Putting it All Together: Teleportation

Let's return to the example of the teleportation circuit defined in Quantum Circuits. We're going to use this to illustrate the concepts we've learned so far. An explanation of quantum teleportation is provided below for those who are unfamiliar with the theory, followed by a walkthrough of the code implementation in Q#.

Quantum Teleportation: Theory

Quantum teleportation is a technique for sending an unknown quantum state (which we'll refer to as the 'message') from a qubit in one location to a qubit in another location (we'll refer to these qubits as 'here' and 'there', respectively). We can represent our message as a vector using Dirac notation:

$$ \ket{\psi} = \alpha\ket{0} + \beta\ket{1} $$

The message qubit's state is unknown to us as we do not know the values of $\alpha$ and $\beta$.

Step 1: Create an entangled state

In order to send the message we need for the qubit here to be entangled with the qubit there. This is achieved by applying a Hadamard gate, followed by a CNOT gate. Let's look at the math behind these gate operations.

We will begin with the qubits here and there both in the $\ket{0}$ state. After entangling these qubits, they are in the state:

$$ \ket{\phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11}) $$

Step 2: Send the message

To send the message we first apply a CNOT gate with the message qubit and here qubit as inputs (the message qubit being the control and the here qubit being the target qubit, in this instance). This input state can be written:

$$ \ket{\psi}\ket{\phi^+} = (\alpha\ket{0} + \beta\ket{1})(\frac{1}{\sqrt{2}}(\ket{00} + \ket{11})) $$

This expands to:

$$ \ket{\psi}\ket{\phi^+} = \frac{\alpha}{\sqrt{2}}\ket{000} + \frac{\alpha}{\sqrt{2}}\ket{011} + \frac{\beta}{\sqrt{2}}\ket{100} + \frac{\beta}{\sqrt{2}}\ket{111} $$

As a reminder, the CNOT gate flips the target qubit when the control qubit is 1. So for example, an input of $\ket{000}$ will result in no change as the first qubit (the control) is 0. However, take a case where the first qubit is 1 - for example an input of $\ket{100}$. In this instance, the output is $\ket{110}$ as the second qubit (the target) is flipped by the CNOT gate.

Let's now consider our output once the CNOT gate has acted on our input above. The result is:

$$ \frac{\alpha}{\sqrt{2}}\ket{000} + \frac{\alpha}{\sqrt{2}}\ket{011} + \frac{\beta}{\sqrt{2}}\ket{110} + \frac{\beta}{\sqrt{2}}\ket{101} $$

The next step to send the message is to apply a Hadamard gate to the message qubit (that's the first qubit of each term).

As a reminder, the Hadamard gate does the following:

Input Output
$\ket{0}$ $\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$
$\ket{1}$ $\frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$

If we apply the Hadamard gate to the first qubit of each term of our output above, we get the following result:

$$ \frac{\alpha}{\sqrt{2}}(\frac{1}{\sqrt{2}}(\ket{0} + \ket{1}))\ket{00} + \frac{\alpha}{\sqrt{2}}(\frac{1}{\sqrt{2}}(\ket{0} + \ket{1}))\ket{11} + \frac{\beta}{\sqrt{2}}(\frac{1}{\sqrt{2}}(\ket{0} - \ket{1}))\ket{10} + \frac{\beta}{\sqrt{2}}(\frac{1}{\sqrt{2}}(\ket{0} - \ket{1}))\ket{01} $$

Note that each term has two $\frac{1}{\sqrt{2}}$ factors. We can multiply these out giving the following result:

$$ \frac{\alpha}{2}(\ket{0} + \ket{1})\ket{00} + \frac{\alpha}{2}(\ket{0} + \ket{1})\ket{11} + \frac{\beta}{2}(\ket{0} - \ket{1})\ket{10} + \frac{\beta}{2}(\ket{0} - \ket{1})\ket{01} $$

The $\frac{1}{2}$ factor is common to each term so we can now take it outside the brackets:

$$ \frac{1}{2}\big[\alpha(\ket{0} + \ket{1})\ket{00} + \alpha(\ket{0} + \ket{1})\ket{11} + \beta(\ket{0} - \ket{1})\ket{10} + \beta(\ket{0} - \ket{1})\ket{01}\big] $$

We can then multiply out the brackets for each term giving:

$$ \frac{1}{2}\big[\alpha\ket{000} + \alpha\ket{100} + \alpha\ket{011} + \alpha\ket{111} + \beta\ket{010} - \beta\ket{110} + \beta\ket{001} - \beta\ket{101}\big] $$

Step 3: Measure the result

Due to here and there being entangled, any measurement on here will affect the state of there. If we measure the first and second qubit (message and here) we can learn what state there is in, due to this property of entanglement.

  • If we measure and get a result 00, the superposition collapses, leaving only terms consistent with this result. That's $\alpha\ket{000} +\beta\ket{001}$. This can be refactored to $\ket{00}(\alpha\ket{0} +\beta\ket{1})$. Therefore if we measure the first and second qubit to be 00, we know that the third qubit, there, is in the state $(\alpha\ket{0} +\beta\ket{1})$.
  • If we measure and get a result 01, the superposition collapses, leaving only terms consistent with this result. That's $\alpha\ket{011} +\beta\ket{010}$. This can be refactored to $\ket{01}(\alpha\ket{1} +\beta\ket{0})$. Therefore if we measure the first and second qubit to be 01, we know that the third qubit, there, is in the state $(\alpha\ket{1} +\beta\ket{0})$.
  • If we measure and get a result 10, the superposition collapses, leaving only terms consistent with this result. That's $\alpha\ket{100} -\beta\ket{101}$. This can be refactored to $\ket{10}(\alpha\ket{0} -\beta\ket{1})$. Therefore if we measure the first and second qubit to be 10, we know that the third qubit, there, is in the state $(\alpha\ket{0} -\beta\ket{1})$.
  • If we measure and get a result 11, the superposition collapses, leaving only terms consistent with this result. That's $\alpha\ket{111} -\beta\ket{110}$. This can be refactored to $\ket{11}(\alpha\ket{1} -\beta\ket{0})$. Therefore if we measure the first and second qubit to be 11, we know that the third qubit, there, is in the state $(\alpha\ket{1} -\beta\ket{0})$.

Step 4: Interpret the result

As a reminder, the original message we wished to send was:

$$ \ket{\psi} = \alpha\ket{0} + \beta\ket{1} $$

We need to get the there qubit into this state, so that the received state is the one that was intended.

  • If we measured and got a result of 00, then the third qubit, there, is in the state $(\alpha\ket{0} +\beta\ket{1})$. As this is the intended message, no alteration is required.
  • If we measured and got a result of 01, then the third qubit, there, is in the state $(\alpha\ket{1} +\beta\ket{0})$. This differs from the intended message, however applying a NOT gate gives us the desired state $(\alpha\ket{0} +\beta\ket{1})$.
  • If we measured and got a result of 10, then the third qubit, there, is in the state $(\alpha\ket{0} -\beta\ket{1})$. This differs from the intended message, however applying a Z gate gives us the desired state $(\alpha\ket{0} +\beta\ket{1})$.
  • If we measured and got a result of 11, then the third qubit, there, is in the state $(\alpha\ket{1} -\beta\ket{0})$. This differs from the intended message, however applying a NOT gate followed by a Z gate gives us the desired state $(\alpha\ket{0} +\beta\ket{1})$.

To summarize, if we measure and the first qubit is 1, a Z gate is applied. If we measure and the second qubit is 1, a NOT gate is applied.

Summary

Shown below is a text-book quantum circuit that implements the teleportation. Moving from left to right you can see:

  • Step 1: Entangling here and there by applying a Hadamard gate and CNOT gate.
  • Step 2: Sending the message using a CNOT gate and a Hadamard gate.
  • Step 3: Taking a measurement of the first and second qubits, message and here.
  • Step 4: Applying a NOT gate or a Z gate, depending on the result of the measurement in step 3.

`Teleport(msg : Qubit, there : Qubit) : Unit`

Quantum Teleportation: Code

We have our circuit for quantum teleportation:

`Teleport(msg : Qubit, there : Qubit) : Unit`

We can now translate each of the steps in this quantum circuit into Q#.

Step 0: Definition

When we perform teleportation, we must know the message we wish to send, and where we wish to send it (there). For this reason, we begin by defining a new Teleport operation that is given two qubits as arguments, msg and there:

operation Teleport(msg : Qubit, there : Qubit) : Unit {
    body (...) {

We also need to allocate a qubit here which we achieve with a using block:

        using (here = Qubit()) {

Step 1: Create an entangled state

We can then create the entangled pair between here and there by using the H and CNOT operations:

            H(here);
            CNOT(here, there);

Step 2: Send the message

We then use the next $\operatorname{CNOT}$ and $H$ gates to move our message qubit:

            CNOT(msg, here);
            H(msg);

Step 3 & 4: Measuring and interpreting the result

Finally, we use M to perform the measurements and perform the necessary gate operations to get the desired state, as denoted by if statements:

            // Measure out the entanglement
            if (M(msg) == One)  { Z(there); }
            if (M(here) == One) { X(there); }

This finishes the definition of our teleportation operator, so we can deallocate here, end the body, and end the operation.

        }
    }
}