  It is known that the matrices that can be exactly represented by a multiqubit
circuit over the Toffoli+Hadamard, Clifford+$T$, or, more generally,
Clifford-cyclotomic gate set are precisely the unitary matrices with entries in
the ring $\mathbb{Z}[1/2, \zeta_k]$, where $k$ is a positive integer that
depends on the gate set and $\zeta_k$ is a primitive $2^k$-th root of unity. In
the present paper, we establish an analogous correspondence for qutrits. We
define the multiqutrit Clifford-cyclotomic gate set of degree $3^k$ by
extending the classical qutrit gates $X$, $CX$, and $CCX$ with the Hadamard
gate $H$ and the $T_k$ gate $T_k=\mathrm{diag}(1,\omega_k, \omega_k^2)$, where
$\omega_k$ is a primitive $3^k$-th root of unity. This gate set is equivalent
to the qutrit Toffoli+Hadamard gate set when $k=1$, and to the qutrit
Clifford+$T_k$ gate set when $k>1$. We then prove that a $3^n\times 3^n$
unitary matrix $U$ can be represented by an $n$-qutrit circuit over the
Clifford-cyclotomic gate set of degree $3^k$ if and only if the entries of $U$
lie in the ring $\mathbb{Z}[1/3,\omega_k]$.
