Synthesis and Arithmetic of Single Qutrit Circuits

Quantum 9, 1647 (2025).

https://doi.org/10.22331/q-2025-02-26-1647

In this paper we study single qutrit circuits consisting of words over the Clifford$+mathcal{D}$ cyclotomic gate set, where $mathcal{D}=text{diag}(pmxi^{a},pmxi^{b},pmxi^{c})$, $xi$ 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 $Hmathcal{D}z$ 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.