Introduction to Quantum Computing with Qiskit (with IBM Quantum)

1. 00:01Welcome back everyone!
2. 00:02I'm very happy that now that you learned the basics, we can actually start
3. 00:06implementing our very first quantum algorithm.
4. 00:09So, before I go into the implementation
5. 00:12with Qiskit, I would first like to show you how how the algorithm works and what
6. 00:17the goal of it is and how we can get the quantum advantage.
7. 00:21So, the Deutsche-Jozsa algorithm was
8. 00:23proposed in 1992 and the way it works is that suppose you are given a function f,
9. 00:30which takes as an input, an n bit string, and then it outputs either a constant or
10. 00:36a balanced function, so we can see it as a black box.
11. 00:39We do not know how that function operates, but we are promised that the output is
12. 00:44either constant, so it's given by a single bit.
13. 00:46So it could be either constantly zero or it could be constantly one,
14. 00:50or it could be balanced, which means that half of the inputs lead
15. 00:55to a zero and half of the inputs lead to a one.
16. 00:58Now, the goal is that we want to determine if f is constant or balanced.
17. 01:04So, if we look at a classical algorithm,
18. 01:06well, let's say we call the function once and we get a zero,
19. 01:10then we have no idea whether the function now is constant or balanced.
20. 01:14We actually need to call it again.
21. 01:16If we get a one, then we know, OK, we have two different outputs,
22. 01:19so it cannot be constant, but it has to be balanced.
23. 01:22So we need at least two periods.
24. 01:24However, in the case where we get again
25. 01:26a zero, we still do not know whether it's constant or balanced.
26. 01:29We need to ask again.
27. 01:31And if we get a zero again, then again, and so on, at most we need to ask it.
28. 01:36Well, we need to give it half of the input
29. 01:38bits plus one, half of the possible bit strings that we can feed it plus one.
30. 01:44And well, if we look at n bit strings,
31. 01:46then in total we could have two to the power n different ones.
32. 01:50So two to the power n minus one is then half of it plus one.
33. 01:55Now, the quantum algorithm needs only
34. 01:57exactly one query, and so this algorithm was one of the very
35. 02:02first examples to show the speed up that quantum algorithms can give.
36. 02:06You can see here also the circuit
37. 02:08representation, as you remember, hopefully we start from the left to the right.
38. 02:12So you can see we have here some qubits,
39. 02:15we can have n qubits, and they're all in the state zero initially.
40. 02:21Then, as I told you,
41. 02:22every quantum algorithm starts with a bunch of Hadamard gates.
42. 02:25So, on each of these qubits we apply a Hadamard gate given by the orange H.
43. 02:31Then we have our oracle, the function UF here.
44. 02:35That's a big box.
45. 02:37You can think of it as this black box
46. 02:39that I'm giving to you, that implements the function that I
47. 02:41promise to you is either constant or balance, but that you do not really know
48. 02:44what it is that you want to figure out what it does.
49. 02:48Then we apply again a bunch of Hadamard
50. 02:50gates and in the end we apply measurements.
51. 02:53So the green boxes correspond to measurements, so each of the qubits we
52. 02:56measure and then even if we are in a superposition before,
53. 02:59each of the measurements will give us either a zero or one.
54. 03:03So, the output bit string, which we call y would be an n bit string.
55. 03:09Now, if we go through the mathematics, we can see that the outcome of this
56. 03:13circuit equals exactly the zero bit string.
57. 03:18So, all outputs will be zero if and only if the function f is constant.
58. 03:23If the output is anything different, then we know it is balanced.
59. 03:28We can unfortunately not go through all
60. 03:30the mathematics now, but you just have to believe me that
61. 03:33now let us look at how we implemented with Qiskit.
62. 03:39So we're using the IBM quantum lab that you have seen in previous videos.
63. 03:44And while in the very beginning what we
64. 03:47have to do is import our basic Qiskit libraries.
65. 03:51So, we run the very first cell where we
66. 03:53just have some basic imports and then we define our oracle.
67. 03:58We call it the Deutsche-Jozsa Oracle.
68. 04:00We define it as a function that depends
69. 04:02on the number of bits n and on the oracle number.
70. 04:06So, that can be just some integer.
71. 04:09Now, we actually imported here also a function called Deutsche-Jozsa problem oracle.
72. 04:15And this function is really nice because
73. 04:17it just directly gives us a random oracle or well actually it's not random,
74. 04:21it gives us an oracle, one out of four different oracles that we
75. 04:25can specify by giving an oracle number between one and four.
76. 04:29Now we call this oracle the Deutsche-Jozsa Oracle.
77. 04:33This is just the name that it displays when we display the circuit.
78. 04:36And then we return this oracle.
79. 04:39Let's run that function, just the definition.
80. 04:42Then we define the complete algorithm,
81. 04:45the Deutsche-Jozsa algorithm, which again depends on n and on the oracle number.
82. 04:49However, I put here oracle number equals
83. 04:51zero, which means in the case where you don't specify the number,
84. 04:55so where you do not put any number between one and four in there,
85. 04:58but leave it just blank, in that case we will have the case zero
86. 05:03by default, which I later on defined to just give me a random oracle.
87. 05:07So, in that case you can test,
88. 05:08if you don't know what your oracle is, you can test it.
89. 05:12So, okay, let us start by creating the circuit.
90. 05:16We start by creating a quantum circuit just with a basic command quantum circuit
91. 05:20and we need n plus one qubits, where I did not yet tell you that,
92. 05:25but so for the circuit that you saw before, we have this function that I
93. 05:29called UF, that I told you, it's just this black box acting as an oracle.
94. 05:33And so this function.
95. 05:35Actually we add an additional qubit
96. 05:37on that so that we can turn our function rather than having a bit
97. 05:43oracle, which the function usually would be that we can turn it into a phase oracle.
98. 05:49So, we will encode the outcome
99. 05:50of the Oracle rather as instead of having it as a bit, we will encode it as a phase.
100. 05:56So, if we input a superposition,
101. 05:57we will have different phases for the different states.
102. 06:00But don't worry about that too much, it's just why I have here n plus one.
103. 06:03It's because of this special extra qubit that we need for the implementation.
104. 06:06And then the second number n here
105. 06:09shows us how many classical bits we need, which is for the measurement. I told you we
106. 06:14want to measure in the end all of our N normal qubits.
107. 06:17So for those we need those N outcomes, classical registers to store them.
108. 06:24Now, in the beginning of the circuit we do Hadamard gates, which you just do by calling
109. 06:30our circuit H for Hadamard and then identify the number of the qubits that we want
110. 06:35to apply to and we want to apply to all of our N qubits.
111. 06:38That's why we say make a for loop for range n.
112. 06:42Then we set up this special output qubit, the one that we don't care about too much,
113. 06:47the one that just turns our bit oracle into phase oracle.
114. 06:51We need to prepare the minus state.
115. 06:53So we need to prepare first apply an X gate and then an H gate.
116. 06:57But as I said, don't worry about that too
117. 06:58much, it's just to get the superposition to then get some phases.
118. 07:02Now, what we do next is here,
119. 07:04I mentioned before, if we do not specify the Oracle number
120. 07:07but we want to have a surprise, then we just do nothing.
121. 07:12We do not identify a number here, we just set at zero.
122. 07:15And in that case we take a random number
123. 07:17that gives us either a constant or a balanced oracle.
124. 07:21Now, of course we've added the Hadamard gates on all our qubits.
125. 07:28What we need to do next is we need to call
126. 07:30our function the Oracle, the function UF that we defined before.
127. 07:34So, we just call the show the Oracle
128. 07:37with our inputs and then we appended to the circuit.
129. 07:41In the end, if you remember the picture I showed you before we had Hadamard gates
130. 07:45the function UF, you have again Hadamard gates and then measurements.
131. 07:48So, now we have the Hadamard gates and the Oracle.
132. 07:51We add again a bunch of Hadamard gates and measurements on each qubit.
133. 07:56Each qubit means here we measure a qubit I
134. 07:59and we store it in the classical register I.
135. 08:01That's why we have two I's here.
136. 08:03And well, in the end we return the circuit.
137. 08:07And now we have defined this function, but now we would like to see doesn't
138. 08:11actually look what we expect it to look like.
139. 08:13So, we specify the number of qubits n equal to four.
140. 08:18And then we choose the Oracle number,
141. 08:21which we can either, for example, choose to be random between one and four.
142. 08:25So, if I put one comma five,
143. 08:27since it's with Python, you don't go to the last number.
144. 08:30So, that would mean we have a random number between one and four.
145. 08:33But I can also just say, okay, I want to have Oracle number one as I wish.
146. 08:37I call the circuit and I just say circuit.draw
147. 08:40and then it draws the circuit and well, here we go with our nice circuit.
148. 08:45We can see the Hadamard gates, as expected on the four qubits.
149. 08:48Then the oracle.
150. 08:49Here this extra qubit that we don't care
151. 08:51about, we don't even measure it, which we needed for the phases.
152. 08:55Then again, a bunch of Hadamard gates and the measurements.
153. 08:59So, I hope this already gave you a good
154. 09:02idea of how we can actually implement a given circuit where we know
155. 09:05theoretically how it looks like on a quantum computer.
156. 09:09Now, in the next video, you will see how we can then run it both
157. 09:13on the simulator and on an actual quantum device.
158. 09:16So stay tuned for the next video.

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