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arXiv:2510.05816

Optimal ancilla-free Clifford+T synthesis for general single-qubit unitaries

Hayata Morisaki, Sano, Kaoru, Seiseki Akibue

Published Oct 7, 2025• 0 citations
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Abstract

We propose two Clifford+$T$ synthesis algorithms that are optimal with respect to $T$-count. The first algorithm, called deterministic synthesis, approximates any single-qubit unitary by a single-qubit Clifford+$T$ circuit with the minimum $T$-count. The second algorithm, called probabilistic synthesis, approximates any single-qubit unitary by a probabilistic mixture of single-qubit Clifford+$T$ circuits with the minimum $T$-count. For most of single-qubit unitaries, the runtimes of deterministic synthesis and probabilistic synthesis are $\varepsilon^{-1/2 - o(1)}$ and $\varepsilon^{-1/4 - o(1)}$, respectively, for an approximation error $\varepsilon$. Although this complexity is exponential in the input size, we demonstrate that our algorithms run in practical time at $\varepsilon \approx 10^{-15}$ and $\varepsilon \approx 10^{-22}$, respectively. Furthermore, we show that, for most single-qubit unitaries, the deterministic synthesis algorithm requires at most $3\log_2(1/\varepsilon) + o(\log_2(1/\varepsilon))$ $T$-gates, and the probabilistic synthesis algorithm requires at most $1.5\log_2(1/\varepsilon) + o(\log_2(1/\varepsilon))$ $T$-gates. Remarkably, complexity analyses in this work do not rely on any numerical or number-theoretic conjectures.

DOI
10.48550/arxiv.2510.05816

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We propose two Clifford+TTT synthesis algorithms that are optimal with respect to TTT-count. The first algorithm, called deterministic synthesis, approximates any single-qubit unitary by a single-qubit Clifford+TTT circuit with the minimum TTT-count. The second algorithm, called probabilistic synthesis, approximates any single-qubit unitary by a probabilistic mixture of single-qubit Clifford+TTT circuits with the minimum TTT-count. For most of single-qubit unitaries, the runtimes of deterministic synthesis and probabilistic synthesis are ε−1/2−o(1)\varepsilon^{-1/2 - o(1)}ε−1/2−o(1) and ε−1/4−o(1)\varepsilon^{-1/4 - o(1)}ε−1/4−o(1), respectively, for an approximation error ε\varepsilonε. Although this complexity is exponential in the input size, we demonstrate that our algorithms run in practical time at ε≈10−15\varepsilon \approx 10^{-15}ε≈10−15 and ε≈10−22\varepsilon \approx 10^{-22}ε≈10−22, respectively. Furthermore, we show that, for most single-qubit unitaries, the deterministic synthesis algorithm requires at most 3log⁡2(1/ε)+o(log⁡2(1/ε))3\log_2(1/\varepsilon) + o(\log_2(1/\varepsilon))3log2​(1/ε)+o(log2​(1/ε)) TTT-gates, and the probabilistic synthesis algorithm requires at most 1.5log⁡2(1/ε)+o(log⁡2(1/ε))1.5\log_2(1/\varepsilon) + o(\log_2(1/\varepsilon))1.5log2​(1/ε)+o(log2​(1/ε)) TTT-gates. Remarkably, complexity analyses in this work do not rely on any numerical or number-theoretic conjectures.
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For the standard Clifford+TTT gate set, deterministic, ancilla-free synthesis now attains the minimal TTT-count for general single-qubit unitaries (Morisaki et al., arXiv:2510.05816). The T\sqrt{T}T​ gate rotates by half the angle of TTT, generating a finer lattice of implementable operations. Access to this magic state is expected to lower the cost of deterministic and ancilla-free synthesis of general single-qubit unitaries (Kliuchnikov et al., arXiv:2203.10064), but no direct Clifford+T\sqrt{T}T​ algorithm existed for this case. We provide one by extending the integer lattice-point enumeration method of Morisaki et al. We adopt a resource state cost model based on the magic-state catalysis approach of Gidney and Fowler (arXiv:1812.01238). On Haar-random targets synthesized to precisions ranging from ε=10−3\varepsilon=10^{-3}ε=10−3 to 10−810^{-8}10−8, the cost of Clifford+T\sqrt{T}T​ circuits scales as 2.4log⁡2(1/ε)2.4\log_2(1/\varepsilon)2.4log2​(1/ε) compared to 3.0log⁡2(1/ε)3.0\log_2(1/\varepsilon)3.0log2​(1/ε) for TTT-count-optimal Clifford+TTT circuits. Once a one-time catalyst state is amortized, the Clifford+T\sqrt{T}T​ circuits are never costlier than their Clifford+TTT counterparts.
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We study ancilla-free approximation of single-qubit unitaries U∈SU(2)U\in {\rm SU}(2)U∈SU(2) by gate sequences over Clifford+GGG, where G∈{T,V}G\in\{T,V\}G∈{T,V} or their generalization. Let ppp denote the characteristic factor of the gate set (e.g., p=2p=2p=2 for G=TG=TG=T and p=5p=5p=5 for G=VG=VG=V). We prove three asymptotic bounds on the minimum GGG-count required to achieve approximation error at most ε\varepsilonε. First, for Haar-almost every UUU, we show that 3log⁡p(1/ε)3\log_{p}(1/\varepsilon)3logp​(1/ε) GGG-count is both necessary and sufficient; moreover, probabilistic synthesis improves the leading constant to 3/23/23/2. Second, for unitaries whose ratio of matrix elements lies in a specified number field, 4log⁡p(1/ε)4\log_p(1/\varepsilon)4logp​(1/ε) GGG-count is necessary. Again, the leading constant can be improved to 222 by probabilistic synthesis. Third, there exist unitaries for which the GGG-count per log⁡p(1/ε)\log_{p}(1/\varepsilon)logp​(1/ε) fails to converge as ε→0+\varepsilon\to 0^+ε→0+. These results partially resolve a generalized form of the Ross--Selinger conjecture.
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