approach can be a powerful tool for precisely classifying
non-equilibrium universal physics in quantum systems.
