  Sarnak's conjecture in quantum computing concerns when the groups
$\operatorname{PU}_2$ and $\operatorname{PSU}_2$ over cyclotomic rings
$\mathbb{Z}[\zeta_n, 1/2]$ with $\zeta_n=e^{2\pi i/n}$, $4|n$, are generated by
the Clifford-cyclotomic gate set. We previously settled this using
Euler-Poincar\'{e} characteristics. A generalization of Sarnak's conjecture is
to ask when these groups are generated by torsion elements. An obstruction to
this is provided by the corank: a group $G$ has $\operatorname{corank} G >0$
only if $G$ is not generated by torsion elements. In this paper we study the
corank of these cyclotomic unitary groups in the families $n=2^s$ and $n=3\cdot
2^s$, $n\geq 8$, by letting them act on Bruhat-Tits trees. The quotients by
this action are finite graphs whose first Betti number is the corank of the
group. Our main result is that for $n=2^s$ and $n=3\cdot 2^s$ the corank groups
doubly exponentially in $s$ as $s\rightarrow \infty$; it is $0$ precisely when
$n=8,12, 16,24$ and indeed the cyclotomic unitary groups are generated by
torsion elements (in fact by the Clifford-cyclotomic gates) for these $n$. We
give explicit lower bounds for the corank in two different ways. The first is
to bound the isotropy subgroups in the action on the tree by explicit
cyclotomy. The second is to relate our graphs to Shimura curves over
$F_n=\mathbb{Q}(\zeta_n)^+$ via interchanging local invariants and applying a
result of Selberg and Zograf. We show that the cyclotomy arguments give the
stronger bounds. In a final section we execute a program of Sarnak to show that
our results for the $n=2^s$ and $n=3\cdot 2^s$ families are sufficient to give
a second proof of Sarnak's conjecture.
