1
00:00:00,000 --> 00:00:07,009
Okay. So now, let's try to understand some
specific unitary transformations on single

2
00:00:07,009 --> 00:00:15,005
qubits. Some specific transformations that
we'll use all the time. And, in fact,

3
00:00:15,005 --> 00:00:22,000
it's, it's useful to, you know, the way
they think about these unitary

4
00:00:22,000 --> 00:00:29,003
transformations on single qubits as, as,
is as quantum gates. So, I'm sure many of

5
00:00:29,003 --> 00:00:35,000
you know about classical circuits where
you have wires that carry bits of

6
00:00:35,000 --> 00:00:40,076
information and then you have gates that
transform these bits in someway. So, in

7
00:00:40,076 --> 00:00:47,006
the quantum manual analog of that, we
think of we think of a wire as carrying a

8
00:00:47,006 --> 00:00:55,000
quantum bit. And then we have a gate that
performs a unitary transformation on that,

9
00:00:55,000 --> 00:01:01,000
on that qubit. And the output is carried
on the output wire. And that's, that's the

10
00:01:01,000 --> 00:01:06,000
output qubit after the unitary
transformation has been performed on it.

11
00:01:06,000 --> 00:01:11,003
Later in the course, we'll, we'll think
about quantum circuits, where we have

12
00:01:11,003 --> 00:01:17,001
many, many wires carrying many cubits. And
then we'll have gates which, which act on

13
00:01:17,001 --> 00:01:23,001
one or two qubits, on multiple qubits. And
the cubits get transformed along the way.

14
00:01:23,001 --> 00:01:28,007
Okay, but for now, we are just thinking
about single qubits, and how they get

15
00:01:28,007 --> 00:01:33,082
transformed as, as quantum gates are
applied to them. Okay. So, so the first

16
00:01:34,008 --> 00:01:41,070
gate that we'll consider, which is a
particularly simple one, is a bit flip.

17
00:01:41,070 --> 00:01:49,075
So, what the bit flip does is, it's given
by this transformation X here. And now,

18
00:01:49,075 --> 00:01:56,039
of, of course we, we'd want to check that
X is a unitary transformation. So, what we

19
00:01:56,039 --> 00:02:06,036
want to do is look at what's X dagger. But
now, since X is real, of course and, and

20
00:02:06,036 --> 00:02:13,005
it's symmetric, when you do the conjugate
transpose, you'll just get back X itself.

21
00:02:18,005 --> 00:02:29,051
And in fact, you can even check that x^2 ,
X, X dagger = X dagger X Well, it's, it's

22
00:02:29,051 --> 00:02:36,055
just identity. So, it's, it is in fact a
unitary transformation. And, okay, so,

23
00:02:36,055 --> 00:02:43,026
what does, what does x do? If you start
off in the zero state, then you, x<i>0 is,</i>

24
00:02:43,026 --> 00:02:48,084
is the one state. And if you start in one,
then you, you end up in< /i> the zero

25
00:02:48,084 --> 00:02:55,046
state. And so, it's a bit flip. But, of
course, you might start, the qubit might

26
00:02:55,046 --> 00:03:01,002
start in a superpos ition of zero and one.
And so, if you apply X to this

27
00:03:01,002 --> 00:03:07,004
superposition, then zero and one swap
places. So, you end up in, well, the

28
00:03:07,004 --> 00:03:13,007
amplitude of zero is now alpha one, and
the amplitude of one is alpha zero because

29
00:03:13,007 --> 00:03:21,009
zero and one, swap positions. Here's
another gate that that's, that's

30
00:03:21,009 --> 00:03:28,065
interesting. It called the phase flip
gate. It's given by this transformation Z

31
00:03:28,065 --> 00:03:36,003
and once again, Z dagger is equal to Z.
And so Z times Z, Z, Z dagger is just Z^2

32
00:03:36,003 --> 00:03:43,004
which is the identity. So, this is another
unitary transformation. And what it does

33
00:03:43,004 --> 00:03:50,005
is it, it leaves the zero state alone, but
applies a phase of -one to the one state.

34
00:03:51,005 --> 00:03:57,006
Now, of course, an overall phase is never,
never, you know, it's, it's irrelevant.

35
00:03:57,006 --> 00:04:04,004
Well, what this means is that if you start
in the superposition of zero and one, then

36
00:04:04,004 --> 00:04:10,006
the relative phases of these two get
changed by the Z gate. And so you, end up

37
00:04:10,006 --> 00:04:21,071
in alpha 0,0 - alpha 1,1. Okay. It's, it's
simple to think about this geometrically.

38
00:04:21,071 --> 00:04:29,047
So, the bit flip, one way you can think
about it is, that, it's a reflection of

39
00:04:29,047 --> 00:04:40,058
this picture, the 0,1, about the 45-degree
line, okay, about this 45 degree access.

40
00:04:40,058 --> 00:04:46,093
But now, if you were living in
3-dimensional space, then you could also

41
00:04:46,093 --> 00:04:53,046
think about it as a rotation. A 180 degree
or a pi rotation about this 45 degree

42
00:04:53,046 --> 00:04:58,044
axis. Because what you would do is, you
would, you would rotate about this axis

43
00:04:58,044 --> 00:05:04,068
and zero would, let's say, come off the
plane and one would go under the plane and

44
00:05:04,068 --> 00:05:10,000
they'd keep rotating until they'd be in
the orthogonal plane. And then, they'd

45
00:05:10,000 --> 00:05:16,045
keep rotating until they, they switch
places when you, when you go all the way

46
00:05:16,045 --> 00:05:25,027
around. But now, remember, we are working
in not the 2-dimensional real space but we

47
00:05:25,027 --> 00:05:30,080
are working in a complex 2-dimensional
space. And so, what happens is in this

48
00:05:30,080 --> 00:05:38,009
complex 2-dimensional space, you actually
have enough room to do this rotation about

49
00:05:38,009 --> 00:05:43,083
this 45 degree axis. You have these, you
know, the imaginary directions where you

50
00:05:43,083 --> 00:05:49,098
can actually do this rotation. And so, so
X is really a rotation, a pi rotation

51
00:05:49,098 --> 00:05:57,056
around this 45-degree axis. Similarly, for
the phas e flip, well, the phase flip,

52
00:05:57,056 --> 00:06:06,059
what does it do. Well, if you, if you were
to draw a zero and one here, then what it

53
00:06:06,059 --> 00:06:19,038
does is, it leaves the, the zero axis
alone but it flips one to -one. And so,

54
00:06:19,038 --> 00:06:28,047
it's a 180 degree, well, you could think
of it as a reflection about this, this,

55
00:06:28,047 --> 00:06:36,034
this, the, the zero vector or again, in
the complex plane, you can think of it as,

56
00:06:36,034 --> 00:06:43,049
it's a rotation through 180 degrees or pi
about this axis, about the axis of the

57
00:06:43,049 --> 00:06:50,098
x-axis. Here's another way to think about,
about the phase flip. So, what does the

58
00:06:50,098 --> 00:07:00,020
phase flip do to the plus state? Well,
remember the plus state is one / square

59
00:07:00,020 --> 00:07:10,000
root two zero + one / square root two one.
And so what it does is, it leaves zero

60
00:07:10,000 --> 00:07:20,003
alone, but it applies a phase of -one to
one. And that' exactly the minus state.

61
00:07:20,006 --> 00:07:26,007
And similarly, Z, when you applied it to
the minus state, gives you plus. And so,

62
00:07:26,007 --> 00:07:32,009
what it's doing is, if you look at the
plus and minus states, then it's swapping

63
00:07:32,009 --> 00:07:41,004
these two. Now, that's still, this 180
degree rotation about the, about the zero

64
00:07:41,004 --> 00:07:49,005
axis. Okay. So, now, let's look at the
third gate, which is called the Hadamard

65
00:07:49,005 --> 00:07:55,008
gate, or the Hadamard transform. This is
one of the most basic quantum gates. And

66
00:07:55,008 --> 00:08:01,005
what it is, is it's given by this
transformation, which, in which all the

67
00:08:01,005 --> 00:08:07,043
entries are one / square root two, except
the lower right entry, which is -one /

68
00:08:07,043 --> 00:08:20,057
square root two. So, once again, H dagger
is equal to H. You take the conjugate and

69
00:08:20,057 --> 00:08:28,073
then transform. You get back the same
matrix and you can verify that H^2 is the

70
00:08:28,073 --> 00:08:35,081
identity, H, H dagger is the identity. So,
it's a unitary transformation. And now,

71
00:08:35,081 --> 00:08:42,003
let's see what happens if you start with
the zero state and apply this gate to it.

72
00:08:42,003 --> 00:08:48,025
Well, if you start with zero, you get the
first column, which is one / square root

73
00:08:48,025 --> 00:08:54,084
2,0 + one / square root 2,1, which is a
plus state. And if you apply H to one, get

74
00:08:54,084 --> 00:09:02,006
the second column, which is one / square
root 2,0 - one / square root 2,1, which is

75
00:09:02,006 --> 00:09:15,066
the minus state. And similarly, H of plus
is zero and H of minus is one. Now, if you

76
00:09:15,066 --> 00:09:25,029
look at this a little more closely when
you apply H to zero, well, you get an

77
00:09:25,029 --> 00:09:30,077
equal superposition of zero and one. So,
if you were to measure after applying H to

78
00:09:30,077 --> 00:09:36,043
zero, you would see zero and one with
equal probability. And similarly if you

79
00:09:36,043 --> 00:09:42,062
applied H to one, and then measure, you'd
see zero and one with equal probability.

80
00:09:42,062 --> 00:09:48,082
So, it seems as though the information
about whether you started with zero or one

81
00:09:48,082 --> 00:09:53,095
is lost, when you, when you, when you
apply the Hadamard transform. But

82
00:09:53,095 --> 00:09:59,036
actually, it's not. The information is all
put into the sine here, whether the phase

83
00:09:59,036 --> 00:10:06,033
is plus or minus. And in fact, the way we
know that the information is not lost is

84
00:10:06,033 --> 00:10:13,028
because when we apply the Hadamard gate
twice, we get back to the, the, the bit

85
00:10:13,028 --> 00:10:19,082
that we started from. Okay. So, you know,
we saw how, how to think about the,

86
00:10:19,082 --> 00:10:26,076
geometrically think about the bit flip,
the gate, the X gate and the Z gate. Now,

87
00:10:26,076 --> 00:10:32,055
let's look at the Hadamard gate and try to
understand what it looks like

88
00:10:32,055 --> 00:10:38,033
geometrically. Well, so, remember, what
the, what the Hadamard gate does is it, it

89
00:10:38,033 --> 00:10:44,094
swaps between zero and plus, and it swaps
one and minus. And so, one way you can

90
00:10:44,094 --> 00:10:52,071
understand it is, if you take this axis
which bisects zero and plus, so this angle

91
00:10:52,071 --> 00:11:03,033
will now be pi by eight. Then what, what
the Hadamard gate, you can think of it as

92
00:11:03,033 --> 00:11:11,087
is a, is a reflection about this axis, the
pi by eight axis. But again, the right way

93
00:11:11,087 --> 00:11:18,048
to think about it in the complex
2-dimensional space is as a rotation

94
00:11:18,048 --> 00:11:24,042
through pi by eight, through 180 degrees
of pi, about this pi by eight axis. So,

95
00:11:24,042 --> 00:11:31,013
it's a rotation about this axis where,
where everything is rotating by pi. Okay,

96
00:11:31,013 --> 00:11:38,045
so finally, let's think about the
following question. So, suppose that you

97
00:11:38,045 --> 00:11:46,061
wanted to implement a phase flip gate. So,
suppose I wanted to implement the phase

98
00:11:46,061 --> 00:11:53,029
gate or Z. But suppose that I happen to
have no phase flip gates around. But I had

99
00:11:53,029 --> 00:12:00,017
a big flip gate and a Hadamard gate. Would
I be able to implement a phase flip gate?

100
00:12:00,017 --> 00:12:06,014
Well, here's a way of doing it. So,
remember what the phase flip does. So, if

101
00:12:06,014 --> 00:12:13,014
you, if you, if you write it out, what Z
does is, if you give it as input, the plus

102
00:12:13,014 --> 00:12:20,091
state. Then it outputs minus, and if you
give it as input minus, then it outputs

103
00:12:20,091 --> 00:12:28,057
plus. And if I can design some gate which,
which transforms plus to minus, minus, and

104
00:12:28,057 --> 00:12:35,023
minus to plus, then it must be the phase
gate. Okay, so how do we, how do we write

105
00:12:35,023 --> 00:12:42,034
out, how do we create a, a gate that
transforms plus to minus and minus to plus

106
00:12:42,034 --> 00:12:49,081
which swaps plus and minus. Well, what we
can do, if you are not given Z, is we can

107
00:12:49,081 --> 00:12:59,034
first start by transforming plus 2,0 and
-2,1. We can do that by applying a

108
00:12:59,034 --> 00:13:09,022
Hadamard gate. Okay. Now that we've done a
Hadamard gate, we can, of course, apply an

109
00:13:09,022 --> 00:13:15,045
x gate, and swap zero with one, and one
with zero, alright? So if, if you now

110
00:13:15,045 --> 00:13:20,073
apply an x gate, then zero gets
transformed to one and one gets

111
00:13:20,073 --> 00:13:29,001
transformed to zero. And now, if we apply
Hadamard again, what, what happens? Well

112
00:13:29,001 --> 00:13:38,050
if you apply Hadamard again, we transform
one to minus, and zero to plus. Okay. So,

113
00:13:38,050 --> 00:13:45,059
what, what, what is, what is this sequence
doing for us? It's transforming, if you

114
00:13:45,059 --> 00:13:52,041
start with a plus state, then our output
is a minor state. And if you start with a

115
00:13:52,041 --> 00:13:59,047
minor state, output is a plus state, which
is exactly what we wanted. So, what we can

116
00:13:59,047 --> 00:14:07,062
say is that Z = HXH. Okay? In fact, here's
another way to, to picture it. Okay. So

117
00:14:07,062 --> 00:14:14,099
what, what this is, you know, what this
diagram is showing you is that, what the

118
00:14:14,099 --> 00:14:21,082
Hadamard does is it swaps zero and plus.
It swaps one and minus. X swaps between

119
00:14:21,082 --> 00:14:28,050
zero and one. Z swaps between plus and
minus. And so, if you wanted to perform Z,

120
00:14:28,050 --> 00:14:36,025
where you want to swap these two, another
way you can do that is perform HXH. If you

121
00:14:36,025 --> 00:14:42,069
want to perform X, another way to achieve
that is you, you do HZH. Okay, what does

122
00:14:42,069 --> 00:14:50,032
this correspond to in terms of linear
algebra? Well, what you're really doing is

123
00:14:50,032 --> 00:15:00,053
we are saying that Z, the transformation Z
is the same transformation as X if you, if

124
00:15:00,053 --> 00:15:05,068
you, if you do the transformation in the
Hadamard basis. So, this is really basis

125
00:15:05,068 --> 00:15:12,054
stage. What we're really saying is that if
we perform the Hadamard transform, then we

126
00:15:12,054 --> 00:15:18,072
perform X and now we perform the inverse
of the Hadamard transform to move back to

127
00:15:18,072 --> 00:15:28,000
our original basis then we get, that's,
that's the Z transformation, But of

128
00:15:28,000 --> 00:15:39,030
course, remember that H inverse = H. And
so, so, of course, we can write this as Z

129
00:15:39,030 --> 00:15:40,004
= HXH.