U1L4m.txt

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So the next thing I want to do is ask, what happens here?

We apply H-- apply H to both of these
and control NOT and then another Hadamard to each of them.
What is this?

Let's see.

So this is one of the nicest identities
that we'll see in quantum computing.
And let's see.

Well, I don't know any good easy way to see this.
Do you, Ike?
I like to do it in block matrix form.
OK, well, so what I was going to do is just
say what happens when you input each of the--
put in some input.
So suppose you put in 0, 0 here.
What would you get?
Here you'd get 0, 0 plus 0, 1 plus 1, 0 plus 1, 1.
Now the controlled NOT takes the first qubit and [INAUDIBLE]
with a second qubit.
So you get 0, 0; 0, 1; 1, 0; 1, 1 here.
And they're all pluses, because these were all pluses.
And now-- actually, maybe I should--

1, 0 goes to 1, 1, and 1, 1 goes to 1, 0.
But this is the same as this.
So when you apply two more H's there, it goes to 0, 0.
So that means you take an inputted 0, 0
to an outputted 0, 0.
And that's a 1 there.

And now I'm going to be very annoying for those
who take notes and use Blackboard arithmetic.
So what would happen if we put in 0, 1?

Well, what would happen is that this would change to a minus,
and this would change to a minus.

And of course, this would change to a minus.
This would change to a minus.
And now if you look at this, this is 0, 0 minus 1, 0 and--
or 0, 0 minus 1, 0 and minus 0, 1 and 1, 1 minus 0, 1 minus 1,
0 plus 1, 1, which is just 0 minus 1, 0 minus 1.
And this state is--
when you take the Hadamard, this turns into a 1,
this turns into a 1, and you get 1, 1.

So 0, 1 goes to 1, 1.

So you stick 0, 1 in.
You get 1, 1 out.
And that's a sky.

OK, now-- well, I'm want to say this is completely symmetric.
So if we'd stuck a 0--
stuck a 1, 1 in, we would have gotten a 0, 1 out.
Because you flip this circuit right to left,
you get the same thing.
So if you have an input that leads to some output,
then the output has to leave back to the input.
But we can also check back with plus.
So this a minus.
This is a plus--

oh, wait, yeah.
An 0, 0 minus 0, 1; 1, 0 minus 1, 1, that is 0,
1 when you are undo the Hadamard.
So this is right.
And then, of course, this last thing has to be a 1 here.

Well, actually it doesn't.
As it was, it might have been a minus 1,
but we can check that too, 0, 0; 0, 1.
So this last thing is 1, 0.
So that means 0 minus 1 here.
This is a minus plus.
This is a minus.
This is a minus.
This is a plus.
So we have 0, 0 plus 0, 1 minus 1, 0 minus 1 here 0, 0 plus 0,
1 minus 1, 0 minus 1, 1 here.
This is the same state as that.
So we get that, 1, 0; 1, 0.
So what does this do?
Well, if the second--

I want to say if the second element is a 0,
it leaves the first bit alone.
If the second bit is a 1, it flips the first bit, because 1,
1 goes to 1, 0, and 1, 0 goes to 1, 1.

So what does that mean?
Well, that means this is our control NOT
gate going the other way.
This takes the first qubit and [INAUDIBLE] it
into the second qubit.
And if you-- well, either if you change bases to plus minus
or if you put a Hadamard on both ends,
this takes the second qubit and [INAUDIBLE] it
into the first qubit.

So this is actually an example of a fundamental phenomenon
in quantum mechanics, that if A does something to B,
then B does something to A. So it's called--

it's called back reaction.
Back action?
Back action.
Back action, right.
Back action.

So if A affects B, then B must effect A.
So I don't think we'll explain this much in this class,
except we'll probably see some more instances of this,
but this is a general phenomenon in quantum mechanics.
OK, are there any questions?