Controlled operations are fundamental building blocks of quantum algorithms. Decomposing n-control-NOT gates (Cn(X)) into arbitrary single-qubit and CNOT gates, is a crucial but non-trivial task. This study introduces Cn(X) circuits outperforming previous methods in the asymptotic and non-asymptotic regimes. Three distinct decompositions are presented: an exact one using one borrowed ancilla with a circuit depth \(\Theta (\log {(n)}^{3})\) , an approximating one without ancilla qubits with a circuit depth \({{{{{{{\mathcal{O}}}}}}}}(\log {(n)}^{3}\log (1/\epsilon ))\) and an exact one with an adjustable-depth circuit which decreases with the number m≤n of ancilla qubits available as \({{{{{{{\mathcal{O}}}}}}}}(\log {(n/\lfloor m/2\rfloor )}^{3}+\log (\lfloor m/2\rfloor ))\) . The resulting exponential speedup is likely to have a substantial impact on fault-tolerant quantum computing by improving the complexities of countless quantum algorithms with applications ranging from quantum chemistry to physics, finance and quantum machine learning.
