QWorld Bronze: Quantum Teleportation

(01/15/2021)

Quantum Teleportation

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Prepare a few blank sheets of paper

• to draw the circuit of the following protocol step by step and
• to solve some of tasks requiring certain calculations.

Asja wants to send a qubit to Balvis by using only classical communication.

Let $\big\lvert v\rangle = a \ b \in \mathbf{R}^2$​ be the quantum state.

Discussion: If Asja has many copies of this qubit, then she can collect the statistics based on these qubits and obtain an approximation of and , say $\tilde a, \tilde b$, respectively. After this, Asja can send $\tilde a$ and by using many classical bits, the number of which depends on the precision of the amplitudes.

On the other hand, If Asja and Balvis share the entangled qubits in state $\sqrt{2}\big\lvert 00\rangle + \sqrt{2}\big\lvert 11\rangle$​ in advance, then it is possible for Balvis to create $\lvert v\rangle$​ in his qubit after receiving two bits of information from Asja.

Protocol

The protocol uses three qubits as specified below:

Asja has two qubits and Balvis has one qubit.

Asja's quantum message (key) is $|v\rangle = a \ b = a|0\rangle + b|1\rangle$​.

The entanglement between Asja's second qubit and Balvis' qubit is $\sqrt{2}|00\rangle + \sqrt{2}|11\rangle$​.

So, the quantum state of the three qubits is

$a|0\rangle + b|1\rangle \\ \sqrt{2}|00\rangle + \sqrt{2}|11\rangle = \sqrt{2} \big( a|000\rangle + a|011\rangle + b|100\rangle + b|111\rangle \big)$​​

CNOT operator by Asja

Asja applies CNOT gate to her qubits where is the control qubit and is the target qubit.

Calculate the new quantum state after this CNOT operator.

Asja applies Hadamard gate to .

Calculate the new quantum state after this Hadamard operator.

Verify that the resulting quantum state can be written as follows:$\frac{1}{2} |00\rangle \big( a|0\rangle+b|1\rangle \big) + \frac{1}{2} |01\rangle \big( a|1\rangle+b|0\rangle \big) + \frac{1}{2} |10\rangle \big( a|0\rangle-b|1\rangle \big) + \frac{1}{2} |11\rangle \big( a|1\rangle-b|0\rangle \big)$​​

Measurement by Asja

Asja measures her qubits. With probability , she can observe one of the basis states.

Depeding on the measurement outcomes, Balvis' qubit is in the following states:

1. "00": $|{v_{00}}\rangle = a|0\rangle + b |{1}\rangle$​​
2. "01": $|{v_{01}}\rangle = a|{1}\rangle + b |{0}\rangle$
3. "10": $|{v_{10}}\rangle = a|{0}\rangle - b|{1}\rangle$​​
4. "11": $|{v_{11}}\rangle = a|{1}\rangle - b |{0}\rangle$

As can be observed, the amplitudes and are "transferred" to Balvis' qubit in each case.

If Asja sends the measurement outcomes, then Balvis can construct $|{v}\rangle$​ exactly.

Asja sends the measurement outcomes to Balvis by using two classical bits: and .

For each pair, determine the quantum operator(s) that Balvis can apply to obtain $|{v}\rangle = a|{0}\rangle+b|{1}\rangle$​ exactly.

Create a quantum circuit with three qubits as described at the beginning of this notebook and two classical bits.

Implement the protocol given above until Asja makes the measurements (included).

• The state of can be set by the rotation with a randomly picked angle.
• Remark that Balvis does not make the measurement.

At this point, read the state vector of the circuit by using "statevector_simulator".

_When a circuit having measurement is simulated by "statevector_simulator", the simulator picks one of the outcomes, and so we see one of the states after the measurement._

Verify that the state of Balvis' qubit is in one of these: $|{v_{00}}\rangle$​, $|{v_{01}}\rangle$​, $|{v_{10}}\rangle$​, and $|{v_{11}}\rangle$​.

Guess the measurement outcome obtained by "statevector_simulator".

See the related project for the solution.

q = QuantumRegister(3)c2 = ClassicalRegister(1,'c2')c1 = ClassicalRegister(1,'c1')qc = QuantumCircuit(q,c1,c2)...qc.measure(q[1],c1)...qc.x(q[0]).c_if(c1,1) # x-gate is applied to q[0] if the classical bit c1 is equal to 1
Read the state vector and verify that Balvis' state is $ab$​ after the post-processing.