Quantum Teleportation

Qiskit Simulation Of The Quantum Teleportation Process Of A Single Quantum Bit Over A Fully-Entangled Channel.

Open in Google Colab: https://colab.research.google.com/drive/1ke1t24idpjMXDOrEvY7IDFalxEdMPm8Y

Open in IBM Quantum Composer: https://shorturl.at/hik36

Problem Statement

Lets say a person called Alice wants to send the qubit state $\vert\psi\rangle = \alpha\vert0\rangle + \beta\vert1\rangle$ to Bob. This entails passing on information about $\alpha$ and $\beta$ to Bob. Alice can't simply generate a copy of $\vert\psi\rangle$ and give the copy to Bob because of the no-cloning theorem, which states that you cannot simply make an exact copy of an unknown quantum state. We are limited to copying only classical states (not superpositions). However, by taking advantage of two classical bits and an entangled qubit pair, Alice can transfer her state $\vert\psi\rangle$ to Bob. After the transfer, Bob will have $\vert\psi\rangle$ and Alice won't. This is called Quantum Teleportation.

Simulating the Quantum Teleportation Process

To transfer a quantum bit, Alice and Bob must use a third party to send them an entangled qubit pair. Alice then performs some operations on her qubit, sends the results to Bob over a classical communication channel, and Bob then performs some operations on his end to receive Alice’s qubit. The process is as follows:

Step 1

A third party, Telamon, creates an entangled Bell pair of qubits and gives one to Bob and one to Alice. In quantum circuit language, the way to create a Bell pair between two qubits is to first transfer one of them to the X-basis ($|+\rangle$ and $|-\rangle$) using a Hadamard gate, and then to apply a CNOT gate onto the other qubit controlled by the one in the X-basis.

Step 2

Let's say Alice owns $q_1$ and Bob owns $q_2$ after they part ways. Alice applies a CNOT gate to $q_1$, controlled by $\vert\psi\rangle$ (the qubit she is trying to send Bob). Then Alice applies a Hadamard gate to $|\psi\rangle$. In our quantum circuit, the qubit ($|\psi\rangle$) Alice is trying to send is $q_0$

Step 3

Next, Alice applies a measurement to both qubits that she owns, $q_1$ and $\vert\psi\rangle$, and stores this result in two classical bits. She then sends these two bits to Bob.

Step 4

Bob, who already has the qubit $q_2$, then applies the following gates depending on the state of the classical bits:

00 $\rightarrow$ Do nothing 01 $\rightarrow$ Apply $X$ gate 10 $\rightarrow$ Apply $Z$ gate 11 $\rightarrow$ Apply $ZX$ gate

(Note that this transfer of information is purely classical.)

By running the Jupyter notebook, we can see that, the statevector, $|q_2\rangle$, obtained from the aer simulator is in the same state of $|\psi\rangle$ we had initially wanted to teleport while the states of $|q_0\rangle$ and $|q_1\rangle$ collapse to either $|0\rangle$ or $|1\rangle$. The state $|\psi\rangle$ has been teleported from qubit 0 to qubit 2. Alice's qubit has now teleported to Bob.