Einführung in das Quantencomputing - Teil 1

1. 00:00Welcome, ladies and gentlemen to the course, Fundamentals of Quantum Computing 1.
2. 00:06And we're getting into the gates now, the quantum gate, but first let's take a look.
3. 00:12Where do we actually stand in the overall course?
4. 00:17We have quantum registers, completed knowledge of what a quantum register is, and now come to the point that are on how
5. 00:23bits operate in quantum registers.
6. 00:25And we will consider there three gates, first operating on a quantum bit, the Pauli X, Pauli Z, and Hadamar.
7. 00:34So this video is about the Pauli X gate.
8. 00:37Why is this called Pauli X?
9. 00:39It is also often simply called the X gate.
10. 00:43It is named after the physicist Wolfgang Pauli.
11. 00:51What does the Pauli X gate do?
12. 00:53The Pauli X gate wants a quantum bit as input,
13. 00:59it then swaps the amplitudes of this quantum bit, and this can be imagined now in quite different ways.
14. 01:09So it swaps the amplitudes, that is, if you apply it to a quantum bit like Alfa times 0 plus, beta
15. 01:16times 1, the amplitudes are interchanged, comes out beta times zero plus alpha times 1.
16. 01:24This is called as the mathematical modeling.
17. 01:28You can calculate very well with this.
18. 01:36Here is an example.
19. 01:38The example is also clearly illustrated graphically by applying Pauli X to a quantum bit in the minus half state
20. 01:47times zero,
21. 01:49so minus a half times zero that is negative, here is a small red square with the edge length a half
22. 01:59plus, here stands now root three half, because the sum of minimum one half to the square and what stands here must
23. 02:06result in one.
24. 02:07You remember quantum gates, a quantum bit is always alpha times zero plus beta times one, so alpha squared plus beta squared
25. 02:15one is. So here we have square root of three halves, that's a square with side length root of three halves and if you
26. 02:22add the areas of these two squares of the red and green comes out as area one and Pauli-X applied to it
27. 02:29now swap the two small squares, i.e. the large green one moves to zero and the small red one moves from zero
28. 02:37away to one or in forms, it returns square root of three half times zero minus half times one.
29. 02:51Now one can consider what X, applied to the zero state, delivers, well the zero state is once the zero state plus zero
30. 03:01swapping times the one state gives zero times the zero state and once the one state, i.e. the one
31. 03:08status.
32. 03:11You can now think about what happens, if you apply the pauli X gate to the one
33. 03:17state to the other base state.
34. 03:24Correctly seen, the zero state comes out
35. 03:32from a computational view point.
36. 03:36If you are a photographer and think in photons, then you can also say yes pauli X is the reflection at a 45 degree
37. 03:44straight line, so what if for them a quantum bit is a photon polarised in some direction,
38. 03:53for example we have one here that's polarised in the fixed north-west-west direction and we're applying it to the Pauli
39. 04:00X then what you get is if you mirror this at the 45 degrees straight.
40. 04:10It comes out, notice a mirroring is not a measurement, you can mirror quantum bits and it's not that you can
41. 04:17measure them.
42. 04:18If you measured it, it will always come out zero or one.
43. 04:22Now we've seen three ways that you can think of that Pauli X gave, which is the first way is that you say it's
44. 04:29the amplitudes are reversed mathematically in the formula.
45. 04:34The second way is to say the two squares are interchanged in the graphical representation, and the third way is to say
46. 04:41says the quantum bit is mirrored 45 degree straight lines,
47. 04:48all three are equivalent.
48. 04:49Choose for yourself what suits you best.
49. 05:02What happens now if you apply Pauli-X to a quantum bit?
50. 05:07Pauli X is only ever applied to one quantum bit, but this one quantum bit is now in a register of several quantum bits.
51. 05:15Here, for example, we apply Pauli- X to the second quantum bit in a quantum register of two quanta
52. 05:26bits.
53. 05:28Then Pauli-X swaps the amplitudes of the states that differ in exactly this quantum bit in the whole register.
54. 05:42So now here we have a register state, if you prefer to think graphically.
55. 05:47We have a register state green, green is a small, red in small, and we apply Pauli-X
56. 05:54to the second QuBit, the second QuBit of the QuBit responsible for top and bottom, this means that along each of the top and bottom
57. 06:02axes is exchanged.
58. 06:05You see, here it is swapped, the bigger green moves down, the smaller moves up, and it is swapped.
59. 06:13The small red one moves up, the small green one moves down.
60. 06:17This is how Pauli X can be thought of in a register, when applied to a quantum bit in a register, it becomes
61. 06:24in a sense, the top and the bottom are swapped when applied to a register of two quantum bits.
62. 06:34In formulae:
63. 06:37The state before in the example before Pauli-X was a half times the 00 state one by root two slightly more than the
64. 06:4601 state minus one by root eight, which is the red the 10 state, plus one by root eight times the 11 state and after
65. 06:56Pauli-X, the amplitudes are swapped where the Second QuBit, differs. It is applied to the second QuBit
66. 07:07so here 00 and 01 is swapped and 01 and 00 is swapped and here also 10 and 11 and 11 and 10 is swapped.
67. 07:21Another example:
68. 07:24For the same effect, but now in a register of three quantum bits.
69. 07:31If now Pauli-X on the third quantum bit at that is responsible for front back on the third quantum bit in a
70. 07:39register of three quantum bits.
71. 07:44This means that the amplitudes along all the edges for which the third quantum be is responsible, i.e. along all the
72. 07:50front-back edges are swapped.
73. 07:54Along each edge is swapped,
74. 07:55this means in plain language, the back side of the cube comes to the front and the front side comes to the back.
75. 08:02Here is an example, the front is not so important, what is important is that you understand, the back comes to the front,
76. 08:07the demand to the back here for example the green comes to the back and the red comes to the front.
77. 08:12And that applies along all these quantum bits.
78. 08:16So you can graphically imagine how Pauli-X works.
79. 08:28In the calculation sits the question, there are two possibilities, either you say you swap the amplitudes, or you
80. 08:34say you swap the base states.
81. 08:37That's for those of you who are now saying that was funny just now, maybe they'll come better that way.
82. 08:43Sure, we have a state now, there are no fractions, they are rounded values,
83. 08:470.5 , 000 that's the state here, approximately to foundational precision minus 0.612 , 001 and so on and we're now going to apply
84. 08:57Pauli-X on that three.
85. 08:58Third quantum bit is applied to each base state, that is, where the base state of the third quantum bit is applied.
86. 09:05bit was zero, it becomes one here zero becomes one,
87. 09:11here too, zero becomes one.
88. 09:14And where the base state of the third quantum bit was one, it becomes zero. One becomes zero.
89. 09:24Decide for yourself which is the better option for you to imagine, whether you say you are swapping the amplitudes
90. 09:31or whether you say you apply Pauli-X on every base state, and then the amplitudes are swapped with it because the
91. 09:38amplitude, this one of 0.5, so the 0.5 , 000 it's now in front of 001, as it's better for it.
92. 09:48That is all for the Pauli-X gate.
93. 09:50Choose your variant The Pauli-X, swaps the amplitudes along all the edges in the cube for which the corresponding
94. 09:59quantum be is responsible for, to which the Pauli X gate is applied.

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