# Interconnection clarification?

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

1. two hadamard gates on one qubit

2. Hadamard gates on two qubits

3. Interconnected Hadamard gates on two qubits

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

I think you have been confused by controlled gates like CNOT, which is a Pauli-X gate applied to a target qubit, provided that the state of the control qubit is $$|1\rangle$$.
• Hi Michele, did you have a look at the magic-state-distillation on quirk from the P.S with series of interconnected Pauli-Z gates Jul 5 '20 at 14:46
• just click on magic-state-distillation from the question to see it (right in the P.S) Jul 5 '20 at 18:00