Exercise 2.52#
The Hadamard gate is defined as
(1)#\[\begin{split}
H = \frac{1}{\sqrt{2}}\begin{pmatrix}
1 & 1 \\
1 & -1
\end{pmatrix}
\end{split}\]
Thus, we could calculate
(2)#\[\begin{split}
H^2 = \frac{1}{2}\begin{pmatrix}
1 & 1 \\
1 & -1
\end{pmatrix}\begin{pmatrix}
1 & 1 \\
1 & -1
\end{pmatrix} = \frac{1}{2}\begin{pmatrix}
2 & 0 \\
0 & 2
\end{pmatrix} = I
\end{split}\]