  The Clifford+T gate set is a topological generating set for PU(2), which has
been well-studied from the perspective of quantum computation on a single
qubit. The discovery that it generates a full S-arithmetic subgroup of PU(2)
has led to a fruitful interaction between quantum computation and number
theory, leading in particular to a proof that words in these gates cover PU(2)
in an almost-optimal manner.
  In this paper we study an analogue gate set for PU(3) called
Clifford+$\mathcal{D}$. We show that this set generates a full S-arithmetic
subgroup of PU(3), and satisfies a slightly weaker almost-optimal covering
property. Our proofs are different from those for PU(2): while both gate sets
act naturally on a (Bruhat-Tits) tree, in PU(2) the generated group acts
transitively on the vertices of the tree, and this is a main ingredient in
proving both arithmeticity and efficiency. In the PU(3)
(Clifford+$\mathcal{D}$) case the action on the tree is far from being
transitive. This makes the proof of arithmeticity considerably harder, and the
study of covering rate by automorphic representation theory becomes more
involved and results in a slower covering rate.
