Contributions to the Theory of Clifford-Cyclotomic Circuits
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908496382918656 |
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| author | Dinh, Linh Ross, Neil J. |
| author_facet | Dinh, Linh Ross, Neil J. |
| contents | Let $n$ be a positive integer divisible by 8. The Clifford-cyclotomic gate set $\mathcal{G}_n$ consists of the Clifford gates, together with a $z$-rotation of order $n$. It is easy to show that, if a circuit over $\mathcal{G}_n$ represents a unitary matrix $U$, then the entries of $U$ must lie in $\mathcal{R}_n$, the smallest subring of $\mathbb{C}$ containing $1/2$ and $\mathrm{exp}(2πi/n)$. The converse implication, that every unitary $U$ with entries in $\mathcal{R}_n$ can be represented by a circuit over $\mathcal{G}_n$, is harder to show, but it was recently proved to be true when $n=2^k$. In that case, $k-2$ ancillas suffice to synthesize a circuit for $U$, which is known to be minimal for $k=3$, but not for larger values of $k$. In the present paper, we make two contributions to the theory of Clifford-cyclotomic circuits. Firstly, we improve the existing synthesis algorithm by showing that, when $n=2^k$ and $k\geq 4$, only $k-3$ ancillas are needed to synthesize a circuit for $U$, which is minimal for $k=4$. Secondly, we extend the existing synthesis algorithm to the case of $n=3\cdot 2^k$ with $k\geq 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14674 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Contributions to the Theory of Clifford-Cyclotomic Circuits Dinh, Linh Ross, Neil J. Quantum Physics Let $n$ be a positive integer divisible by 8. The Clifford-cyclotomic gate set $\mathcal{G}_n$ consists of the Clifford gates, together with a $z$-rotation of order $n$. It is easy to show that, if a circuit over $\mathcal{G}_n$ represents a unitary matrix $U$, then the entries of $U$ must lie in $\mathcal{R}_n$, the smallest subring of $\mathbb{C}$ containing $1/2$ and $\mathrm{exp}(2πi/n)$. The converse implication, that every unitary $U$ with entries in $\mathcal{R}_n$ can be represented by a circuit over $\mathcal{G}_n$, is harder to show, but it was recently proved to be true when $n=2^k$. In that case, $k-2$ ancillas suffice to synthesize a circuit for $U$, which is known to be minimal for $k=3$, but not for larger values of $k$. In the present paper, we make two contributions to the theory of Clifford-cyclotomic circuits. Firstly, we improve the existing synthesis algorithm by showing that, when $n=2^k$ and $k\geq 4$, only $k-3$ ancillas are needed to synthesize a circuit for $U$, which is minimal for $k=4$. Secondly, we extend the existing synthesis algorithm to the case of $n=3\cdot 2^k$ with $k\geq 3$. |
| title | Contributions to the Theory of Clifford-Cyclotomic Circuits |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2508.14674 |