Asymptotic Gate Count Bounds for Ancilla-Free Single-Qubit Synthesis with Arithmetic Gates
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914082305605632 |
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| author | Sano, Kaoru Morisaki, Hayata Akibue, Seiseki |
| author_facet | Sano, Kaoru Morisaki, Hayata Akibue, Seiseki |
| contents | We study ancilla-free approximation of single-qubit unitaries $U\in {\rm SU}(2)$ by gate sequences over Clifford+$G$, where $G\in\{T,V\}$ or their generalization. Let $p$ denote the characteristic factor of the gate set (e.g., $p=2$ for $G=T$ and $p=5$ for $G=V$). We prove three asymptotic bounds on the minimum $G$-count required to achieve approximation error at most $\varepsilon$. First, for Haar-almost every $U$, we show that $3\log_{p}(1/\varepsilon)$ $G$-count is both necessary and sufficient; moreover, probabilistic synthesis improves the leading constant to $3/2$. Second, for unitaries whose ratio of matrix elements lies in a specified number field, $4\log_p(1/\varepsilon)$ $G$-count is necessary. Again, the leading constant can be improved to $2$ by probabilistic synthesis. Third, there exist unitaries for which the $G$-count per $\log_{p}(1/\varepsilon)$ fails to converge as $\varepsilon\to 0^+$. These results partially resolve a generalized form of the Ross--Selinger conjecture. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_07627 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic Gate Count Bounds for Ancilla-Free Single-Qubit Synthesis with Arithmetic Gates Sano, Kaoru Morisaki, Hayata Akibue, Seiseki Quantum Physics Mathematical Physics Number Theory We study ancilla-free approximation of single-qubit unitaries $U\in {\rm SU}(2)$ by gate sequences over Clifford+$G$, where $G\in\{T,V\}$ or their generalization. Let $p$ denote the characteristic factor of the gate set (e.g., $p=2$ for $G=T$ and $p=5$ for $G=V$). We prove three asymptotic bounds on the minimum $G$-count required to achieve approximation error at most $\varepsilon$. First, for Haar-almost every $U$, we show that $3\log_{p}(1/\varepsilon)$ $G$-count is both necessary and sufficient; moreover, probabilistic synthesis improves the leading constant to $3/2$. Second, for unitaries whose ratio of matrix elements lies in a specified number field, $4\log_p(1/\varepsilon)$ $G$-count is necessary. Again, the leading constant can be improved to $2$ by probabilistic synthesis. Third, there exist unitaries for which the $G$-count per $\log_{p}(1/\varepsilon)$ fails to converge as $\varepsilon\to 0^+$. These results partially resolve a generalized form of the Ross--Selinger conjecture. |
| title | Asymptotic Gate Count Bounds for Ancilla-Free Single-Qubit Synthesis with Arithmetic Gates |
| topic | Quantum Physics Mathematical Physics Number Theory |
| url | https://arxiv.org/abs/2510.07627 |