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IEEE International Conference on Quantum Computing and Engineering (QCE)

LCU-Based Synthesis of Rotation Gates for Fault-Tolerant Quantum Computing

Soshun Naito, Yasunari Suzuki, Yuuki Tokunaga

Published Aug 30, 2025• 0 citations
Publisher

Abstract

Implementing arbitrary-angle rotation gates is a fundamental task in Fault-Tolerant Quantum Computing (FTQC). Over the past decade, researchers have focused on minimizing the$\mathbf{T}$-count in the Clifford+T decomposition based on the Solovay-Kitaev theorem. Recently, however, the advances in T-gate preparation techniques known as the zero-level distillation have significantly reduced the cost of$T$gates, and the main research focus has been shifted to simpler and more efficient synthesis protocols. In this paper, we propose a method to prepare the magic state for arbitrary-angle$\mathbf{R Z}$gates using the Linear Combination of Unitaries (LCU) framework, where the RZ gate is represented as a linear combination of$I$and iZ with positive integer coefficients. The unique feature of our protocol is that for angles whose tangent is rational, our protocol can prepare the magic state with no approximation error. Using this property, our protocol is expected to improve the accuracy of high-precision quantum algorithms. Moreover, the number of Clifford+T gates is proportional to the number of ancilla qubits, which is the same order as the previous methods. We also address the issue of the probabilistic behavior of the magic state preparation and injection and show that these issues can be mitigated by parallelizing the preparation process and combining our protocol with other approximation methods.

DOI
10.1109/qce65121.2025.10432

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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 θθθ, requiring O(log⁡1/δ)O(\log 1/δ)O(log1/δ) T gates. We show that it is possible to do much better for small angles, reducing the T cost to O~(θ2/δ)\tilde{O}(θ^2/δ)O~(θ2/δ), and returning to existing O(log⁡1/δ)O(\log1/δ)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 θθθ-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 θθθ, 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.
Quantum error correction is essential for achieving practical quantum computing but has a significant computational overhead. Among fault-tolerant (FT) gate operations, non-Clifford gates, such as T, are particularly expensive due to their reliance on magic state distillation. These costly T gates appear frequently in FT circuits as many quantum algorithms require arbitrary single-qubit rotations, such as Rx and Rz gates, which must be decomposed into a sequence of T and Clifford gates. In many quantum circuits, Rx and Rz gates can be fused to form a single U3 unitary. However, existing synthesis methods, such as gridsynth, rely on indirect decompositions, requiring separate Rz decompositions that result in a threefold increase in T count.
Approximate synthesis of general single-qubit unitaries over the Clifford+T\sqrt{T}T​ gate set
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.
We introduce STAR-magic mutation, an efficient protocol for implementing logical rotation gates on early fault-tolerant quantum computers. This protocol judiciously combines two of the latest state preparation protocols: transversal multi-rotation protocol and magic state cultivation. It achieves a logical rotation gate with a favorable error scaling of O(θL2(1−Θ(1/d))pph)\mathcal{O}(θ_L^{2(1-Θ(1/d))}p_{\text{ph}})O(θL2(1−Θ(1/d))​pph​), while requiring only the ancillary space of a single surface code patch. Here, θLθ_LθL​ is the logical rotation angle, pphp_{\text{ph}}pph​ is the physical error rate, and ddd is the code distance. This scaling marks a significant improvement over the previous state-of-the-art, O(θLpph)\mathcal{O}(θ_L p_{\text{ph}})O(θL​pph​), making our protocol particularly powerful for implementing a sequence of small-angle rotation gates, like Trotter-based circuits. Notably, for θL≲10−5θ_L \lesssim 10^{-5}θL​≲10−5, our protocol achieves a two-order-of-magnitude reduction in both the execution time and the error rate of analog rotation gates compared to the standard TTT-gate synthesis using cultivated magic states. Building upon this protocol, we also propose a novel quantum computing architecture designed for early fault-tolerant quantum computers, dubbed ``STAR ver.~3". It employs a refined circuit compilation strategy based on Clifford+TTT+φφφ gate set, rather than the conventional Clifford+TTT or Clifford+φφφ gate sets. We establish a theoretical bound on the feasible circuit size on this architecture and illustrate its capabilities by analyzing the spacetime costs for simulating the dynamics of quantum many-body systems. Specifically, we demonstrate that our architecture can simulate biologically-relevant molecules or lattice models at scales beyond the reach of exact classical simulation, with only a few hundred thousand physical qubits, even assuming a realistic error rate of pph=10−3p_{\text{ph}}=10^{-3}pph​=10−3.
For universal quantum computation, a major challenge to overcome for practical implementation is the large amount of resources required for fault-tolerant quantum information processing. An important aspect is implementing arbitrary unitary operators built from logical gates within the quantum error correction code. A synthesis algorithm can be used to approximate any unitary gate up to arbitrary precision by assembling sequences of logical gates chosen from a small set of universal gates that are fault-tolerantly performable while encoded in a quantum error-correction code. However, current procedures do not yet support individual assignment of base gate costs and many do not support extended sets of universal base gates. We analysed cost-optimal sequences using an exhaustive search based on Dijkstra’s pathfinding algorithm for the canonical Clifford+ T set of base gates and compared them to when additionally including Z -rotations from higher orders of the Clifford hierarchy. Two approaches of assigning base gate costs were used. First, costs were reduced to T -counts by recursively applying a Z -rotation catalyst circuit. Second, costs were assigned as the average numbers of raw (i.e. physical level) magic states required to directly distil and implement the gates fault-tolerantly. We found that the average sequence cost decreases by up to 54±3% when using the Z -rotation catalyst circuit approach and by up to 33±2% when using the magic state distillation approach. In addition, we investigated observed limitations of certain assignments of base gate costs by developing an analytic model to estimate the proportion of sets of Z -rotation gates from higher orders of the Clifford hierarchy that are found within sequences approximating random target gates.