The Hadamard Operator on one qubit is:

\begin{align*}
H = \frac{1}{\sqrt{2}}\left[(|0\rangle + |1\rangle)\langle 0|+(|0\rangle - |1\rangle)\langle 1|\right]
\end{align*}

Show that:
\begin{align*}
H^{\otimes n} = \frac{1}{\sqrt{2^n}}\sum_{x,y}(-1)^{x \cdot y}|x\rangle \langle y|
\end{align*}

I can evaluate things like $H \otimes H$ in practice, but I don't know how to get a general formula for $H^{\otimes n}$. Are there any tricks I could use?

This post imported from StackExchange Physics at 2014-03-09 09:18 (UCT), posted by SE-user user82235