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 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.
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 an approach to unitary synthesis that implements an arbitrary n-qubit unitary operator U by a Clifford+T circuit with T-count O(2ndFC(U)), where dFC(U) is the Frobenius norm distance of U to the Clifford group. The T-count is shown to be near-optimal when dFC(U) is a constant. Our approach improves the previous best upper bound O(24n/3) due to Tan (2025) for a large class of unitary operators U as long as dFC(U)≪2n/3.
We study the Clifford+T synthesis of quantum multiplexers U=⨁i=12nUi: an n-qubit control register selects which single-qubit gate Ui∈SU(2) acts on the target qubit. For each block Ui, consider the minimum even T-count needed to approximate it within diamond distance ε without ancillae, and let m be the largest of these costs. For an arbitrary multiplexer U, we construct an approximation of U within the same distance using at most m+O(2n(m+1))T gates. With one control qubit, the cost is m+O(1) and no ancillae are required. We also prove a lower bound that allows the Clifford+T circuit to use any number of clean ancillae initialized to zero and returned exactly to zero. For fixed n and independent Haar-random blocks, the minimum T-count among these circuits lies between (3−δ)log2(1/ε) and (3+δ)log2(1/ε) for every fixed δ>0, except with probability O(εcn,δ) as ε→0. The optimal leading coefficient is therefore the same as for a single Haar-random SU(2) gate. In the lower-bound proof, unitarity forces pairs of off-diagonal blocks describing transitions in opposite directions to satisfy cancellation relations with residuals of order ε2. We combine these relations with the arithmetic of Clifford+T circuits to bound the Haar measure of targets compatible with each pair of off-diagonal blocks. As an application, we obtain ancilla-free approximations of Haar-random SU(4) gates within diamond distance ε, using at most (9+δ)log2(1/ε)T gates for every fixed δ>0, except with probability O(εcδ) with cδ>0.
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.
Unitary synthesis underlies universal quantum computation by translating unitary transformations into circuits of elementary gates. We show that for any budget m≥0 of ancilla, every n-qubit unitary can be exactly implemented using a quantum circuit consisting of O(4n) CNOT and single-qubit gates in O(n+4n/(n+m)) layers attaining the asymptotically optimal depth--ancilla tradeoff while simultaneously achieving optimal circuit size. The key idea is to encode computational basis states as wavefunctions sharing a Gaussian envelope, converting unitary action on amplitudes into coordinate transformations that admit shallow circuit implementations. The continuous encoding is then discretized, with approximation errors removed by exact residual correction. The depth-ancilla tradeoff follows by recursively applying the quantum Shannon decomposition, synthesizing its smaller unitary factors, and reusing ancillary qubits.