Polylogarithmic-depth controlled-NOT gates without ancilla qubits.

Controlled operations are fundamental building blocks of quantum algorithms. Decomposing n-control-NOT gates (C(X)) into arbitrary single-qubit and CNOT gates, is a crucial but non-trivial task. This study introduces C(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 , an approximating one without ancilla qubits with a circuit depth and an exact one with an adjustable-depth circuit which decreases with the number m≤n of ancilla qubits available as . 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.