More efficient Clifford+T synthesis for small-angle rotations and application to Trotterization

• URL: http://arxiv.org/abs/2605.31544v1
• Date: Fri, 29 May 2026 16:59:34 GMT
• Title: More efficient Clifford+T synthesis for small-angle rotations and application to Trotterization
• Abstract summary: We show that it is possible to do much better for small angles, reducing the T cost to $tilde O(2/)$, and returning to existing $O(log1/)$ results in the worst case.<n>Our work dispels the widely-stated claim that Clifford+T rotation synthesis has a high cost independent of $$, and further develops a scalable quasi-probability method for rotation synthesis.
• Score: 1.3048920509133806
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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