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Quantum teleportation with Google Cirq

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Category: Quantum Mechanics

Last Updated: 26 May 2020

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      In this article, our aim is to expose the teleportation algorithm introduced in The quantum teleportation algorithm but this time as implemented with Google Cirq API.

As a quick reminder, we show again the quantum circuit, and which operations/gates Bob has to apply after Alice's measurements:

Alice's measurements and Bob's operations

Note that after the measurement, we have classical information, which we represent in circuit diagrams with two lines (instead of one line for qubits)

To get a better idea of the possible outcomes Alice can get when she will measure, we can group terms containing the same state for the first two terms.
There are four possibilities for the first two qubits, namely |00>,|01>,|10>, and|11>.  Grouping these terms together, we can rewrite the state

Now Alice measures her two qubits (the first two qubits). 

- Suppose she gets the |00> outcome. Then, Bob’s qubit is in the state α|0> +β|1>, which is |ψ> exactly. In this case, we are done, and Alice’s unknown qubit |ψ> has been teleported to Bob.

- Suppose instead Alice measures |01>. By looking at the above expression, we see that Bob’s qubit in this case would be α|1>+β|0>. This is almost |ψ>, but the amplitudes are flipped.  How can we get|ψ> exactly from this state?  We can perform a NOT gate X(α|1> +β|0>) = α|0> +β|1> =|ψ>.

- If Alice measures |10⟩, Bob’s qubit is in the state α|0⟩−β|1⟩. Further, he can obtain |ψ⟩ by performing a Pauli-Z gate on his qubit. As we have seen in the previous article Introduction to quantum logic gates that will have for effect to negate the |1> basis state to -|1>. That is, Z(α|0⟩−β|1⟩) =α|0⟩+β|1⟩ = |ψ⟩.

- And finally in the case of Alice measuring |11>, Bob would have to apply both the NOT gate and the Pauli-Z gates.That is ZX(α|1⟩−β|0>) =α|0⟩+β|1⟩ = |ψ⟩.

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