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bilanush
bilanush
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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 tointo 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?

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 to 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?

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?

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Sanchayan Dutta
Sanchayan Dutta
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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 to 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.?

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 to 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.

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 to 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?

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bilanush
bilanush
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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 to 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.