Synthesis and Arithmetic of Single Qutrit Circuits

In this paper we study single qutrit circuits consisting of words over the Clifford$+D$ cyclotomic gate set, where $D=\text{diag}(\pmξ^{a},\pmξ^{b},\pmξ^{c})$, $ξ$ 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 $\mathbb{Z}[ξ, \frac{1}χ]$ based on the possibility of reducing their smallest denominator exponent (sde) with respect to $χ:= 1 - ξ,$ by acting an appropriate gate in Clifford$+D$. We do this by studying the notion of `derivatives mod $3$' of an arbitrary element of $\mathbb{Z}[ξ]$ and using it to study the smallest denominator exponent of $HDz$ where $H$ is the qutrit Hadamard gate and $D$. 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$+D$ gates naturally arise as gates with sde $0$ and $3$ in the group $U(3,\mathbb{Z}[ξ, \frac{1}χ])$ of $3 \times 3$ unitaries with entries in $\mathbb{Z}[ξ, \frac{1}χ]$. We illustrate the general applicability of these methods to obtain an exact synthesis algorithm for Clifford$+R$ and recover the previous exact synthesis algorithm in \cite{kmm}. The framework developed to formulate qutrit gate synthesis for Clifford$+D$ extends to qudits of arbitrary prime power.