Quantum Logic Gates

Not gate / X gate

$$NOT|0⟩=|1⟩$$ $$NOT|1⟩=|0⟩$$ $$NOT(a|0⟩+b|1⟩)=b|0⟩+a|1⟩$$ $$X=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$$

Hadamard gate / H gate

by expanding the range of states that we can access, beyond what is possible on a classical computer, it becomes possible to take shortcuts in our computation

$$H|0⟩=\frac{|0⟩+|1⟩}{\sqrt{2}}$$ $$H|1⟩=\frac{|0⟩-|1⟩}{\sqrt{2}}$$ $$H(\alpha |0⟩+\beta |1⟩)=\alpha (\frac{|0⟩+|1⟩}{\sqrt{2}})+\beta (\frac{|0⟩-|1⟩}{\sqrt{2}})=\frac{\alpha +\beta }{\sqrt{2}}|0⟩+\frac{\alpha -\beta }{\sqrt{2}}|1⟩$$ $$H=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]$$

cancellation of $|1⟩$ and reinforcement of $|0⟩$

$$\begin{array}{rl}& H(H|0⟩)=H(\frac{|0⟩+|1⟩}{\sqrt{2}})\\ & =\frac{1}{\sqrt{2}}(\frac{|0⟩+|1⟩}{\sqrt{2}})+\frac{1}{\sqrt{2}}(\frac{|0⟩-|1⟩}{\sqrt{2}})\\ & =\frac{1}{\sqrt{2}}(\frac{|0⟩}{\sqrt{2}}+\frac{|0⟩}{\sqrt{2}})\\ & =|0⟩\end{array}$$