When running the following operations on 3 qubits:
H(qubits[0]);
CNOT(qubits[0], qubits[1]);
CNOT(qubits[0], qubits[2]);
CNOT(qubits[1], qubits[2]);
I get these results: qubit 0 is in a superposition qubit 1 is the same as qubit 0 qubit 2 is the same as qubit 0 half the time. e.g. superposition-like values.
Why does running CNOT on qubit 2 with both other qubits after running CNOT on qubit 1 with qubit 0 cause qubit 2 to enter a state of superposition between qubit 0 and not qubit 0?
If you do some quantum computing maths, you will find out that you end up in the following state:
|ψ 〉 = (|000〉 + |011〉) / √2
This is essentially a superposition between qubit 0 and entangled qubits 1 and 2.
|ψ 〉 = |0〉 ⊗ (|00〉 + |11〉) / √2
You can do this maths with IBM QISKit in Python:
from qiskit import QuantumProgram
from math import sqrt
import numpy as np
qp = QuantumProgram()
cname = '3-qubit'
num_qubits = 3
qr = qp.create_quantum_register('qr', num_qubits)
cr = qp.create_classical_register('cr', num_qubits)
qc = qp.create_circuit(cname, [qr], [cr])
qc.h(qr[0])
qc.cx(qr[0], qr[1])
qc.cx(qr[0], qr[2])
qc.cx(qr[1], qr[2])
qc.measure(qr, cr)
results = qp.execute(cname)
print(results.get_counts(cname))
This would give you a result similar to the following:
{'000': 530, '011': 494}
You can also explicitly obtain this state |ψ〉 by taking the unitary matrix of your circuit and applying it to your initial state |000〉, that is a vector [1,0,0,0,0,0,0,0]:
results = qp.execute(cname, backend='local_unitary_simulator', shots=1)
data = results.get_data(cname)
u = np.real(data['unitary'] * sqrt(2.0)).astype(int)
psi = np.zeros(2**num_qubits, dtype=np.int_)
psi[0] = 1
u @ psi
The result is
array([1, 0, 0, 1, 0, 0, 0, 0])
The 0-th entry is |000〉, the 3-rd entry is |011〉.