# 问题：

The Hadamard operator on one qubit may be written as $H=\frac{1}{\sqrt{2}}[(|0\rangle+|1\rangle)\langle 0|+(|0\rangle-|1\rangle)\langle 1|]$;

Show explicitly that the Hadamard transform on $n$ $qubits$, $H^{⊗n}$, may be written as$H^{\otimes n}=\frac{1}{\sqrt{2^{n}}} \sum_{x, y}(-1)^{x \cdot y}|x\rangle\langle y|$;

Write out an explicit matrix representation for $H^{⊗2}$;

## 解答

$H^{⊗2}= H \otimes H=\frac{1}{\sqrt{2}}\left[\begin{array}{ll}1 * H & 1 * H \\ 1 * H & -1 * H\end{array}\right]=\frac{1}{2}\left[\begin{array}{cccc}1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1\end{array}\right]$.

#### 参考

[1]www.qtumist.com