Teleportation
If you've ever watched Star Trek, you might have wondered if it's possible to actually build a teleporter.
For things that are as big as humans, this will probably remain a science fiction plot device. For qubits however, teleportation is real! In fact, quantum technologies are now used to teleport qubits every day. We'll explore quantum teleportation in this section.
Alice, Bob, and Joey
In order to discuss quanutm teleportation, let's use a slightly silly example.
The year is 2101, and humans and otters are living harmoniously. Together, they created a second home on Mars! Joey (the otter) lives on Earth and would like to go visit Bob (the human), who lives on Mars.
Alice (the human) is also friends with Joey and Bob, and decides to help Joey get to Mars. A major issue is that Joey is afraid of spaceships, so they will have to find a different way to get Joey to Mars.
Teleporting Joey as Qubit
To get to Mars, Alice and Joey first send one of the qubits from a Bell state to Mars, so that there are entangled particles spread between the two planets.
Next, we'll see how Alice, Joey, and Bob can team up to get Joey to Mars.
We'll first need to turn Joey into a qubit. Otters, like all living things, cannot turn into a qubit. However, let's imagine a scenario where Joey can do this (superpowers may be involved - which is why we describe this story as a silly scenario).
Joey is now recorded as the state of the qubit. He might be the up, down, left or right states we've learned about. For Joey to safely reach Mars, we need to make sure the correct Joey state ends up at Mars.
Alice has two qubits: one holds the Joey state and the other is one half of the entangled Bell pair shared with Bob (the other half is on Mars!). The first step to teleport the Joey state to Mars is for Alice to perform a special type of measurement called a Bell measurement on her qubits. A Bell measurement is done on two qubits at once, and Alice will get some measurement outcomes which are a pair of classical bits, for example 00 or 01.
After Alice does this measurement, the qubit over on Mars will be in one of the states up, down, left, right. We know from the rules of entanglement that the state of Bob's qubit on Mars will be related to the state of Alice's qubit on Earth, because they were part of an entangled Bell pair. But wait! Just because the two states are related doesn't mean they're exactly the same. Maybe Alice's qubit is in the opposite state to Bob's. If that's the case, we'll need to fix it so that Alice and Bob end up with exactly the same state.
We can fix Bob's qubit on Mars by using classically controlled gates. Remember that a classically controlled gate applies a quantum gate to a qubit depending on the value of a classical bit. In this case, we'll use Alice's measurement outcomes to control the gates that Bob applies to his qubit. So, Alice sends her bits over to Bob. If Bob applies the right gate, Joey
Fixing up Joey
Our goal will be to understand how Bob can fix up his qubit using Alice's measurement outcomes, so that Joey is successfully transported to Mars.
When Alice measures the two qubits, the two bits she gets out are:
00 with 25% chance
01 with 25% chance
10 with 25% chance
11 with 25% chance
For each different outcome, Bob will have to do different gates to fix up the Joey state.
Here is an almost working teleporter:
Joey turns into a qubit (using the Qubit-izer), and then the Joey qubit and one other qubit get measured. Some classical bits come out. Then, when the last qubit tries to turn back into Joey, something goes (horrifically) wrong!
If we understand the right classically controlled gates to put at the end, we can fix this and save Joey!
Fixing up Joey ... from Scratch
Bob needs to do some classically controlled gates using Alice's qubits so that Joey can safely arrive on Mars.
Your job, as a quantum engineer, is to build the fixer-upper! You can build it in this circuit simulator:
When you've built the fixer-upper, it should let Joey arrive safely on Mars three times in a row.
To figure out what classically controlled gates to use, we can put the teleportation circuit in repair mode.
In repair mode, we can set the initial state of the Joey qubit by clicking on the Bloch sphere above the Source button.
Now hit the Send Qubit button a bunch of times until you get enough data to fill in the table on the right. We want the output to be the same as the input for any combination of Alice's measurement outcomes. But notice that in some cases the output and input don't match!
Try adding classically controlled Z and X gates to the first qubit (left wire) and the second qubit (right wire) and generating some new data. See if you can find the right combination of gates so that the input always matches the output. Note: they'll have to match for EVERY input state: up, down, left, and right!
Try this on your own first, then take a look at the key below to find how we can achieve the right states! Once you have the answer, switch over to Test mode and see if you can successfully teleport Joey!
Answer Key
Bob applies gates on his qubit using the results of Alice's measurements:
00 → No gate
01 → Apply 𝑋 gate
10 → Apply 𝑍 gate
11 → Apply 𝑍𝑋 gate
In our simulation, the above key translates into applying a controlled Z gate on the first qubit (left wire) and a controlled X gate on the second qubit (right wire). Once these gates are applied, we can reconstruct the teleported qubit - in other words, Bob can fix up Joey!
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