在阅读该页内容之前,我们向量子计算的开创者费曼和Deutsch致敬,同时向三位量子信息学的奠基人Charles H. Bennett, David Deutsch, Peter Shor表示敬意.
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
运用数学归纳法，$n=1$已由题目给出；
假设$n$的情况已知，则$n+1$的情况为$ H \otimes H^{\otimes n}=\frac{1}{\sqrt{2^{n+1}}} \sum_{x y}(-1)^{x \cdot y}(|0 x\rangle\langle 0 y|+| 1 x\rangle\langle 0 y|+| 0 x\rangle\langle 1 y|-| 1 x\rangle\langle 1 y|)= H^{\otimes n+1}$；
$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
参与者
作者：HKL, W65
贡献者:Dingyan, Wjw,Wxw
