Exact Synthesis of Multiqutrit Clifford-Cyclotomic Circuits

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 $\mathbb{Z}[1/3,{\omega }_k]$.

In Proceedings QPL 2024, arXiv:2408.05113

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Cited by in corpus (4)

• Synthesis and Arithmetic of Single Qutrit Circuits
• Contributions to the Theory of Clifford-Cyclotomic Circuits
• Transversal AND in Quantum Codes
• Synthesis of Single Qutrit Circuits from Clifford+R