Suppose that I have a Quantum Circuit with $N$ qubits initialized to $|0\rangle$. The Quantum circuit is described by the followin Figure:

The angles of the rotations $R_y$ are random. At which point of the circuit I can say that in general the generic qubit $i$ is entangled with the generic qubit $j$ (Regardless of $N$) ? How can I prove it?

(Of course there are come specific case where we have trivial state due to some particular rotations, but I'm not interested in them)

  • $\begingroup$ Your statement will be probabilistic. I.e. with high probability i and j are entangled and a necessary condition is that there is some path in the circuit that connects qubits i and j where the cnot gates connect the different wires. I guess how deep the circuit needs to be will depend on how close the qubits initially were. $\endgroup$
    – Rammus
    Dec 3, 2022 at 13:59


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