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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 bythe Clifford-cyclotomic gate set. We previously settled this usingEuler-Poincar\'{e} characteristics. A generalization of Sarnak's conjecture isto ask when these groups are generated by torsion elements. An obstruction tothis 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 thecorank of these cyclotomic unitary groups in the families $n=2^s$ and $n=3\cdot2^s$, $n\geq 8$, by letting them act on Bruhat-Tits trees. The quotients bythis action are finite graphs whose first Betti number is the corank of thegroup. Our main result is that for $n=2^s$ and $n=3\cdot 2^s$ the corank groupsdoubly 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 bytorsion elements (in fact by the Clifford-cyclotomic gates) for these $n$. Wegive explicit lower bounds for the corank in two different ways. The first isto bound the isotropy subgroups in the action on the tree by explicitcyclotomy. The second is to relate our graphs to Shimura curves over$F_n=\mathbb{Q}(\zeta_n)^+$ via interchanging local invariants and applying aresult of Selberg and Zograf. We show that the cyclotomy arguments give thestronger bounds. In a final section we execute a program of Sarnak to show thatour results for the $n=2^s$ and $n=3\cdot 2^s$ families are sufficient to givea second proof of Sarnak's conjecture.

Sarnak's conjecture in quantum computing, cyclotomic unitary group coranks, and Shimura curves