Update note quantum30/Quantum Logic Gates.md

src/site/notes/quantum30/Quantum Logic Gates.md

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+---
+{"dg-publish":true,"permalink":"/quantum30/quantum-logic-gates/"}
+---
+
+## Not gate / X gate 
+
+$$
+{NOT|0\rangle = |1\rangle}
+$$
+$$
+{NOT|1\rangle = |0\rangle}
+$$
+$$
+{NOT( a|0\rangle + b|1\rangle ) = b|0\rangle + a|1\rangle}
+$$
+$$
+{X} =\begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix}
+$$
+
+## 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
+
+$$
+H|0\rangle = \dfrac{|0\rangle + |1\rangle}{\sqrt2}
+$$
+$$
+H|1\rangle = \dfrac{|0\rangle - |1\rangle}{\sqrt2}
+$$
+$$
+H(\alpha|0\rangle + \beta|1\rangle) = \alpha(\dfrac{|0\rangle + |1\rangle}{\sqrt2}) + \beta(\dfrac{|0\rangle - |1\rangle}{\sqrt2})
+= \dfrac{\alpha + \beta}{\sqrt2}|0\rangle + \dfrac{\alpha - \beta}{\sqrt2}|1\rangle
+$$
+$$
+{H} = \dfrac{1}{\sqrt2}\begin{bmatrix} 1 & 1 \\ 1 & -1\end{bmatrix}
+$$
+
+### cancellation of $|1\rangle$ and reinforcement of $|0\rangle$ 
+$$
+\begin{aligned}
+&H(H|0\rangle) = H(\dfrac{|0\rangle + |1\rangle}{\sqrt2}) \\
+&                        =  \dfrac{1}{\sqrt2}(\dfrac{|0\rangle + |1\rangle}{\sqrt2}) +  \dfrac{1}{\sqrt2}(\dfrac{|0\rangle - |1\rangle}{\sqrt2}) \\
+& = \dfrac{1}{\sqrt2}(\dfrac{|0\rangle}{\sqrt2} + \dfrac{|0\rangle}{\sqrt2}) \\
+& = |0\rangle
+\end{aligned}
+$$