## 2023年8月29日火曜日

### Reconfirmation of quantum teleportation using Qni

Quantum teleportation has already been covered in detail in this article. This time, I would like to reconfirm the point by executing it with the quantum circuit simulator Qni.

In the previous article, Bob applied his I-, X-, Z-, and Y-gates to his qubits, corresponding to the classical 2-bit information 00, 01, 10, and 11 received from Alice. This allowed Bob to obtain the state of the qubit sent by Alice. Logically this is fine. However, when running on the Qni, I found it easier to apply the X and Z gates in succession instead of the Y gate. The description will be made accordingly.

(1) Use of CX (Controlled X) and CZ (Controlled Z)
Considering the gate that Bob should apply to his qubit according to 00, 01, 10, 11 mentioned above, we arrive at the following simple conclusion :

step1: If the lower bit of the received 2-bit information is 1, the X gate is applied.
step2: If the upper bit of the received 2-bit information is 1, the Z gate is applied.

According to this, the CX gate and CZ gate can be applied as follows. Note that either |0> or |1> is input to the CX and CZ control lines (other qubits are not input).
As an example in the above diagram, ① is set with the RX(π/3) gate, but any state is acceptable. There are two points worth noting here. First, the state of qubit ① that Alice wants to send to Bob vanishes in Alice's measurement shown in the center, resulting in ②. Nevertheless, the state of ① appeared in ③ by applying the gate by Bob shown on the far right. It matches the No Cloning Theorem. This is exactly what quantum teleportation is.

Second, regardless of the measurement results (classical 2-bit information) by Alice in the central part of the figure, Bob on the far right only needs to apply CX and CZ continuously. However, when Bob is far away from Alice, it is difficult for us to imagine Bob receiving classical 2-bit information from Alice through normal communication. To clarify the image, the following alternative (explicitly checking the classical bits) can also be used.

(2) Explicitly examine classical 2-bit information and apply necessary gates
In Qiskit, etc., classical bit registers can be referenced, but in Qni, the corresponding processing can be realized by adding an execution condition (if) to the gate as shown in the figure below.

In this example, since the measurement result by Alice in the center of the figure is "10", it can be seen that the X gate applied by Bob does not work (it is dark) and only the Z gate is applied.

In this article, I referred to  for the principle and mechanism of quantum teleportation,  for the Qni simulation, and  for the Qiskit simulation.

References
 Chris Bernhardt, “Quantum Computing for Everyone”, MIT Press, 2020
https://www.chrisbernhardt.info/
 Qni; https://github.com/qniapp/qni
 Satoyuki Tsukano, "Inside the mind of a quantum computer", Gijutsu-hyoron-sha, June 2023. (in Japanese)