1. The problem statement, all variables and given/known data I am supposed to construct a controlled Hadamard gate using only single qubit and CNOT gates. 2. Relevant equations We know that any arbitrary unitary Operator U can be written as the Martrix product U=AXBXC, where X is the NOT-Matrix and ABC=1 (identity matrix) I've already shown that any arbitrary controlled operator can be written as CU=Cphase* (A⊗1)*CNOT*(B⊗1)*CNOT*(C⊗1), with ABC=1 The other relevant equations are the standard equations from quantum mechanics 3. The attempt at a solution I've tried everything, LU-Decomposition, QR, Diagonalization etc. I've read something about Sine-Cosine-Decomposition, I think this might be the right direction. I've also read something about Lie-Groups, though unfortunately, the Hadamard gate is not part of SO(2), only of O(2), there's way more explanation on the generators of SO (2), especially SU (2), and unfortunately I don't know that much about group theory.