Okay, so quantum teleportation might sound too exciting and too science fiction to be
of any actual interest, technically but it's, it's actually a very fundamental
primitive in, in computation. And, so in this video I'll, I'll say in a few
minutes, you know I'll give you a little sketch of how teleportation gets used in
computation. This might be a little more advanced but it's really a point into the
literature for those of you who are interested. Okay so, so let's look at how,
how telepretation proceeds, right? So Alice has a qubit which is in this state
let's. Let's say this is, she has this qubit which is in the state alpha zero
plus beta one. It's in an unknown, unknown state. And then Alice wants to send this
qubit over to Bob. And, all they have is an, entangled pair of qubits which they
share a Bell State. And so the teleportation is a, is a protocol where
Alice measures her two qubits in some appropriate basis, sends the two classical
qubits, classical bits across to Bob who makes, makes some unitary correction to
his qubit. And the qubit that Alice destroyed by making a measurement
magically appears on Bob's side. Okay, so that's teleportation. Now, let's step back
and think about it for a moment. So, what is it that, you know what, what
determined, what basis, the qubit psi reappeared on Bob's side? Well, if you
think about it for, for a little while, you'll realize that what determined the
basis was the fact that the Bell State happened to be of the form (00) +
(eleven). What if you had used (01) + (ten). Then if Alice's qubit had been a
zero, it would have appeared on Bob's side as a one and vice versa. So now, why not
take this a step further? Why not take Bob's , qubit and apply a unitary rotation
to it? You could apply any arbitrary unitary rotation to it. And now when Alice
thus had teleportation step, wouldn't you expect it to arrive on Bob's side in this
rotated basis? But if that were the case then, what you would have managed is, if
Alice wanted to apply some unitary rotation to, to her qubit, it would appear
on Bob's side, pre-rotated. Okay, but then you could say about what's the point of
doing something like this? After all you know what does it matter? Alex applies the
unitary rotation, the gate. Bob applies the gate, you have to apply it anyway. So
here is the interesting part. The interesting part is that what this says is
that you could apply a rotation, you could apply the gate before even touching the
qubit. So in other words, suppose that our, our computation is unreliable.
Suppose we don't quite have a handle on it. Sometimes it works, sometimes it
doesn't. And on the other hand the, the cubit that we want to apply the
computation to that's valuable. If we, if we were to spoil it and have to make it
all over again, that would take a lot of effort to, to create. What we could do is
apply the gate in advance. We'd apply it to Bob's qubit in this Bell State. And if
you spoil that computation, we could just discard that Bell State and start all over
again with a new fresh Bell State. And we could keep trying until we succeed in
that, in that, in doing that rotation to our satisfaction. Once we are done with,
with doing the rotation to our satisfaction. Then we do our teleportation
protocol and lo and behold the qubit reappears on Bob's side properly rotated.
Okay, that's the idea at least behind computation by teleportation. Okay, but to
really make it happen, we need to do something a little more interesting. Okay
so, so this is a, this is a schematic of what we would like to happen. So we are
trying to do teleportation before we even start the teleportation circuit, we apply
a single qubit unit, or AU. A quantum gate u to Bob's qubit. And once we are
satisfied we've managed to apply it, we want to teleport psi to Bob. And what we
are expecting is Bob receives u psi. Okay, so let's see whether this, this is going
to work out as expected. Okay, so what, what happens? Well, here's the actual
teleportation circuit. Remember what, what Alice does is she does a CNOT from into,
into, her, her, her part of the Bell State and then she does a Bell basis
measurement. Which is, she applies a Hadamard here, measures these two qubits
and calls up Bob, who applies either a x or a z correction or both. Okay so now,
what, what are we plan, planning to do instead? Now, what we've done is before
any of this teleportation circuit started, what we're doing is we're applying u to
Bob's qubit and what would we like to happen? Alright, what, what, what actually
have we managed to do? Well, what was suppose to happen was without the u here,
it's after we apply the x and z corrections that we got psi. Meaning
whatever the state was, let's say we only applied how, how to apply the x
correction. So it was after applying the x correction, you know we got the correct
bit sign and having applied x, we then wanted to apply u to it. Right? So if you
apply this operator u followed by x, then we got the correct output. This, this
created the output. It would be equal to u psi which is what we wanted. What do we
actually end up applying? Well, we end up applying u first and then x. Okay, now in
general, you know that matrices don't commute. And so u followed by x is in
general, not going to be equal to x followed by u. And so what we would have
to do in order to actually do computation by teleportation is we'd actually have to
apply some sort of correction after having done this. So we, we did the, we applied
our gate too soon. We applied u and then x instead of x and then u. Okay, so it turns
out that there are certain kinds of single qubit gates which actually commute with x
so you can apply them in any order. For, for these kinds of gates, it's actually
very good. For simple gates, we might actually have equality here and then,
those gates we can just do simply by, by teleportation. But there are other gates
where they don't commute but you can apply a very simple unitary correction so you,
you do get equality if you apply some unitary correction a. And if a is simple
enough, then this is a worthwhile process to carry out. And thi s is what
computation by teleportation does. So if you want to read more about this, here's
the paper that, that introduced this concept. It's, I think, a very beautiful
paper and there are lots of follow up works based on it having to do with how
you carry out quantum computation reliably in the presence of noise? So it's called
fault-tolerant quantum computing. There are ways of using what, what this also
shows is that entanglement can be used as a resource in quantum computation. And
then there's, you know this has been developed in very interesting ways in
what's called quantum computation by measurement.