Controlled gates are key components in various quantum algorithms. Improving on the prior work of Gosset et al., we show that, for an allowed error $\varepsilon$, $3\log_2(1/\varepsilon) + o(\log(1/\varepsilon))$ $T$ gates are sufficient to approximate most multi-qubit controlled SU(2)s. We also show that this T-count matches the lower bound when the use of an almost controlled gate is prohibited. As an application, general controlled gate synthesis and efficient SU(4) gate synthesis are given.
Controlled gates are key components in various quantum algorithms. Improving on the prior work of Gosset et al., we show that, for an allowed error ε, 3log2(1/ε)+o(log(1/ε))T gates are sufficient to approximate most multi-qubit controlled SU(2)s. We also show that this T-count matches the lower bound when the use of an almost controlled gate is prohibited. As an application, general controlled gate synthesis and efficient SU(4) gate synthesis are given.
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 study ancilla-free approximation of single-qubit unitaries U∈SU(2) by gate sequences over Clifford+G, where G∈{T,V} or their generalization. Let p denote the characteristic factor of the gate set (e.g., p=2 for G=T and p=5 for G=V). We prove three asymptotic bounds on the minimum G-count required to achieve approximation error at most ε. First, for Haar-almost every U, we show that 3logp(1/ε)G-count is both necessary and sufficient; moreover, probabilistic synthesis improves the leading constant to 3/2. Second, for unitaries whose ratio of matrix elements lies in a specified number field, 4logp(1/ε)G-count is necessary. Again, the leading constant can be improved to 2 by probabilistic synthesis. Third, there exist unitaries for which the G-count per logp(1/ε) fails to converge as ε→0+. These results partially resolve a generalized form of the Ross--Selinger conjecture.
T-efficient unitary synthesis via block-sparse Hamiltonian simulation
We construct Clifford+T circuits implementing arbitrary n-qubit unitaries with worst-case T-count O(N8/7) at constant operator-norm error, where N=2n, improving the previous exponent 5/4. The construction uses O(N6/7) clean ancillas. The main idea is to reduce the data addressed by each query to the block-encoding circuit for a sparse piece while controlling the additional simulation work needed to combine the pieces. We simulate a Hermitian logarithm of the target by flattening its traceless part with a two-layer random similarity transformation, randomly partitioning the transformed generator into block-sparse Hermitian pieces, and combining their evolutions using qDRIFT. For a target operator-norm error ε∈(0,1/2), let L=log2(N/ε), the qubit count plus the requested bits of accuracy. By successively correcting the approximation error, we obtain a circuit using O(N8/7L3/7+NL)T gates and O(N) clean ancillas whenever L≤N. We also prove that, at constant precision, the exponent 8/7 is optimal within a sequential block-SELECT model.
We determine the non-Clifford T-gate cost of constructing block encodings of structured fermionic and spin Hamiltonians in a unitary Clifford+T model, when arbitrarily many clean ancillas and unrestricted block-encoding subnormalization are allowed, but without mid-circuit measurements or classical feed-forward. Our main technical tool is an ancilla-compression theorem: any block encoding of an n-qubit operator with a clean ancillas and at most sT gates can be compressed to use at most min{a,n+2s} ancillas, without increasing the absolute error or T-count. For general second-quantized Hamiltonians with bounded one- and two-body coefficients, at operator-norm block-encoding error ε, a volume-covering argument combined with circuit counting gives the worst-case lower bound Ω(n2log(n4/ε)), matching the existing upper bound at fixed precision. For the bond-dependent Kitaev honeycomb family on n spins, we obtain independent lower bounds Ω(n) from stabilizer nullity and Ω(log(1/ε)) from one-qubit state preparation, established using different Hamiltonian instances. Together with an explicit LCU construction, they give the tight worst-case scaling Θ(n+log(1/ε)). As an application, we evaluate the T-count of a Hamiltonian simulation circuit based on quantum singular value transformation, with each block-encoding query compiled separately. When phase synthesis and controlled queries add at most constant-factor overhead, the simulation T-count scales as the query count times the optimal T-count per query.
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.
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 ε−1/2−o(1) and ε−1/4−o(1), respectively, for an approximation error ε. Although this complexity is exponential in the input size, we demonstrate that our algorithms run in practical time at ε≈10−15 and ε≈10−22, respectively. Furthermore, we show that, for most single-qubit unitaries, the deterministic synthesis algorithm requires at most 3log2(1/ε)+o(log2(1/ε))T-gates, and the probabilistic synthesis algorithm requires at most 1.5log2(1/ε)+o(log2(1/ε))T-gates. Remarkably, complexity analyses in this work do not rely on any numerical or number-theoretic conjectures.