# 量子计算（一）

（1）量子非门（Quantum NOT Gate）是把 α0|0⟩+α1|1⟩ 映射成 α1|0⟩+α0|1⟩，也就是把 α0 和 α1 交换顺序。

（2）Quantum Controlled NOT Gate 是把 α0|00⟩+α1|01⟩+α2|10⟩+α3|11⟩ 映射成 α0|00⟩+α1|01⟩+α3|10⟩+α2|11⟩，也就是把 α2 和 α3 交换顺序。

（3）一个很著名的计算单元是 Hadamard Gate，输入 α0|0⟩+α1|1⟩，输出 2^{-1/2}(α0+α1)|0⟩+2^{-1/2}(α0-α1)|1⟩ 。Hadamard Gate 就是把经典的状态 |0⟩ 和 |1⟩ 转换成 |0⟩ 和 |1⟩ 的“halfway” 状态。不要小看这个操作，即使仅仅对 n 个量子比特中的第一位进行了 Hadamard gate 运算，所有的 2^{n} 个系数都会改变，证明如下：

Theorem. (No-Cloning Theorem, Wootters and Zurek, Dieks) An unknown quantum system cannot be cloned by unitary transformations.

Proof. By contradiction, there exists a unitary transformation U that makes a clone of a quantum system. That means for any state $|\varphi\rangle$,

$U: |\varphi 0\rangle \rightarrow |\varphi\varphi\rangle$.

Consider two linear independent states $|\varphi\rangle$ and $|\phi\rangle$. Then we have $U|\varphi 0\rangle=|\varphi\varphi\rangle$, $U|\phi 0\rangle=|\phi\phi\rangle$ from the assumption of U. Let $|\psi\rangle=\frac{1}{\sqrt{2}}(|\varphi\rangle +|\phi\rangle)$, we get

$U|\psi 0\rangle=\frac{1}{\sqrt{2}}(U|\varphi 0\rangle+U|\phi 0\rangle)$.

However,

$U|\psi 0\rangle=|\psi\psi\rangle=\frac{1}{2}(|\varphi\varphi\rangle+|\varphi\phi\rangle+|\phi\varphi\rangle+|\phi\phi\rangle)$

which contradicts the previous result. Therefore, there does not exist a unitary cloning transformation.