QWorld Bronze: Quantum Teleportation

Published by strangeworks (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 v=a bR2\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 a and b, say a~,b~\tilde a, \tilde b, respectively. After this, Asja can send a~\tilde a and b~\tilde b 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 200+211\sqrt{2}\big\lvert 00\rangle + \sqrt{2}\big\lvert 11\rangle​ in advance, then it is possible for Balvis to create v\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=a b=a0+b1|v\rangle = a \ b = a|0\rangle + b|1\rangle​.

The entanglement between Asja's second qubit and Balvis' qubit is 200+211\sqrt{2}|00\rangle + \sqrt{2}|11\rangle​.

So, the quantum state of the three qubits is

a0+b1200+211=2(a000+a011+b100+b111)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 q[2] is the control qubit and q[1] is the target qubit.

Task 1

Calculate the new quantum state after this CNOT operator.

Hadamard operator by Asja

Asja applies Hadamard gate to q[2].

Task 2

Calculate the new quantum state after this Hadamard operator.

Verify that the resulting quantum state can be written as follows:1200(a0+b1)+1201(a1+b0)+1210(a0b1)+1211(a1b0)\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 41, she can observe one of the basis states.

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

  1. "00": v00=a0+b1|{v_{00}}\rangle = a|0\rangle + b |{1}\rangle ​​
  2. "01": v01=a1+b0|{v_{01}}\rangle = a|{1}\rangle + b |{0}\rangle
  3. "10": v10=a0b1|{v_{10}}\rangle = a|{0}\rangle - b|{1}\rangle ​​
  4. "11": v11=a1b0|{v_{11}}\rangle = a|{1}\rangle - b |{0}\rangle

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

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

Task 3

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

For each (x,y) pair, determine the quantum operator(s) that Balvis can apply to obtain v=a0+b1|{v}\rangle = a|{0}\rangle+b|{1}\rangle ​ exactly.

Task 4

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 q[2] 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: v00|{v_{00}}\rangle​, v01|{v_{01}}\rangle​, v10|{v_{10}}\rangle​, and v11|{v_{11}}\rangle​.

Guess the measurement outcome obtained by "statevector_simulator".

See the related project for the solution.

Task 5

Implement the protocol above by including the post-processing part done by Balvis, i.e., the measurement results by Asja are sent to Balvis and then he may apply X or Z gates depending on the measurement results.

We use the classically controlled quantum operators.

Since we do not make measurement on q[0], we define only 2 classical bits, each of which can also be defined separated.

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 abab​ after the post-processing.

See the related project for the solution.

​Source: https://gitlab.com/qkitchen/basics-of-quantum-computing/-/blob/master/bronze/B56_Quantum_Teleportation.ipynb