# If CNOTs and single qubit gates are universal then why do we need to prove that controlled U operations can be composed by them as well?

In the book by Chuang and Nielsen they prove that controlled U operations can be made out of CNOTs and single qubit gates. But then they go on to prove that they are universal by showing that every n by n matrix can be decomposed into two level matrices and then to CNOTs and single qubit gates. But if so, then why can't we prove this way that controlled U can be too, decomposed to them. Since a controlled U is after all an n by n matrix. Why is there a separate proof for them?

• Because it might be more insightful? These gates might be more relevant? Because the construction you get is simpler? Or just for educational purposes? This is a bit like asking why a math textbook first proves Cauchy-Schwarz and then later Hölder's inequality, given that the former is a special case. Feb 17, 2019 at 21:15
• But is this indeed the case here? They give a pretty lengthy and complicated proof for this . What's the point if later it can be proven by the universality. Why do they have to show controlled U and not other special cases? Feb 18, 2019 at 9:50
• I thought, but I am not sure. Maybe their universality proof which is based on factorization of unitary into two level matrices doesn't hold for controlled U just like it doesn't hold for CNOT Feb 18, 2019 at 9:53
• I don't think the comparison to Cauchy Schwar is valid. As it's a theorem. But here what's the point Feb 19, 2019 at 13:37
• Unlike the universality result, which is a random claim? Feb 19, 2019 at 14:35