## Nielsen and Chuang Exercise 2.33

Exercise 2.33The Hadamard operator on one qubit may be written as:Show explicitly that the Hadamard transform on n qubits, , may be written as

Write out an explicit matrix representation for .

The matrix representation for is given by

We will prove that inductively. First, for ,

which has the proper form. Next, given , we can show that

which completes the proof.

I cannot think of any way to prove this other than inductively. If anyone has a way, please leave a comment.

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