I just want to clear what seems to be a bit confusing to me with quantum gates and their corresponding output. below are 3 scenarios implementing two hadamard gates
with the first scenario, I am sure the case is $$ H \cdot H = H^2 = \begin{pmatrix} \frac{\sqrt{2}}{2}&\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2}&\frac{-\sqrt{2}}{2} \\ \end{pmatrix}. \begin{pmatrix} \frac{\sqrt{2}}{2}&\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2}&\frac{-\sqrt{2}}{2} \\ \end{pmatrix} = \begin{pmatrix} 1&0 \\ 0&1 \\ \end{pmatrix} = I $$
with the second scenario, I am sure the case is
$$H \otimes H = H_2 = \begin{pmatrix}
\frac{\sqrt{2}}{2}&\frac{\sqrt{2}}{2} \\
\frac{\sqrt{2}}{2}&\frac{-\sqrt{2}}{2} \\
\end{pmatrix} \otimes \begin{pmatrix}
\frac{\sqrt{2}}{2}&\frac{\sqrt{2}}{2} \\
\frac{\sqrt{2}}{2}&\frac{-\sqrt{2}}{2} \\
\end{pmatrix} = \begin{pmatrix}
\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2} \\
\frac{1}{2}&\frac{-1}{2}&\frac{1}{2}&\frac{-1}{2} \\
\frac{1}{2}&\frac{1}{2}&\frac{-1}{2}&\frac{-1}{2} \\
\frac{1}{2}&\frac{-1}{2}&\frac{-1}{2}&\frac{1}{2} \\
\end{pmatrix}$$
What seems to be the case with scenario 3 or what does the interconnection do to the qubits?
P.S I haven't noticed Hadamard gates in a circuit as in scenario 3 but have seen Pauli-X and Pauli-Z gates used in this manner, for example the magic-state-distillation on Quirk