Exact Synthesis of Multiqutrit Clifford-Cyclotomic Circuits

• URL: http://arxiv.org/abs/2405.08136v5
• Date: Mon, 12 Aug 2024 11:20:24 GMT
• Title: Exact Synthesis of Multiqutrit Clifford-Cyclotomic Circuits
• Authors: Andrew N. Glaudell, Neil J. Ross, John van de Wetering, Lia Yeh,
• Abstract summary: We prove that a $3ntimes 3n$ unitary matrix $U$ can be represented by an $n$-qutrit circuit over the Clifford-cyclotomic gate set of degree $3k$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\mathbb{Z}[1/3,\omega_k]$.

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