Einführung in das Quantencomputing - Teil 2

1. 00:00a warm welcome ladies and gentlemen to the course basics
2. 00:03of quantum computing part two and now we're going to talk about quantum
3. 00:07Adder We can all do addition we learn it in elementary school
4. 00:12and now the question is quantum adder
5. 00:15first look where are we in the course we are in the course always
6. 00:20still at the classical functions and we have already
7. 00:25videos that in principle you can use any classical function
8. 00:28can be recreated on a quantum computer but there was a
9. 00:31Disclaimer this is not how it is done in practice and I would like to
10. 00:34now show you quantum adders that are done in practice.
11. 00:38built like this and are an incredibly important application for
12. 00:42all kinds of quantum algorithms so addition is a
13. 00:47incredibly important application for arithmetic and quantum adder
14. 00:50are very important for the more complicated quantum
15. 00:54algorithms let us first see what a classical adder
16. 00:59so an adder does not make now in the decimal system like we do
17. 01:02learn it in school but in the binary system that means
18. 01:05the adder has two input bits x one and x two which can be respectively
19. 01:09be zero or one and it calculates the sum x one plus
20. 01:14x two in binary representation so that it really means the sum
21. 01:18that means if both are zero, zero comes out the binary-
22. 01:24representation is 00 if one of them is one comes one plus
23. 01:30zero comes out so 1 the binary representation is zero one but
24. 01:36if both are one plus one is two and two has the
25. 01:41The binary representation 10 so this is what a classical
26. 01:45adder does calculates two input bits both zero or one
27. 01:49the real sum
28. 01:52and therefore has two output bits which are typically called
29. 01:57c for carry thus the carry bit and s for sum the sum
30. 02:03Function values as written even more
31. 02:08zero zero becomes zero zero zero one and one zero that the sum
32. 02:13one becomes no carry but the sum is one and one plus
33. 02:17one is a carry and the sum is zero some of them
34. 02:21feel at the written adder added with the carry
35. 02:24exactly the same that is the classic adder has
36. 02:29two input bits 2 classic input bits what does the
37. 02:34Quantum adder quantum adder has four input bits x one and
38. 02:45x two and two more answer qubits that are initialized to zero
39. 02:50can typically be and typically are
40. 02:53Will be and it has four output qubits x one x two c
41. 02:59and s are called the okay Quantum circuit with four
42. 03:06qubits x one x two and two answer qubits has a total of 16 possible
43. 03:11states sixteen base states two to the power of 4 I want to give them
44. 03:18now not show what the quantum adder on
45. 03:20does on all base states but only on those where the
46. 03:24where the two output bits are initialized to zero that would have
47. 03:30we would like it to do that here is the input x one x two 00 01 10 11
48. 03:34Answer qubits are
49. 03:38initialized to zero and at the end after the calculation shall be
50. 03:43the x-en remain as they are so again the input
51. 03:48that makes then the backward calculation
52. 03:50and the two answer qubits are supposed to carry the carry-
53. 03:56bit and the sum bit the quantum circuit sees
54. 04:02as follows first look Answer-Qubits are both with zero
55. 04:10initialized so and now the idea is to say well if
56. 04:16x one and x two are both one then let the carry be one
57. 04:23be we see exactly when the two are one then the carry is one at all
58. 04:30something is done to the carry bit yes and the sum is calculated
59. 04:37by looking is an x one one if yes with CNOT the
60. 04:41Answer qubit flip no then it does nothing is
61. 04:45x two one if yes with CNOT flip the answer qubit and
62. 04:50if now it was the two one then the two one lift
63. 04:52CNOTs otherwise if both were zero it is sum bit zero
64. 04:57and if one was one and the other was zero then it is sum bit
65. 05:01one so this is what a quantum adder looks like for two classical
66. 05:10bits looks like for the time being now, of course, you can and that will
67. 05:14of course also used in quantum algorithms in the quantum
68. 05:18Adders also enter superpositions not only base states
69. 05:24what happens if you put a superposition into the
70. 05:27quantum adder so the answer qubits are again
71. 05:30initialized with zero and the superposition has the form
72. 05:33somehow often times 00 to somehow often times 01 alpha one
73. 05:37plus alpha two times one zero alpha three times one
74. 05:40one tensor the answer qubits are first of all not correlated
75. 05:46so that 00 are the answer qubits multiplied out looks
76. 05:53the state looks like that the base states in the superposition
77. 05:56still get two zeros on it we had
78. 06:06looked at how the base states |0000⟩ |0100⟩ |1000⟩ |1100⟩
79. 06:11so the four that occur here like the ones through
80. 06:18the quantum circuit here it is written again
81. 06:22and the superposition is seen now is changed in the same way
82. 06:27with the same amplitudes so that is this input state
83. 06:30transitions to α_0 |0000⟩ + α_1 |0101⟩ + α_2 |1001⟩ + α_3 |1110⟩
84. 06:36α_0 |0000⟩ + α_1 |0101⟩ + α_2 |1001⟩ + α_3 |1110⟩
85. 06:42α_0 |0000⟩ + α_1 |0101⟩ + α_2 |1001⟩ + α_3 |1110⟩
86. 06:46α_0 |0000⟩ + α_1 |0101⟩ + α_2 |1001⟩ + α_3 |1110⟩
87. 06:48this is what the quantum adder looks like and this is what you can do now
88. 07:01times differently write I would like to show them still the know
89. 07:04we already know the quantum adder has four input bits and it has now
90. 07:08two more answer qubits and they are now not going to be
91. 07:11initialized to zero but we want to make the quantum circuit
92. 07:14as such and then you don't necessarily have to
93. 07:18be initialized to anything but we just look at
94. 07:20the quantum circuit so it has four input bits which are called
95. 07:23x one x two and answer qubits and has four output qubits
96. 07:28x1 x2 c and s how to write c and s no yes
97. 07:36x one and x two come out unchanged
98. 07:39quantum circuit out again x three is the carry-
99. 07:45Bit we have seen that becomes x three plus the carry of
100. 07:49x one and x two this plus here is the exclusive or or
101. 07:55this here is to say that x three is reversed if the carry is
102. 07:59of x one x two is one we have already seen if
103. 08:07x three was initialized with zero then the carry bit is
104. 08:13yes 0 or 1 depending on what the sum of x one x two is
105. 08:16yes and if x three was initialized with 1 then it will be yes
106. 08:20is also reversed exactly when the two inputs
107. 08:22bits were one then the sum is now zero the exclusive or
108. 08:30and this complicated situation
109. 08:34is reversed exactly when the carry is one can be
110. 08:36one can write as formula briefly so x three xor carry of
111. 08:40x one and x two and exactly the same
112. 08:47you can also do with the sum bit that is x four the
113. 08:50what before was zero or one xor the sum bit of
114. 08:54x one and x two here is normal for them as a foil
115. 08:58this is the xor the value is one exactly if exactly one
116. 09:01of the two summands is one and that is called flipping
117. 09:09Example now let's take a base state |0110⟩
118. 09:15where now not both answer qubits were at zero
119. 09:18but are initialized to one zero and if they want to
120. 09:21see what the output state is of this quantum-
121. 09:26circuit so you can follow it piece by piece
122. 09:31what the Toffoli gate and the two CNOTs are doing
123. 09:36feel free to stop and try important it comes
124. 09:43out zero one one one the carry bit is not reversed
125. 09:50because we are not at inputs one and the sum bit
126. 09:54but is reversed because one of them is one
127. 09:58Quantum adders as said have an enormous importance
128. 10:06for many algorithms often is run in before one bit
129. 10:10runs a qubit in also just changes it somehow
130. 10:13so that you can calculate some kind of weighted sum with it
131. 10:16but you can see that now simply is a circuit of a
132. 10:21classical function calculated on quantum and really so maybe
133. 10:29one more remark you see with this circuit he uses
134. 10:32only Toffoli gate and and CNOT he could use x as well
135. 10:36and this leads to the fact that base states in base states
136. 10:40it is nice property if you think of classical
137. 10:43boolean functions thinks quantum circuit the base state
138. 10:46is not transferred to a base state, for example, in the case of the
139. 10:49Hadamard uses it is not appropriate to use the classical Boolean
140. 10:52function but here we have base state purely
141. 10:55base state out superposition in exactly the same superposition out

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