Resource-efficient and high-precision approximate synthesis of quantum circuits expressed in the Clifford+T gate set is vital for Fault-Tolerant quantum computing.Efficient optimal methods are known for single-qubit R Z unitaries, otherwise the problem is generally intractable.Search-based methods, like simulated annealing, empirically generate low resource cost approximate implementations of general multi-qubit unitaries so long as low precision (Hilbert-Schmidt distances of ε ≥ 10 -2 ) can be tolerated.These algorithms build up circuits that directly invert target unitaries.We instead leverage search-based methods to first approximately diagonalize a unitary, then perform the inversion analytically.This lets difficult continuous rotations be bypassed and handled in a postprocessing step.Our approach improves both the implementation precision and run time of synthesis algorithms by orders of magnitude when evaluated on unitaries from real quantum algorithms.On benchmarks previously synthesizable only with analytical techniques like the Quantum Shannon Decomposition, diagonalization uses an average of 95% fewer non-Clifford gates.
Resource-efficient and high-precision approximate synthesis of quantum circuits expressed in the Clifford+T gate set is vital for Fault-Tolerant quantum computing.Efficient optimal methods are known for single-qubit R Z unitaries, otherwise the problem is generally intractable.Search-based methods, like simulated annealing, empirically generate low resource cost approximate implementations of general multi-qubit unitaries so long as low precision (Hilbert-Schmidt distances of ε ≥ 10 -2 ) can be tolerated.These algorithms build up circuits that directly invert target unitaries.We instead leverage search-based methods to first approximately diagonalize a unitary, then perform the inversion analytically.This lets difficult continuous rotations be bypassed and handled in a postprocessing step.Our approach improves both the implementation precision and run time of synthesis algorithms by orders of magnitude when evaluated on unitaries from real quantum algorithms.On benchmarks previously synthesizable only with analytical techniques like the Quantum Shannon Decomposition, diagonalization uses an average of 95% fewer non-Clifford gates.
We present a simple algorithm that implements an arbitrary n -qubit unitary operator using a Clifford + T circuit with T-count O ( 2 4 n / 3 n 2 / 3 ) and ancilla-count O ( 2 2 n / 3 n 1 / 3 ) . The previous best T-count for unitary synthesis was O ( 2 3 n / 2 n ) with O ( 2 n / 2 ) ancillae by [Low , ]. There is a fundamental tradeoff between T gates and ancillae, and our algorithm extends the achievable regime of this tradeoff, reducing the exponent in T-count from 3 n / 2 to 4 n / 3 . The best known lower bound remains Ω ( 2 n ) , so the optimal T-count for general unitary synthesis is still open. Our construction is based on a recursive application of the cosine-sine decomposition, together with a generalization of the optimal diagonal unitary synthesis method by [D. Gosset , ] to multi-controlled k -qubit unitaries.
Approximate synthesis of general single-qubit unitaries over the Clifford+T gate set
For the standard Clifford+T gate set, deterministic, ancilla-free synthesis now attains the minimal T-count for general single-qubit unitaries (Morisaki et al., arXiv:2510.05816). The T gate rotates by half the angle of T, 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 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 to 10−8, the cost of Clifford+T circuits scales as 2.4log2(1/ε) compared to 3.0log2(1/ε) for T-count-optimal Clifford+T circuits. Once a one-time catalyst state is amortized, the Clifford+T circuits are never costlier than their Clifford+T counterparts.
We developed a general framework for synthesizing target gates by using a finite set of basic gates, which is a crucial step in quantum compilation. When approximating a gate in SU(n), a naive brute-force search requires a computational complexity of O(1/ε(n2−1)) to achieve an approximation with error ε. In contrast, by using our method, the complexity can be reduced to O(−n2logε/ε((n2−1)/2)). This method requires almost no assumptions and can be applied to a variety of gate sets, including Clifford+T and Clifford+V. Further, we introduce a suboptimal but short run-time algorithm for synthesizing multi-qubit controlled gates. This approach highlights the role of subgroup structures in reducing synthesis complexity and opens a new direction of study on the compilation of multi-qubit gates. The framework is broadly applicable to different universal gate sets, and our analysis suggests that it can serve as a foundation for resource-efficient quantum compilation in near-term architectures.
We show how to directly and efficiently approximate arbitrary one-qubit unitaries, bypassing the Euler decomposition and the magnitude approximation problem, at the cost of one ancillary qubit. Our technique also applies to approximating unitaries with multi-qubit gate sets such as Clifford and CS, or Clifford and CCZ, as well as to approximating orthogonal matrices using multi-qubit gate sets such as Real Clifford and CCZ. The key tools are repeat-until-success circuits, lattice-based exact synthesis algorithms, integer point enumeration in convex sets, and relative norm equations.
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(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), 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.
Abstract In quantum computing there are quite a few universal gate sets, each having their own characteristics. In this paper we study the Clifford+CS universal fault-tolerant gate set. The CS gate is used is many applications and this gate set is an important alternative to Clifford+T. We introduce a generating set in order to represent any unitary implementable by this gate set and with this we derive a bound on the CS-count of arbitrary multi-qubit unitaries. Analysing the channel representation of the generating set elements, we infer JnCS⊂JnT J n CS ⊂ J n T , where JnCS J n CS and JnT J n T are the set of unitaries exactly implementable by the Clifford+CS and Clifford+T gate sets, respectively. We develop CS-count optimal synthesis algorithms for both approximately and exactly implementable multi-qubit unitaries. With the help of these we derive a CS-count-optimal circuit for Toffoli, implying JnTof=JnCS J n Tof = J n CS , where JnTof J n Tof is the set of unitaries exactly implementable by the Clifford+Toffoli gate set. Such conclusions can have an important impact on resource estimates of quantum algorithms.