Please, pardon the caption. No intention of making new theories. I am a bit confused about the quantum teleportation. I missed a hadamard gate before CNOT gate for creating the EPR pair. But the circuit seems to bee working and copying q0 to q2.

Circuit without entanglement and initial Hadamard gate:

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Results consistent with q0 wave function: This is measurement of q2 for multiple shots

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I am new to quantumn mechanics domain, any guide or help would mean a lot. Thank you for your help :).

  • $\begingroup$ It seems that there is a missing Hadamard on qubit q55_1 necessary to create Bell state. In this configuration second and third CNOT work as fan out and simply "copy" state from q55_0 to q52_0. $\endgroup$ – Martin Vesely Jan 15 at 15:37
  • $\begingroup$ Yeah exactly my point, this is not the correct teleportation protocol. But how do we end up having a measurement consistent with teleportation? $\endgroup$ – Sayan Dey Jan 15 at 15:58
  • $\begingroup$ @SayanDey Are you sure that is the right result? Also, without the Hadamard gate in front of q_551 you can essentially removed the CNOT between q_551 and q_552 since they always start in the $|0\rangle$ state so a CNOT here would not do anything $\endgroup$ – KAJ226 Jan 15 at 16:13
  • $\begingroup$ @KAJ226 I understand your point. Also I am not claiming the states are same. But one thing to be precise, if you see the wave function in the 1st pic, it is consistent with their squared value which are 3/4 and 1/4 shown as probabilities in the second pic. Wave function belongs to q550 and probabilities belong to q552. $\endgroup$ – Sayan Dey Jan 15 at 16:44
  • $\begingroup$ I'll second KAJ226 - I don't think this output can match this circuit, since no gates act on q2 that can put it in superposition. Are you quite sure the histogram is built on this circuit for qubit q2? $\endgroup$ – Mariia Mykhailova Jan 15 at 18:47

Your circuit gets lucky on the probabilities for this one particular value, but doesn't work for any other. It also produces the wrong state, as you can confirm in this adjusted version of the quirk teleportation circuit. Compared to the original initialized with your value your phase is off.

The difference becomes even clearer when you use the rotating message from the original quirk teleportation circuit and remove the the Hadamard as in your circuit. To no great surprise, there is a value at which the probabilities are correct, but as you can see from the receiving qubit sadly moving up and down the Z axis the state is not correctly teleported.

There are infinitely many states that produce the same probability distribution, because:

$$a * a^{\ast} = (\alpha + i \beta) * (\alpha - i \beta) = \alpha^2 + \beta^2$$

Also recall that you can multiply with an arbitrary global phase and that shifts along the x or y axis are orthogonal and thus don't influence the outcome when measuring along the z axis. The QFT is a nice example of this property, where, depending on the input, the different qubits have a different phase (essentially rotating on the x,y-circle), but they have equal probability for all inputs.

  • $\begingroup$ Yes I was sure the wave function was not the same. Thanks for the explanation, exactly the one I needed. This makes everything clear now. $\endgroup$ – Sayan Dey Jan 16 at 5:50

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