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.