Synthesis and Arithmetic of Single Qutrit Circuits
- URL: http://arxiv.org/abs/2311.08696v3
- Date: Mon, 1 Jan 2024 04:39:46 GMT
- Title: Synthesis and Arithmetic of Single Qutrit Circuits
- Authors: Amolak Ratan Kalra, Dinesh Valluri, Michele Mosca
- Abstract summary: We study single qutrit quantum circuits consisting of words over the Clifford+ $mathcalD$ gate set.
We characterize classes of qutrit unit vectors $z$ with entries in $mathbbZ[xi, frac1chi]$.
- Score: 0.9208007322096532
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: In this paper we study single qutrit quantum circuits consisting of words
over the Clifford+ $\mathcal{D}$ gate set, where $\mathcal{D}$ consists of
cyclotomic gates of the form $\text{diag}(\pm\xi^{a},\pm\xi^{b},\pm\xi^{c}),$
where $\xi$ 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
$HDz$ where $H$ is the qutrit Hadamard gate and $D \in \mathcal{D}.$ 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 $\mathbb{Z}[\xi, \frac{1}{\chi}]$
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