  Let $n\geq 8$ be divisible by 4. The Clifford-cyclotomic gate set
$\mathcal{G}_n$ is the universal gate set obtained by extending the Clifford
gates with the $z$-rotation $T_n = \mathrm{diag}(1,\zeta_n)$, where $\zeta_n$
is a primitive $n$-th root of unity. In this note, we show that, when $n$ is a
power of 2, a multiqubit unitary matrix $U$ can be exactly represented by a
circuit over $\mathcal{G}_n$ if and only if the entries of $U$ belong to the
ring $\mathbb{Z}[1/2,\zeta_n]$. We moreover show that $\log(n)-2$ ancillas are
always sufficient to construct a circuit for $U$. Our results generalize prior
work to an infinite family of gate sets and show that the limitations that
apply to single-qubit unitaries, for which the correspondence between
Clifford-cyclotomic operators and matrices over $\mathbb{Z}[1/2,\zeta_n]$ fails
for all but finitely many values of $n$, can be overcome through the use of
ancillas.
