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Hello, everyone. Today we have a very
interesting lecture, it's straight out of

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science fiction so we'll talk about
Quantum Teleportation. But before we get

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there, we have to say something about,
quantum circuits. Okay so, lets see.

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Remember, last time we talked about
one-qubit gates. So one-qubit gates, we

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think of as you know, we have a wire that
carries a qubit of information. So let's

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say, α0(0) + α1(1). And now, we have a
gate which, which acts on it and remember

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what a gate does is it takes this two
dimensional complex vector space. The

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state is a unit vector in that space and
then it takes the space and we rotate it.

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Okay, this rotation in the complex vector
space is called the Unitary Transformation

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U, which in this case is a two by two
unitary transformation. So U is two by two

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unit rate transformation. And, the fact
that this unitary is given by the

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condition U times U [inaudible] is, is U
[inaudible] times U is identity. So we saw

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several examples of, of, of such gates and
you know, they included x, the nought

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gate, Z, the face flip, H, the Hadamard
gate. Okay, so what, what these do is,

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they take the input qubit and convert it
into, into some new qubit, alpha nought

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prime zero plus alpha one prime one. Now
of course, you remember that you could,

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you could describe the you know, the,
this, this rotation, the unitary rotation

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by either specifying the unitary matrix,
or you could just say zero gets mapped to

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what and one gets mapped to what, right?
So if you, if you specify these, these

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four numbers which are that zero gets,
gets mapped to a(0) + b(1) and one gets

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back to c(0) + d(1) then you specify it,
what the, what the mapping is because it's

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a linear map. So now if you have α(0) +
β(1), what does it go to? Well, it goes

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to alpha times a(0) + b(1) + beta times
c(0) + d(1). Also, the unitary matrix

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that, that, that describes how these you
know, the mapping is, is just given by a,

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b, c, d. Okay, so that's, that's what we
did last time. And now, this time we are

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going to talk about two-qubit ga tes. So,
so here is our, here is our picture. So,

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we have two wires that are coming in, each
carrying a qubit. And then, this

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encounters a gate U and, then you have two
wires coming out, which, which, which are

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not carrying two qubits, which are in a
different state. So, lets, what is the

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state of our initial qubits? Well remember
that, the state of two-qubits is described

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by four complex numbers, the four complex
amplitudes, alpha 00 through alpha eleven.

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And output qubits are described by four
new complex numbers, alpha 00 prime

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through alpha eleven prime. So now, the
way that this output state is related to

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the input state, is, of course, given by
this unitary transformation U, which is a

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linear transformation. It's, have what,
what's the dimension of this

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transformation. Well, clearly, U is four
dimensional. Right? Meaning, if we write

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it out, it would, looked like, this kind
of matrix. Okay. And of course, you

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satisfy this condition, uu dagger equal to
u dagger u is the identity. And once

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again, you can specify the action of U by
just saying, how does it map 00? How does

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it map 01? Etcetera. How does it map
eleven? So, so if you say, how does it map

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each of these four states, then you, then
you have specified U as well. Okay? So,

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let's, let's say that it maps (00) to
a(00) + b(01) + c(10) + d(11). Then we

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know that the first column of this matrix
is going to be a, b, c, d. Right? So check

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that you understand why this is, right?
Because 00 is this vector. Okay? So if you

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want to know what, what this u map 00 to
u, you multiply this matrix u by this

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vector. And what do you get? You get this
column, a, b, c, d, which is exactly what

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this was. Okay, so make sure you
understand all this. Okay. So, now, let's

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look at an example of, of a two qubit
gate. This is the most basic of the

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two-qubit gates. So it takes as input two
qubits, outputs two qubits and it's

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denoted by this funny symbol where there's
a dot on the one side align which with

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this, with this with this circular round,
which denotes, an XOR. Okay, so if this wa

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s a classical gate, so if the inputs a and
b were classical, then outputs would look

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like this, the first input is let through
unchanged and the second bit, so the first

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bit is let through unchanged and the
second bit is flipped, if and only if the

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first is one. Okay, so the first bit, this
is called, the control. And that bit is

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called the Target Bit. Okay, so the CNOT
stands for Controlled NOT. Alright. Okay,

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so now remember what the rules were that
if you want to describe what the CNOT gate

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is, it's sufficient to describe what it
does on input 00, 01, ten, eleven. So

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let's see, what, how does the CNOT gate
map 00? So now the control bit is zero so

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the target bit remains unchanged so it
gets mapped to 00. How about 01? Well, the

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control bit is twenty so the target bit is
unchanged so, 01. How about ten? Well, now

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the control bit is a one, so the target
bit flips and so it gets mapped to one,

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one. How about one, one, one? Okay, so
this time, okay again the controlled bit

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is one, so the target bit flips. So the
output is one, zero. So now you know

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exactly how CNOT works because if you are
given any super position of this four

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states. So if you are alpha (00). Okay, so
if you had this superposition alpha 00(00)

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+ alpha 01(01), + alpha ten(10) + alpha
eleven(11). So what does it get mapped to

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under CNOT? Well answer is simple. What,
what it does is zero, zero stays unchaged,

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so it goes to alpha 00(00). Zero, one
unchanged so it's 01(01). Now, one, one

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gets mapped to one, zero so you would have
alpha eleven(10) and one, zero gets mapped

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to one, one so you get alpha ten(11).
Okay. We can also write out what the

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matrix is, corresponding to CNOT. And CNOT
is given by this matrix. So, make sure you

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see why this matrix is giving you this
transformation. Right? Because this one

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gives you the first column, which is one,
zero, zero, zero. This gives you the

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second column, this gives you the third
column, where you get a one, in the last

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entry, because every one, one. This gives
you the fourth column, where you have one

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in the third entry. And of course, its
unitary to check that you, just check that

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CNOT times CNOT conjugate transpose is the
identity. In fact, again, what you have is

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that the conjugate transpose is itself and
so CNOT is it's own inverse. If you apply

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it twice, you get back to where you
started from.