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So now we are ready to look at our first
quantum circuit. So, how does a quantum

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circuit work? Quantum circuit consists of
wires and gates. It's just like a

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classical circuit except that the gates
are now quantum gates and the wires carry

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bits of information. The one other
property of quantum circuits is every gate

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has the same number of wires leaving it as
the number that enter it, okay. Let's make

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up a circuit that consists of two gates,
the Hadamard Gate and the CNOT Gate. So

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remember what the Hadamard Gate does? What
it does is. H(0) =,, + and H(1) =,, -. In

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fact H(+) = (zero) and H(-1) = (one) as
well. So it swaps zero and +, swaps one

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and -. So it's a one qubit gate. Okay, so
now what we want to understand is what's

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the behavior of the simple quantum circuit
where you first perform a Hadamart on the

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first qubit. And then you perform a CNOT
with the first qubit as the source as the,

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as the control bit and the second qubit as
a target bit. So let's try to understand

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what happens if the input bits are both
zero. So, let's follow, follow the state

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as we go along. So initially, the state is
00 but then what happens after the, after

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this first Hadamard? Well the first qubit
now is in the state one / square root two

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(zero) + one / square root two (one). And
the second qubit is still in the state

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zero. So the state here is one / square
root to (00) + one / square root two

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(ten). Okay so now, this is the input
state to the CNOT Gate. So what happens

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after we go through this CNOT Gate? Well,
okay, so you want to understand what the

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state is here. What is the CNOT Gate do to
(00)? Well, the control bit is zero so you

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end up with one / square root two (00).
What about (ten)? Well now the control bit

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is one and so the target bit gets split so
you end up in the state one / square root

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two (eleven). I hope you'll recognize this
state. This is a Bell State. Okay so, here

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is a circuit that starts with two qubits
initially in the zero state and what it

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does is it entangles them into a Bell
State. Okay now, here's the interesting

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thing. If you play with the circuit you
realize that as y ou change he inputs from

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00 to the four possibility, the other
three possibilities 01, ten, and eleven.

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The output state changes between this,
whats called this five plus state which

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was, which was what the, what we were
calling the Bell State but in fact there

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is a whole basis of Bell States. There are
four orthogonal Bell States. These are

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called the Bell Basis States and they look
like this. They, they come in pairs.

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There's phi+ and phi-. Phi+ is 00 +
eleven. Phi- is 00 - eleven. Of course,

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with this normalization, one / square root
two. And then you have psi+ and psi- which

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are 01 + ten and 01 - ten. Okay, so you
should check that these are four mutually

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orthogonal basis states. They are, they
are mutually orthogonal unit vectors. And

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you should also check that as you change
the input bits. Say from 00 to, should we

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do one more example. Let's say, what
happens if you start with ten? So suppose,

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suppose that you start with ten. Now,
what's the output look like? Well, so at

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this point we'll be in the state (one /
square root two (zero) - one / square root

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two (one)) (ten) which is. One / square
root two (00) - one / square root two

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(ten). And now when we feed this through
this CNOT Gate and we are looking at the

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state here. This, the first part goes to
just (00). The second part, the control

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bit is one, so we get (eleven) which is
exactly this one, okay. So check that you

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understand what happens in the case you
have 01 and then eleven as inputs? Okay,

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so now let's look at one other aspect of
this. Remember that we said the CNOT is

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its own inverse, Hadamard is its own
inverse so now, let's think about it this

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way. Let's think the input bits as coming
in from this side and the output bits

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coming out on, on the left. Here. Okay, so
instead of the circuit being one where you

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first do a Hadamard on the first qubit and
then do a CNOT, what if you, first the

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CNOT and then the Hadamard. Okay. So, so
just for convenience let me think of the

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circuit as running from right to left.
Okay but, but remember, we could have done

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exactly the same thing where we flip the
circuit arou nd. So we, we put the, the

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CNOT first on the left, then the Hadamard
and then we put the inputs on the left and

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the output on the right, it's all the same
thing. But for that in a mess, what I'm

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going to do is I'm going to now think of
the input is coming in from the right and

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I'll keep the circuit the way it was.
Except as I think of, think of the bits as

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moving from right to left. Okay. So what
happens? So lets, lets hear that we feed

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in the Bell State (00) + (eleven) We want
to know what's the output going to be?

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What happens? What, what would the state
be out here? Well I claim it's exactly

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what it was, you know so when we started
from 00 and this, ran the circuit forward

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the state out here was one / square root
two (00) + one / square root two (ten).

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And then, when we ran this forward, this,
this state got transformed into the Bell

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State after going through the CNOT. Well,
what happens if we apply CNOT again to

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this state? I plainly come back to where
we started from because you apply the CNOT

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twice, you come back to where you started
from. Okay, so when we move from right to

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left here, we are just retracing our path.
Same goes for the Hadamard. When we apply

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the Hadamard to this state we retrace our
path and we come back to 00. Okay? So, so

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here's the, here's the picture you should
have. If you start with the state 00, you

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apply this circuit to it, it entangles the
qubits and creates this Bell State. If you

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start from ten, it entangles it and
creates this, this other Bell State phi-.

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And similarly for (01), (eleven) creates
this two states. But now you can also run

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the circuit backwards from right to left.
And now, if you feed in a Bell State on

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this side, on the right side what the
circuit does for you as you run it from

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right to left is it analyzes the Bell
State for you. And outputs (00), (01),

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etcetera which tells you which Bell State
it was. Okay, what does this mean? So

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suppose that I, I were to tell you I have
these two qubits. They are in one of the

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four Bell States, one of these four Bell
States, psi+, phi+, psi-, phi- but I don't

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know which one. Can you please help me
figure out which one it's in? This is what

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you would do. You would, you would create
the circuit. And it created, you know to

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look like this you, you know, at first
does a CNOT and then does a Hadamard. And

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now, what this, what this circuit does for
you is, is it analyzes Bell States. What

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that means is after you run the circuit,
if you just measure these two qubits in

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the standard basis. If you get as output
00 you knew that the Bell State was phi+,

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it was (00) + (eleven). If you get (ten)
you know it was phi-, (00) - (eleven),

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etcetera. So this is what's called a Bell
Basis Measurement Circuit