  We consider the group structure of quantum cellular automata (QCA) modulo
circuits and show that it is abelian even without assuming the presence of
ancillas, at least for most reasonable choices of control space; this is a
corollary of a general method of ancilla removal. Further, we show how to
define a group of QCA that is well-defined without needing to use families, by
showing how to construct a coherent family containing an arbitrary finite QCA;
the coherent family consists of QCA on progressively finer systems of qudits
where any two members are related by a shallow quantum circuit. This
construction applied to translation invariant QCA shows that all translation
invariant QCA in three dimensions and all translation invariant Clifford QCA in
any dimension are coherent.
