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Okay, so quantum teleportation might sound
too exciting and too science fiction to be

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of any actual interest, technically but
it's, it's actually a very fundamental

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primitive in, in computation. And, so in
this video I'll, I'll say in a few

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minutes, you know I'll give you a little
sketch of how teleportation gets used in

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computation. This might be a little more
advanced but it's really a point into the

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literature for those of you who are
interested. Okay so, so let's look at how,

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how telepretation proceeds, right? So
Alice has a qubit which is in this state

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let's. Let's say this is, she has this
qubit which is in the state alpha zero

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plus beta one. It's in an unknown, unknown
state. And then Alice wants to send this

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qubit over to Bob. And, all they have is
an, entangled pair of qubits which they

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share a Bell State. And so the
teleportation is a, is a protocol where

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Alice measures her two qubits in some
appropriate basis, sends the two classical

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qubits, classical bits across to Bob who
makes, makes some unitary correction to

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his qubit. And the qubit that Alice
destroyed by making a measurement

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magically appears on Bob's side. Okay, so
that's teleportation. Now, let's step back

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and think about it for a moment. So, what
is it that, you know what, what

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determined, what basis, the qubit psi
reappeared on Bob's side? Well, if you

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think about it for, for a little while,
you'll realize that what determined the

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basis was the fact that the Bell State
happened to be of the form (00) +

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(eleven). What if you had used (01) +
(ten). Then if Alice's qubit had been a

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zero, it would have appeared on Bob's side
as a one and vice versa. So now, why not

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take this a step further? Why not take
Bob's , qubit and apply a unitary rotation

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to it? You could apply any arbitrary
unitary rotation to it. And now when Alice

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thus had teleportation step, wouldn't you
expect it to arrive on Bob's side in this

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rotated basis? But if that were the case
then, what you would have managed is, if

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Alice wanted to apply some unitary
rotation to, to her qubit, it would appear

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on Bob's side, pre-rotated. Okay, but then
you could say about what's the point of

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doing something like this? After all you
know what does it matter? Alex applies the

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unitary rotation, the gate. Bob applies
the gate, you have to apply it anyway. So

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here is the interesting part. The
interesting part is that what this says is

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that you could apply a rotation, you could
apply the gate before even touching the

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qubit. So in other words, suppose that
our, our computation is unreliable.

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Suppose we don't quite have a handle on
it. Sometimes it works, sometimes it

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doesn't. And on the other hand the, the
cubit that we want to apply the

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computation to that's valuable. If we, if
we were to spoil it and have to make it

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all over again, that would take a lot of
effort to, to create. What we could do is

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apply the gate in advance. We'd apply it
to Bob's qubit in this Bell State. And if

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you spoil that computation, we could just
discard that Bell State and start all over

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again with a new fresh Bell State. And we
could keep trying until we succeed in

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that, in that, in doing that rotation to
our satisfaction. Once we are done with,

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with doing the rotation to our
satisfaction. Then we do our teleportation

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protocol and lo and behold the qubit
reappears on Bob's side properly rotated.

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Okay, that's the idea at least behind
computation by teleportation. Okay, but to

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really make it happen, we need to do
something a little more interesting. Okay

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so, so this is a, this is a schematic of
what we would like to happen. So we are

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trying to do teleportation before we even
start the teleportation circuit, we apply

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a single qubit unit, or AU. A quantum gate
u to Bob's qubit. And once we are

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satisfied we've managed to apply it, we
want to teleport psi to Bob. And what we

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are expecting is Bob receives u psi. Okay,
so let's see whether this, this is going

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to work out as expected. Okay, so what,
what happens? Well, here's the actual

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teleportation circuit. Remember what, what
Alice does is she does a CNOT from into,

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into, her, her, her part of the Bell State
and then she does a Bell basis

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measurement. Which is, she applies a
Hadamard here, measures these two qubits

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and calls up Bob, who applies either a x
or a z correction or both. Okay so now,

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what, what are we plan, planning to do
instead? Now, what we've done is before

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any of this teleportation circuit started,
what we're doing is we're applying u to

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Bob's qubit and what would we like to
happen? Alright, what, what, what actually

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have we managed to do? Well, what was
suppose to happen was without the u here,

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it's after we apply the x and z
corrections that we got psi. Meaning

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whatever the state was, let's say we only
applied how, how to apply the x

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correction. So it was after applying the x
correction, you know we got the correct

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bit sign and having applied x, we then
wanted to apply u to it. Right? So if you

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apply this operator u followed by x, then
we got the correct output. This, this

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created the output. It would be equal to u
psi which is what we wanted. What do we

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actually end up applying? Well, we end up
applying u first and then x. Okay, now in

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general, you know that matrices don't
commute. And so u followed by x is in

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general, not going to be equal to x
followed by u. And so what we would have

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to do in order to actually do computation
by teleportation is we'd actually have to

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apply some sort of correction after having
done this. So we, we did the, we applied

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our gate too soon. We applied u and then x
instead of x and then u. Okay, so it turns

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out that there are certain kinds of single
qubit gates which actually commute with x

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so you can apply them in any order. For,
for these kinds of gates, it's actually

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very good. For simple gates, we might
actually have equality here and then,

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those gates we can just do simply by, by
teleportation. But there are other gates

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where they don't commute but you can apply
a very simple unitary correction so you,

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you do get equality if you apply some
unitary correction a. And if a is simple

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enough, then this is a worthwhile process
to carry out. And thi s is what

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computation by teleportation does. So if
you want to read more about this, here's

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the paper that, that introduced this
concept. It's, I think, a very beautiful

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paper and there are lots of follow up
works based on it having to do with how

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you carry out quantum computation reliably
in the presence of noise? So it's called

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fault-tolerant quantum computing. There
are ways of using what, what this also

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shows is that entanglement can be used as
a resource in quantum computation. And

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then there's, you know this has been
developed in very interesting ways in

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what's called quantum computation by
measurement.