Exact Synthesis of Multiqutrit Clifford-Cyclotomic Circuits
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913464182636544 |
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| author | Glaudell, Andrew N. Ross, Neil J. van de Wetering, John Yeh, Lia |
| author_facet | Glaudell, Andrew N. Ross, Neil J. van de Wetering, John Yeh, Lia |
| contents | 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 $\mathbb{Z}[1/2,ζ_k]$, where $k$ is a positive integer that depends on the gate set and $ζ_k$ 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 $T_k=\mathrm{diag}(1,ω_k, ω_k^2)$, where $ω_k$ 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 $\mathbb{Z}[1/3,ω_k]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_08136 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exact Synthesis of Multiqutrit Clifford-Cyclotomic Circuits Glaudell, Andrew N. Ross, Neil J. van de Wetering, John Yeh, Lia Quantum Physics 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 $\mathbb{Z}[1/2,ζ_k]$, where $k$ is a positive integer that depends on the gate set and $ζ_k$ 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 $T_k=\mathrm{diag}(1,ω_k, ω_k^2)$, where $ω_k$ 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 $\mathbb{Z}[1/3,ω_k]$. |
| title | Exact Synthesis of Multiqutrit Clifford-Cyclotomic Circuits |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2405.08136 |