  We introduce a new framework for constructing topological quantum memories,
by recasting error recovery as a dynamical process on a field generating
cellular automaton. We envisage quantum systems controlled by a classical
hardware composed of small local memories, communicating with neighbours, and
repeatedly performing identical simple update rules. This approach does not
require any global operations or complex decoding algorithms. Our cellular
automata draw inspiration from classical field theories, with a Coulomb-like
potential naturally emerging from the local dynamics. For a 3D automaton
coupled to a 2D toric code, we present evidence of an error correction
threshold above 6.1% for uncorrelated noise. A 2D automaton equipped with a
more complex update rule yields a threshold above 8.2%. Our framework provides
decisive new tools in the quest for realising a passive dissipative quantum
memory.
