* Qubit 4: HADAMARD-Gate (10 min)

Construct a gate (2 by 2 matrix) $\hat{H}$ that shifts the basis vectors into superpositions

$$\vert\rangle \to \frac{1}{\sqrt{2}}\left(\vert\rangle + \vert 1\r... ...ert 1\rangle \to \frac{1}{\sqrt{2}}\left(\vert\rangle - \vert 1\rangle \right).$$ (32)

Write down the explicit form of $\hat{H}$.