More efficient Clifford+T synthesis for small-angle rotations and application to Trotterization
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| Autores principales: | , , , , |
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| Formato: | Preprint |
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2026
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| _version_ | 1866917548233064448 |
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| author | Bothe, Marius Sünderhauf, Christoph Witham, Michael J. Campbell, Earl T. Blunt, Nick S. |
| author_facet | Bothe, Marius Sünderhauf, Christoph Witham, Michael J. Campbell, Earl T. Blunt, Nick S. |
| contents | Clifford+T synthesis of rotation gates is an important routine in fault-tolerant quantum compilation. While Clifford+T synthesis is scalable, it has a high overhead of tens of T gates per rotation in practice, translating to high resource estimates for many fault-tolerant algorithms. However, these well-known results, including those using probabilistic mixtures [Quantum 7, 1208 (2023)], are independent of the rotation angle $θ$, requiring $O(\log 1/δ)$ T gates. We show that it is possible to do much better for small angles, reducing the T cost to $\tilde O(θ^2/δ)$, and returning to existing $O(\log1/δ)$ results in the worst case. This is particularly important since many algorithms, such as Trotterization, are dominated by small-angle rotations. Further, we perform a detailed theoretical and numerical study of quasi-probabilities, which can further reduce the total T cost of large circuits by orders of magnitude with only a small overhead in sample complexity. We also develop a scheme based on quasi-probability mixtures of Clifford+T fallback channels. We derive new $θ$-dependent formulas that can be used for resource estimation of fault-tolerant quantum algorithms. As an application of our results, we show that the gate cost of Trotterization circuits compiled to a Clifford+T gate set is constant in the small Trotter step size limit, and can be reduced by orders of magnitude even for large step sizes. The cost of fault-tolerant Trotterization for a variety of applications should be re-examined in light of these results. Our work dispels the widely-stated claim that Clifford+T rotation synthesis has a high cost independent of $θ$, and further develops a scalable quasi-probability method for rotation synthesis. We also expect our results to bring forward useful early fault-tolerant quantum computing by reducing required magic state resources. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_31544 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | More efficient Clifford+T synthesis for small-angle rotations and application to Trotterization Bothe, Marius Sünderhauf, Christoph Witham, Michael J. Campbell, Earl T. Blunt, Nick S. Quantum Physics Clifford+T synthesis of rotation gates is an important routine in fault-tolerant quantum compilation. While Clifford+T synthesis is scalable, it has a high overhead of tens of T gates per rotation in practice, translating to high resource estimates for many fault-tolerant algorithms. However, these well-known results, including those using probabilistic mixtures [Quantum 7, 1208 (2023)], are independent of the rotation angle $θ$, requiring $O(\log 1/δ)$ T gates. We show that it is possible to do much better for small angles, reducing the T cost to $\tilde O(θ^2/δ)$, and returning to existing $O(\log1/δ)$ results in the worst case. This is particularly important since many algorithms, such as Trotterization, are dominated by small-angle rotations. Further, we perform a detailed theoretical and numerical study of quasi-probabilities, which can further reduce the total T cost of large circuits by orders of magnitude with only a small overhead in sample complexity. We also develop a scheme based on quasi-probability mixtures of Clifford+T fallback channels. We derive new $θ$-dependent formulas that can be used for resource estimation of fault-tolerant quantum algorithms. As an application of our results, we show that the gate cost of Trotterization circuits compiled to a Clifford+T gate set is constant in the small Trotter step size limit, and can be reduced by orders of magnitude even for large step sizes. The cost of fault-tolerant Trotterization for a variety of applications should be re-examined in light of these results. Our work dispels the widely-stated claim that Clifford+T rotation synthesis has a high cost independent of $θ$, and further develops a scalable quasi-probability method for rotation synthesis. We also expect our results to bring forward useful early fault-tolerant quantum computing by reducing required magic state resources. |
| title | More efficient Clifford+T synthesis for small-angle rotations and application to Trotterization |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2605.31544 |