What is the difference between the two entanglement circuits?

Question

For a three qubits system to be in an entanglement(GHZ) state, the circuit can be built as per Qiskit,

Here, the control bit of the second X gate is on q0. what if the control bit is shifted to q1. This also leads to entanglement. what is the difference between the two different ways?

Further to my question, I did the same for 5 qubits, I get different results:

which one is correct?

Answer 1

The two circuits are equivalent.

import qiskit.quantum_info as qi
from qiskit.circuit import QuantumCircuit
qc1 = QuantumCircuit(3)
qc1.h(0)
qc1.cx(0,1)
qc1.cx(0,2)
op1 = qi.Operator(qc1)
print(qc1)
     ┌───┐          
q_0: ┤ H ├──■────■──
     └───┘┌─┴─┐  │  
q_1: ─────┤ X ├──┼──
          └───┘┌─┴─┐
q_2: ──────────┤ X ├
               └───┘


qc2 = QuantumCircuit(3)
qc2.h(0)
qc2.cx(0,1)
qc2.cx(1,2)
op2 = qi.Operator(qc1)
print(qc2)

     ┌───┐          
q_0: ┤ H ├──■───────
     └───┘┌─┴─┐     
q_1: ─────┤ X ├──■──
          └───┘┌─┴─┐
q_2: ──────────┤ X ├
               └───┘


print(op1 == op2)
True

---

To see this more explicitly, note the following idenity

Thus you have

Note that $CNOT*CNOT = I$, and when the control qubit is $|0\rangle$ nothing happened so you can remove it.