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.