Okay. So now, let's look at what happens when you, when you have a tensor product
of operators or gates, right? So, here's what I mean. So, let's say, I have two
qubits. And you apply the quantum gate, the U to the first qubit, and V to the
second qubit. So now, what we want to know is what's the [unknown] transformation
we've applied, to the two qubits together, right? So, in other words, what we want to
know is if U was a, b, c, d, and V was e, f, g, h. Then, what's this transformation
we've applied to the two qubits? The way we denote it is, we denote this
transformation is by U tensor V. And it's going to be a four x four matrix. Write
index by zero, zero, zero one, one zero, one, one. And what it looked like is, so,
these first four entries will look just like V, except they'll be scaled by a,
these four entries will look just like V except scaled by b, and so on. Okay, so,
here's what I mean by this. So, let's write out these four entries explicitly.
So, these first entries would be atimes e, a times f, a times g, a times h Okay. So,
let's, let's try to understand, why this is the case. So, first of all, let's just
make sure we're, we're on the same page with respect to convention. So, if you
were to, you know, if, if you were to say that input is 01, right, what does that
mean? Well, it means that this cubit is zero and this cubit is one, right? Okay.
So, so now, let's try to understand what happens when if you apply U to this, to,
to this qubit? Well, what's that doing? So, remember, it's only applying to this,
this first qubit and leaving the second one unchanged, and since it leaves it
unchanged, it's as though. So, it, it applies only to this first cubit and
leaves the second one unchanged, right? So, what are we doing when we leave the
second one unchanged? It means that we go from zero to zero with amplitude a, right?
So, it's as though. All these four entries, whatever happens to them as a
result of the second gate, what the first gate does is it goes from here to this,
this quadrant with, with, with amplitude a and s imilarly this quadrant has amplitude
b, c, and d but that's the effect of U. The effect of V, okay, so if we were to
write down the effect of V, well then what happens there? Well, now V is acting on
this, on this qubit which is, which is in the least significant position. So, okay,
which means that you're going from zero to zero with amplitude e f, g, g, right? And
then, of course, it has to be scaled by whatever amplitude you go from where,
where, the most significant bits, stays as zero. Okay. So, if you put both of these
effects together, what you get is that the, is that the transformation is exactly
this. You get, you get each of these four by, two by two blocks is, is a scaled
version of V, the transformation V. And what scaled version? Well, depends upon
which quadrant you are in. It's either scaled by a, b, c, or d. It's exactly in
the pattern according to you. Okay. So, so, let's, let's do an example. So
suppose, suppose, suppose now, U was equal to the Hadamard transform and V was equal
to X which is zero one, one zero. Okay. So, so now, an H of course, you know is,
is one / square root two, one / square root two, one / square root two - one /
square root two. So, what does H tensor V equal to? Alright. So, what did we get?
Well. We get one / square root 2x, one / square root 2x, one / square root 2x - one
/ square root 2x which if you, if you open this out, it's zero, zero one / square
root two, one / square root two zero, zero one / square root two, one / square root
two, zero, zero one / square root two, one / square root two And, zero, zero - one /
square root two - one / square root two And so, we just, we just took multiples of
X here as each of these four blocks.