Andrew N. Glaudel, Neil J. Ross, John van de Wetering, Lia Yeh (May 15 2024).
Abstract: It is known that the unitary 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 this paper, we establish the analogous correspondence for qutrits. We define the multiqutrit Clifford-cyclotomic gate set of order $3^k$ by extending the classical qutrit gates $X$, $CX$, and Toffoli with the Hadamard gate $H$ and the single-qutrit 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 order $3^k$ if and only if the entries of $U$ lie in the ring $\mathbb{Z}[1/3,\omega_k]$.