Synthesis and Arithmetic of Single Qutrit Circuits

In this paper we study single qutrit circuits consisting of words over the Clifford$+\mathcal{D}$ cyclotomic gate set, where $\mathcal{D}=\text{diag}(\pm\xi^{a},\pm\xi^{b},\pm\xi^{c})$, $\xi$ is a primitive $9$-th root of unity and $a,b,c$ are integers. We characterize classes of qutrit unit vectors $z$ with entries in $\mathbb{Z}[\xi, \frac{1}{\chi}]$ based on the possibility of reducing their smallest denominator exponent (sde) with respect to $\chi := 1 – \xi,$ by acting an appropriate gate in Clifford$+\mathcal{D}$. We do this by studying the notion of `derivatives mod $3$' of an arbitrary element of $\mathbb{Z}[\xi]$ and using it to study the smallest denominator exponent of $H\mathcal{D}z$ where $H$ is the qutrit Hadamard gate and $\mathcal{D}$. In addition, we reduce the problem of finding all unit vectors of a given sde to that of finding integral solutions of a positive definite quadratic form along with some additional constraints. As a consequence we prove that the Clifford$+\mathcal{D}$ gates naturally arise as gates with sde $0$ and $3$ in the group $U(3,\mathbb{Z}[\xi, \frac{1}{\chi}])$ of $3 \times 3$ unitaries with entries in $\mathbb{Z}[\xi, \frac{1}{\chi}]$. We illustrate the general applicability of these methods to obtain an exact synthesis algorithm for Clifford$+R$ and recover the previously known exact synthesis algorithm of Kliuchnikov, Maslov, Mosca (2012). The framework developed to formulate qutrit gate synthesis for Clifford$+\mathcal{D}$ extends to qudits of arbitrary prime power.