Exact Synthesis of Multiqutrit Clifford-Cyclotomic Circuits

TL;DR

This paper characterizes exactly which multiqutrit unitary matrices can be implemented using Clifford-cyclotomic gates by linking their entries to specific algebraic rings, extending known qubit results to qutrit systems.

Contribution

It establishes a precise correspondence between multiqutrit Clifford-cyclotomic circuits and matrices with entries in algebraic rings, generalizing prior qubit-based synthesis results.

Findings

Matrices representable by the gate set have entries in algebraic rings.

Provides a necessary and sufficient condition for exact circuit synthesis.

Extends the qubit synthesis framework to qutrit systems.

Abstract

It is known that the 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 the present paper, we establish an analogous correspondence for qutrits. We define the multiqutrit Clifford-cyclotomic gate set of degree $3^k$ by extending the classical qutrit gates $X$, $CX$, and $CCX$ with the Hadamard gate $H$ and the $T_k$ 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 degree $3^k$ if and only if the entries of $U$ lie in the ring $\mathbb{Z}[1/3,\omega_k]$.