Problem $1$B $($Quantum Teleportation$)$

We will now look into the problem of "destructive" copying of a quantum state called quantum teleportation. Suppose we have a qubit $0$

that is currently a super position $||=0||0+1||1$

and another qubit $1$

initialized to $||0$

$.$ We wish to transform $1$

to be $||$

while $0$

is potentially destroyed in the process. Note that no$-$cloning theorem will not allow us to do this without changing $0$

in some way.

Prove that the following steps will copy $0$

over to $1$

by completing the calculations asked of you.

Let $0$

be in the super position $||$

above and $1$

in initial state $||0$

$.$ The joint system is in the state

$||||0=0||00+1||10$

Apply the Hadamard gate to $1$

$.$ What is the resulting state?

answer?

Apply the controlled$-$X gate to qubit $0$

with control qubit $1$

$.$ What is the resulting state?

answer?

Apply the controlled$-$X gate to qubit $1$

with control qubit $0$

$.$ What is the resulting state?

answer?

Measure the qubit $0$

$.$ Write down the possible results of the measurement along with the probabilities and the states for $1$

after the partial measurement.

answer?

If you followed along the steps above, you will realize that the quantum circuit will cause $0$

to collapse to either a $||0$

or a $||1$

but $1$

will now have $||$

$.$ This is the essence of quantum teleportation. It is important that you do the calculations correctly and obtain the result before you try the problem below.

Implement the function quantum$\_$teleport that takes in an instance of QuantumCircuit and index of two qubits b$0$ and b$1.$ It should implement the above scheme to copy b$0$ into b$1,$ which is initialized to $||0$

$.$ Do not measure b$1.$