The Clifford-cyclotomic group and Euler-Poincar\'e characteristics

Abstract

For an integer $n\geq 8$ divisible by $4$, let $R_n=\mathbb{Z}[\zeta_n,1/2]$ and let $\operatorname{U}_2(R_n)$ be the group of $2\times 2$ unitary matrices with entries in $R_n$. Set $\operatorname{U}_2^\zeta(R_n)=\{\gamma\in\operatorname{U}_2(R_n)\mid \det\gamma\in\langle\zeta_n\rangle\}$. Let $\mathcal{G}_n\subseteq \operatorname{U}_2^\zeta(R_n)$ be the Clifford-cyclotomic group generated by a Hadamard matrix $H=\frac{1}{2}[\begin{smallmatrix} 1+i & 1+i\\1+i &-1-i\end{smallmatrix}]$ and the gate $T=[\begin{smallmatrix}1 & 0\\0 & \zeta_n\end{smallmatrix}]$. We prove that $\mathcal{G}_n=\operatorname{U}_2^\zeta(R_n)$ if and only if $n=8, 12, 16, 24$ and that $[\operatorname{U}_2^\zeta(R_n):\mathcal{G}_n]=\infty$ if $\operatorname{U}_2^\zeta(R_n)\neq \mathcal{G}_n$. We compute the Euler-Poincar\'{e} characteristics of the groups $\operatorname{SU}_2(R_n)$, $\operatorname{PSU}_2(R_n)$, $\operatorname{PU}_2(R_n)$, $\operatorname{PU}^\zeta_2(R_n)$, and $\operatorname{SO}_3(R_n^+)$.