# How to transform $|00\rangle$ into $\frac{1}{2}(|00⟩ + |01⟩ + |10⟩ + |11⟩)$?

I am quite new to Quantum Computing. If we have an input-state $$|00⟩$$. How could you transform it into the state $$\frac{1}{2}(|00⟩ + |01⟩ + |10⟩ + |11⟩)$$

Consider the state $$\frac{1}{2}(|00⟩ + |01⟩ + |10⟩ + |11⟩)$$. This admits a factorisation \begin{align*}\frac{1}{2}(|00⟩ + |01⟩ + |10⟩ + |11⟩)&=\frac12\{|0\rangle\otimes(|0\rangle+|1\rangle)+|1\rangle\otimes(|0\rangle+|1\rangle)\}\\&=\frac12\{(|0\rangle+|1\rangle)\otimes(|0\rangle+|1\rangle)\}\\&=\left(\frac{|0\rangle+|1\rangle}{\sqrt2}\right)\otimes \left(\frac{|0\rangle+|1\rangle}{\sqrt2}\right)\\&=H|0\rangle\otimes H|0\rangle\end{align*} where $$H$$ is the Hadamard gate and $$\otimes$$ is the tensor product.
Thus all you have to do is apply parallel Hadamard gates to two copies of $$|0\rangle$$ and the resulting state is the one you desire.
More compactly $$H(|0\rangle)\otimes H(|0\rangle)=(H\otimes H)(|0\rangle\otimes|0\rangle)=H_2|00\rangle$$ where $$H_2=H \otimes H = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \otimes \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} = \frac{1}{2} \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1 \end{bmatrix}$$