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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
    two hadamard gates on one qubit

  2. Hadamard gates on two qubits
    Hadamard gates on two qubits

  3. Interconnected Hadamard gates on two qubits
    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

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Scenario 3 does not make sense to me. You should provide a reference, if any.

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 $|1\rangle$.

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  • $\begingroup$ 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 $\endgroup$
    – LiNKeR
    Jul 5 '20 at 14:46
  • $\begingroup$ Dear Linker, can you provide a link about that? $\endgroup$ Jul 5 '20 at 16:02
  • $\begingroup$ just click on magic-state-distillation from the question to see it (right in the P.S) $\endgroup$
    – LiNKeR
    Jul 5 '20 at 18:00
  • $\begingroup$ I am sorry but nothing happens when I click it. $\endgroup$ Jul 5 '20 at 18:01
  • $\begingroup$ that feels weird, maybe you should try using a different browser or go to algassert.com/quirk and select it $\endgroup$
    – LiNKeR
    Jul 5 '20 at 18:06

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