2

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?

bilanush
  • 861
  • 7
  • 12
  • 2
    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. – Norbert Schuch Feb 17 '19 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? – bilanush Feb 18 '19 at 09:50
  • 1
    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 – bilanush Feb 18 '19 at 09:53
  • I don't think the comparison to Cauchy Schwar is valid. As it's a theorem. But here what's the point – bilanush Feb 19 '19 at 13:37
  • Unlike the universality result, which is a random claim? – Norbert Schuch Feb 19 '19 at 14:35

1 Answers1

2

Constructing controlled-U out of single qubit rotations and cNOT is part of the proof of universality of single qubit rotations and cNOT.

The bit of Nielsen & Chuang that you're referring to decomposes an arbitrary unitary in terms of gates such as controlled-controlled-....-controlled-U. See, for example, Fig 4.16 (P. 193 of 2002 printing). But that gate is built out of controlled-U. See, for example, Fig. 4.10 (P. 184). (although Exercise 4.28 gives you a different construction without work qubits). So, you need to construction of controlled-U for the whole thing to work.

DaftWullie
  • 57,689
  • 3
  • 46
  • 124
  • Not sure what you mean by figure 4.16 and 4.10. In which pages? I don't find them. According to what I see their proof is based on decomposition to two level matrices and then , each of them they decompose to single qubit matrices and CNOTs. – bilanush Feb 18 '19 at 10:10
  • 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 – bilanush Feb 18 '19 at 10:11
  • what do you understand by "two level matrices"? – DaftWullie Feb 18 '19 at 11:47
  • One would have assumed that the OP would have checked that this is not used in the proof ... – Norbert Schuch Feb 18 '19 at 16:13
  • I am sorry, daftwullie but I do think it isn't used in the proof. So what you said is wrong. As to two level matrix is a matrix which acts non trivially only on two elements – bilanush Feb 19 '19 at 13:33
  • 1
    @bilanush So, according to you, how do N&C go from having a two-level matrix to decomposing it in terms of cNOT and single qubit unitaries? – DaftWullie Feb 19 '19 at 13:59
  • Ha ok. Sorry. So you say they are using a single U operation which is controlled by other qubits. That's a legit answer , upvoted! But don't you think there might be something deeper? I mean, you disagree with me that controlled U operations are not included in the first decomposition to a multiple of two level matrices? Just like CNOT isn't included. CNOT is a 4x4 matrix so this too should have been included in the proof. It isn't. So why couldn't be that U operations aren't included either? – bilanush Feb 19 '19 at 19:19