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I am reading Quantum Computation and Quantum Information by Chuang and Nielsen and they claim that it is easy to show that transformation $U_f: \left| x, y \right\rangle \to \left| x, y \oplus f(x) \right\rangle$ is unitary.

My first instinct is to use the condition $U^\dagger U = I$, but I can’t figure out how to construct the matrix $U_f$. Is there another way?

Norbert Schuch
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Andrej
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2 Answers2

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As it happens, this $U_f$ is its own inverse. So to show that it is unitary, you need to show that

  1. $U_fU_f=I$

  2. $U_f^\dagger = U_f$

(that is, $U_f$ is its own inverse).

Both of these are straightforward (1. ist just arithmetics, for 2. you can use a change of variables).

Norbert Schuch
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First notice that a transformation is unitary if and only if it sends an orthonormal basis to an orthonormal basis. Then conclude that every transformation which permutes the computational basis, such as $U_f$, is unitary.

Adam Zalcman
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