  We consider a graph with a single quantum system at each node. The entire
compound system evolves in discrete time steps by iterating a global evolution
$U$. We require that this global evolution $U$ be unitary, in accordance with
quantum theory, and that this global evolution $U$ be causal, in accordance
with special relativity. By causal we mean that information can only ever be
transmitted at a bounded speed, the speed bound being quite naturally that of
one edge of the underlying graph per iteration of $U$. We show that under these
conditions the operator $U$ can be implemented locally; i.e. it can be put into
the form of a quantum circuit made up with more elementary operators -- each
acting solely upon neighbouring nodes. We take quantum cellular automata as an
example application of this representation theorem: this analysis bridges the
gap between the axiomatic and the constructive approaches to defining QCA.
KEYWORDS: Quantum cellular automata, Unitary causal operators, Quantum walks,
Quantum computation, Axiomatic quantum field theory, Algebraic quantum field
theory, Discrete space-time.
