1
00:00:00,000 --> 00:00:10,083
Okay. So now, let's look at what happens
when you, when you have a tensor product

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of operators or gates, right? So, here's
what I mean. So, let's say, I have two

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00:00:26,048 --> 00:00:35,026
qubits. And you apply the quantum gate,
the U to the first qubit, and V to the

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second qubit. So now, what we want to know
is what's the [unknown] transformation

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we've applied, to the two qubits together,
right? So, in other words, what we want to

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know is if U was a, b, c, d, and V was e,
f, g, h. Then, what's this transformation

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00:01:06,034 --> 00:01:12,080
we've applied to the two qubits? The way
we denote it is, we denote this

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transformation is by U tensor V. And it's
going to be a four x four matrix. Write

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index by zero, zero, zero one, one zero,
one, one. And what it looked like is, so,

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these first four entries will look just
like V, except they'll be scaled by a,

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00:01:59,016 --> 00:02:05,003
these four entries will look just like V
except scaled by b, and so on. Okay, so,

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00:02:05,003 --> 00:02:11,077
here's what I mean by this. So, let's
write out these four entries explicitly.

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00:02:11,077 --> 00:02:23,072
So, these first entries would be atimes e,
a times f, a times g, a times h Okay. So,

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00:02:23,072 --> 00:02:29,085
let's, let's try to understand, why this
is the case. So, first of all, let's just

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00:02:29,085 --> 00:02:35,039
make sure we're, we're on the same page
with respect to convention. So, if you

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00:02:35,039 --> 00:02:41,054
were to, you know, if, if you were to say
that input is 01, right, what does that

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00:02:41,054 --> 00:02:48,002
mean? Well, it means that this cubit is
zero and this cubit is one, right? Okay.

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00:02:48,002 --> 00:02:58,024
So, so now, let's try to understand what
happens when if you apply U to this, to,

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00:02:58,024 --> 00:03:06,056
to this qubit? Well, what's that doing?
So, remember, it's only applying to this,

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00:03:06,056 --> 00:03:14,001
this first qubit and leaving the second
one unchanged, and since it leaves it

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00:03:14,001 --> 00:03:22,009
unchanged, it's as though. So, it, it
applies only to this first cubit and

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00:03:22,009 --> 00:03:28,044
leaves the second one unchanged, right?
So, what are we doing when we leave the

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00:03:28,044 --> 00:03:34,078
second one unchanged? It means that we go
from zero to zero with amplitude a, right?

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00:03:34,078 --> 00:03:41,012
So, it's as though. All these four
entries, whatever happens to them as a

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00:03:41,012 --> 00:03:48,030
result of the second gate, what the first
gate does is it goes from here to this,

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00:03:48,030 --> 00:03:55,078
this quadrant with, with, with amplitude a
and s imilarly this quadrant has amplitude

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00:03:55,078 --> 00:04:05,010
b, c, and d but that's the effect of U.
The effect of V, okay, so if we were to

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00:04:05,010 --> 00:04:13,059
write down the effect of V, well then what
happens there? Well, now V is acting on

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00:04:13,059 --> 00:04:26,093
this, on this qubit which is, which is in
the least significant position. So, okay,

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00:04:26,093 --> 00:04:40,068
which means that you're going from zero to
zero with amplitude e f, g, g, right? And

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00:04:40,068 --> 00:04:49,020
then, of course, it has to be scaled by
whatever amplitude you go from where,

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00:04:49,020 --> 00:04:55,056
where, the most significant bits, stays as
zero. Okay. So, if you put both of these

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00:04:55,056 --> 00:05:01,072
effects together, what you get is that
the, is that the transformation is exactly

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00:05:01,072 --> 00:05:07,005
this. You get, you get each of these four
by, two by two blocks is, is a scaled

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00:05:07,005 --> 00:05:13,006
version of V, the transformation V. And
what scaled version? Well, depends upon

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00:05:13,006 --> 00:05:18,032
which quadrant you are in. It's either
scaled by a, b, c, or d. It's exactly in

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00:05:18,032 --> 00:05:26,030
the pattern according to you. Okay. So,
so, let's, let's do an example. So

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00:05:26,030 --> 00:05:40,058
suppose, suppose, suppose now, U was equal
to the Hadamard transform and V was equal

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00:05:40,058 --> 00:05:52,036
to X which is zero one, one zero. Okay.
So, so now, an H of course, you know is,

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00:05:52,091 --> 00:05:57,066
is one / square root two, one / square
root two, one / square root two - one /

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00:05:57,066 --> 00:06:11,086
square root two. So, what does H tensor V
equal to? Alright. So, what did we get?

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00:06:11,086 --> 00:06:20,000
Well. We get one / square root 2x, one /
square root 2x, one / square root 2x - one

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00:06:20,000 --> 00:06:32,013
/ square root 2x which if you, if you open
this out, it's zero, zero one / square

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00:06:32,013 --> 00:06:33,096
root two, one / square root two zero, zero
one / square root two, one / square root

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00:06:33,096 --> 00:06:36,070
two, zero, zero one / square root two, one
/ square root two And, zero, zero - one /

46
00:06:36,070 --> 00:06:38,012
square root two - one / square root two
And so, we just, we just took multiples of

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00:06:38,012 --> 00:00:00,000
X here as each of these four blocks.