WEBVTT 1 00:00:47.609 --> 00:00:53.579 Okay, to see good afternoon parallel people. 2 00:00:53.579 --> 00:00:57.000 This is class 22, Thursday, April. 3 00:00:57.000 --> 00:01:02.759 15th, and can you hear me. 4 00:01:04.290 --> 00:01:09.209 Um, thank you. 5 00:01:09.209 --> 00:01:17.939 So I want to do today is, I think I've got audio working. You'll have to tell me. 6 00:01:17.939 --> 00:01:21.030 What I've discovered is that I cannot. 7 00:01:21.030 --> 00:01:24.420 The only way I could tell what you can. 8 00:01:24.420 --> 00:01:35.459 See, from my Webex meetings, if I got a separate RCS account and signed on as someone else, it's impossible sign on even 2 different devices as myself. 9 00:01:35.459 --> 00:01:39.060 And to see what another person would see. 10 00:01:39.060 --> 00:01:48.390 Any case we're talking about quantum computing, and I really want to show you some videos because. 11 00:01:48.390 --> 00:01:53.730 They talk better than me, and they have visual assistance and so on. So. 12 00:01:53.730 --> 00:02:03.120 And Scott, the chat window open also, I think, and you can also unmute the audio as you did last time. 13 00:02:03.120 --> 00:02:14.009 So, since we talked about the double slit experiment, I got this little thing talking about it, the Academy, something at dealt because a lot of quantum research and quantum. 14 00:02:16.974 --> 00:02:31.644 Tutorials and class stuff, and so on a very large amount of stuff, and I'm going to run that David thing again, he's 1 of the lead bottom 1 of the pioneers, and there's a very long quantum computing for computer scientists video. I'll show you the 1st, half hour of it. 15 00:02:32.365 --> 00:02:45.324 And then shorter 1, quantum algorithms, and a non technical industry oriented update. And then, as his time left over, I'll go back into my introduction and these things are mostly listed in the introduction introduction talk also. 16 00:02:47.099 --> 00:02:52.740 So, the quantum computing for computer scientists is very technical, which is good because. 17 00:02:52.740 --> 00:03:07.530 My problem is, as I see myself, what I'm teaching as a curator to search around a large amount of material and to find a small fraction of it, which I wish to present to, you. 18 00:03:07.530 --> 00:03:19.349 What makes my job difficult is there's a lot of, of course, high level management oriented stuff, which gives a big non technical picture suitable for a manager. 19 00:03:20.004 --> 00:03:33.895 With no equations and then at the other end, there's a lot of stuff by the physicists, including this business course that I've been teaching quantum physics for a while, but I want to hit in the middle at the computer scientists point of view. 20 00:03:34.770 --> 00:03:46.770 Which is when I created this course, I picked the title quantum computer programming to illustrate what this course is to do. So it's aiming for stuff in the middle. That's what I'm selecting things for. 21 00:03:46.770 --> 00:03:52.620 Okay, I got to see if some of these things. 22 00:03:52.620 --> 00:03:56.490 I work. 23 00:03:57.719 --> 00:04:01.590 And actually, can I see. 24 00:04:01.590 --> 00:04:07.259 Just a 2nd, okay it is. 25 00:04:07.259 --> 00:04:17.759 And is sharing, so we'll see if you get sound and you'll have to tell me because again, the fact that I might hear sound doesn't tell me if you can. 26 00:04:23.160 --> 00:04:30.838 1 of the ways superposition can be visualized is by shooting quantum particles at 2 narrow slits. 27 00:04:30.838 --> 00:04:36.718 A classical particle will always pass either through the top slit or through the bottom slit. 28 00:04:36.718 --> 00:04:45.778 But a quantum particle, if properly prepared can be put in a superposition of 2 paths, 1 path that passes through the top slit. 29 00:04:45.778 --> 00:04:56.848 And 1 path that passes through the bottom slit, this superposition leaves a mark in the form of interference fringes when we later measure to particles position. 30 00:04:58.858 --> 00:05:05.608 Okay, I had that video just because of the point last time. Okay, David, this time. 31 00:05:07.348 --> 00:05:14.218 Maybe quantum computing is becoming quite important in the world to no small degree based on your personal contributions. 32 00:05:14.218 --> 00:05:20.608 I'm not so interested on the applications. Can we factor? Large numbers steal money for. 33 00:05:20.608 --> 00:05:25.408 Transactions or catch spies, but what it is. 34 00:05:25.408 --> 00:05:30.028 Fundamentally, and what it possibly can tell us about the nature of reality. 35 00:05:32.459 --> 00:05:37.048 That's really why I'm interested in it as well. And I came to this from a physics. 36 00:05:37.048 --> 00:05:41.608 Point of view not a computation from so. 37 00:05:42.749 --> 00:05:50.459 1 of the things that really attracts me about the theory of contract consultation is what it tells us about what. 38 00:05:50.459 --> 00:05:53.608 Kind of thing a law of physics is. 39 00:05:53.608 --> 00:05:59.968 It's been a mystery to philosophers and physicists for decades. 40 00:05:59.968 --> 00:06:06.478 Uh, well, I think Eugene called the unreasonable effectiveness of mathematics. 41 00:06:06.478 --> 00:06:10.379 In the natural sciences, especially. 42 00:06:10.379 --> 00:06:14.038 When we realize that the set of computable options. 43 00:06:14.038 --> 00:06:28.439 Which are familiar to us that made of things like additional multiplication and so on from a mathematicians point, they form an infinite testimony, tiny subset of the set of all possible mathematical relationships. And yet. 44 00:06:28.439 --> 00:06:32.069 Physics is made entirely out of votes. 45 00:06:32.069 --> 00:06:36.478 And if it weren't, we wouldn't be able to know any physics. 46 00:06:36.478 --> 00:06:44.069 So, when it became more and more obvious that. 47 00:06:44.069 --> 00:06:57.598 Computation is built into the laws of physics at a fundamental level. A lot of people immediately jump to the conclusion. Oh, well, the reason that mathematics is useful in the physical sciences, is that the world is a computer. 48 00:06:57.598 --> 00:07:04.918 And we are just programs running in that computer or something like that. Or we are just a simulation running in a computer. 49 00:07:04.918 --> 00:07:08.788 Now, it seems to me that that misses the whole point. 50 00:07:08.788 --> 00:07:13.769 Of the, the lesson of the universality of computation for physics. 51 00:07:13.769 --> 00:07:17.639 Because it requires a notion of. 52 00:07:17.639 --> 00:07:24.509 What is original computable that is outside the physical world so that it was it was set by God or something. 53 00:07:24.509 --> 00:07:28.439 To be a certain set and that's why how universe. 54 00:07:28.439 --> 00:07:41.939 Only in instantiate that set of mathematical relationships. Well, then you make that that doesn't solve the problem. You may as well have said that God set up job universe with those relationships. So it gets get out. We remove the middle. 55 00:07:41.939 --> 00:07:46.019 So, I think the, the real, the important lesson of. 56 00:07:46.019 --> 00:07:51.959 Control of of the universality of computation as revealed by quantum computers. 57 00:07:51.959 --> 00:07:56.488 To be part of physics is that computers can be built. 58 00:07:56.488 --> 00:08:00.119 Universal computers can be built within the. 59 00:08:00.119 --> 00:08:04.949 That is really the amazing thing, because however, the universe was, you could imagine. 60 00:08:04.949 --> 00:08:10.949 Some kind of a super computer with, with unknown mathematics that simulated it. 61 00:08:10.949 --> 00:08:15.478 But the amazing thing about our universe is that you can make an object. 62 00:08:15.478 --> 00:08:22.108 Such a computer that can simulate any physical process. That's what universality is. 63 00:08:22.108 --> 00:08:31.949 And this object, the set of all, it's possible motions that is the set of all possible programs that could be programmed into it. 64 00:08:31.949 --> 00:08:36.958 Is in 11 correspondence with the set of all possible motions of anything. 65 00:08:36.958 --> 00:08:41.489 And that is a, that is telling us. 66 00:08:41.489 --> 00:08:47.969 Something about the universe from the inside is telling us something about what laws of nature actually are. 67 00:08:47.969 --> 00:08:51.389 Okay, now how does the quantum. 68 00:08:51.389 --> 00:08:54.899 Part help us when the. 69 00:08:54.899 --> 00:09:02.099 Theory of computation was 1st, discovered by, by and then developed by Alan Turing during the 19 thirties. 70 00:09:02.099 --> 00:09:11.038 It wasn't realized that this was a branch of physics the total it was invented as a branch of mathematics to study mathematical proofs. 71 00:09:11.038 --> 00:09:14.938 And the. 72 00:09:14.938 --> 00:09:18.089 The theory was built up. 73 00:09:18.089 --> 00:09:22.528 From a conjecture that a certain type of. 74 00:09:22.528 --> 00:09:25.649 Abstract object the Turing machine. 75 00:09:25.649 --> 00:09:30.239 Uh, could represent all things that that could be computations now. 76 00:09:30.239 --> 00:09:34.469 What quantum computers and then. 77 00:09:34.469 --> 00:09:41.879 Historically, what happened after that is that people began to worry that the physical world might not be able to instantiate. 78 00:09:41.879 --> 00:09:53.129 These these operations perfectly, and that, therefore, the, the real world might be a weaker kind of computer than the material computer. It might be an idealization. 79 00:09:53.129 --> 00:09:59.129 Now, when we studied this more carefully, and and this is where you just began began to come in. 80 00:09:59.129 --> 00:10:07.379 We found that not only can a universal computer exist physically, but it's more powerful than a jury machine. 81 00:10:07.379 --> 00:10:11.278 And what the mathematicians were doing unconsciously. 82 00:10:11.278 --> 00:10:15.178 Is that they, when they invented these abstract objects, they were. 83 00:10:15.178 --> 00:10:25.918 Applying that intuition about physical object. They didn't know that that's what they were doing. And because they were applying their intuition about physical objects, they got it wrong. 84 00:10:25.918 --> 00:10:33.178 They they thought about computing, right? Making marks of squares of paper. And then as remarked. 85 00:10:33.178 --> 00:10:41.428 They thought they understood paper, but in fact, paper like everything else based quantum mechanics and the real computation in the world. 86 00:10:41.428 --> 00:10:46.889 Is quantum computation the theory of computation is the theory of quantum computation. 87 00:10:46.889 --> 00:10:52.619 And that is a theory of physics. So that means that the theory of computation is irretrievable. 88 00:10:52.619 --> 00:10:56.668 Within physics because of the quantum theory of computation. 89 00:10:56.668 --> 00:11:08.249 Now, what is briefly the quantum theory of computation? How does that work? How is computation and quantum theory quantum mechanics, integrated into a quantum theory of computation. 90 00:11:08.249 --> 00:11:17.158 A theory of computation within any laws of physics is the theory of how you can use physical objects. 91 00:11:17.158 --> 00:11:23.609 To represent abstract objects so you want to represent the the. 92 00:11:23.609 --> 00:11:29.938 123, and you can use physical objects like fingers to. So that will be 1. that's called 2. that's called 3. and so on. 93 00:11:29.938 --> 00:11:33.418 And the computers are ways of. 94 00:11:33.418 --> 00:11:38.428 Instantiating abstract objects and their relationships. 95 00:11:38.428 --> 00:11:42.958 In physical objects and their motion. So now. 96 00:11:42.958 --> 00:11:49.229 What happens with quantum computers is that we simply take the deepest physical theory. We have quantum theory. 97 00:11:49.229 --> 00:12:01.318 And we say what kind of information processing does quantum theory in general allow and what does it not and that's the theory of contract. And when you do that, what do you 5 compared. 98 00:12:01.318 --> 00:12:15.389 With a classical computer, when you make this quantum computer, you find the find a number of similarities, and we find the reasons why the tiering theory worked as well as it did. 99 00:12:15.389 --> 00:12:21.599 And then you find a number of dramatic differences between the 2 columns from computers and. 100 00:12:21.599 --> 00:12:24.989 And classical computers, the 1, that's. 101 00:12:24.989 --> 00:12:29.458 Got the most attention is that for certain types of calculation. 102 00:12:29.458 --> 00:12:33.389 Quantum computer can perform it exponentially faster. 103 00:12:33.389 --> 00:12:37.318 Than any classical computer, so you could have. 104 00:12:37.318 --> 00:12:41.038 For people have a built comfortable because yet, but we open. They soon will. 105 00:12:41.038 --> 00:12:45.089 And when when the contract is built a small quantum computer with. 106 00:12:45.089 --> 00:12:50.249 With a few 1000 cube bits that that's the quantum analog of bits. 107 00:12:50.249 --> 00:13:01.828 Compared to the billions of bits in our normal desktop or even our, or even on mobile right? In other words, a very, very weak, comparatively week quantum computer. 108 00:13:01.828 --> 00:13:05.548 Could could perform more computations simultaneously. 109 00:13:05.548 --> 00:13:14.308 Then could be performed by the entire visible universe if it was all made into a computer. In fact, when I say more, that's an understatement exponentially more. 110 00:13:14.308 --> 00:13:17.519 And that, but only for certain types of. 111 00:13:17.519 --> 00:13:20.578 Computation and that's a token of the fact. 112 00:13:20.578 --> 00:13:27.359 That the whole notion of computation is different in quantum computers. It's not that, like, with all classical computers. 113 00:13:27.359 --> 00:13:30.629 You can say that 1 computer is 10 times as fast as the other. 114 00:13:30.629 --> 00:13:38.759 With quantum computers, they are vastly faster than classical computers for some are computations and the same for others. 115 00:13:38.759 --> 00:13:42.389 And interestingly, they're not slower. 116 00:13:42.389 --> 00:13:45.808 For any of. 117 00:13:45.808 --> 00:13:51.239 Uh, computations because the quantum computer among its abilities is to simulate a classical computer. 118 00:13:51.239 --> 00:14:00.448 Okay, now drilling down with Matt on. 119 00:14:00.448 --> 00:14:06.568 I'm reroute some refresh of stuff. I told yesterday and in your algebra and then getting into. 120 00:14:06.568 --> 00:14:10.678 So this will be a long 1 because I think it's worth it. So. 121 00:14:11.999 --> 00:14:19.499 Again, I'll show the first third you can watch the last on your own if you wish welcome to quantum computer scientists. 122 00:14:19.499 --> 00:14:24.389 Happy valentine's day we will learn about this most dramatic of all subjects. 123 00:14:24.389 --> 00:14:27.509 Quantum computing. This is a. 124 00:14:27.509 --> 00:14:39.899 Talk aimed at computer scientists, we will not go over very much if any physics during this talk, we won't go over the double sided experiment or uncertainty. Principle. We're not we're not learning, but any of that says, we're going to come up with a computational model. 125 00:14:39.899 --> 00:14:44.399 Which you all be find it fairly intuitive, I think, is the state machine. 126 00:14:44.399 --> 00:14:46.104 And we will run a quantum 127 00:16:31.464 --> 00:16:32.933 professor think we lost audio. 128 00:17:30.713 --> 00:17:42.713 It's probably probably is the best probability of changing the world. Sorry programs and search probably won't change the world too much, which is we might be able to have an exponential speed up in simulating quantum mechanical systems. 129 00:17:42.989 --> 00:17:49.348 So things like a drug design, if you seem like like biological molecules and tracking. 130 00:17:49.348 --> 00:17:53.909 Things like that, it could massively speed up a lot of our biological research. 131 00:17:53.909 --> 00:18:01.048 And everyone talks about like, nitrogen fixation for this. That's like part of the main Microsoft sales pitch, but. 132 00:18:01.048 --> 00:18:04.798 This is the 1 that actually motivated me the bottom 1, which I think. 133 00:18:04.798 --> 00:18:08.038 From talking to you before the presentation, we can also motivate you. 134 00:18:08.038 --> 00:18:14.608 It's intellectually interesting and it's interesting because it's just kind of outside my intuition. Like, I think all of us. 135 00:18:14.608 --> 00:18:21.058 At this point in our career is can basically look at any digital or mechanical system and have like, a ballpark idea of how it works. Right? 136 00:18:21.058 --> 00:18:27.179 But if you look at a quantum computer, like, how can a quantum computer outperform classical computation it doesn't make any sense. Like, it's. 137 00:18:27.179 --> 00:18:32.278 There's no way I can even, like, start to guess how it would work. And so I really wanted to learn it. 138 00:18:32.278 --> 00:18:35.729 And I think there's a reason for this, and this is kind of get into it philosophical, but. 139 00:18:35.729 --> 00:18:40.679 Our language, our informal language that we use developed in a very classical world. 140 00:18:40.679 --> 00:18:44.128 It is simply not equipped to deal with the quantum world. 141 00:18:44.128 --> 00:18:48.419 And this is why any pop science article you've read on quantum phenomena. 142 00:18:48.419 --> 00:18:53.249 It doesn't really ring true that all the metaphors. They don't really make sense. 143 00:18:53.249 --> 00:19:00.598 Because we're trying to express it in this language was developed in a classical world. We need to learn a new language, which is the language of mathematics. 144 00:19:00.598 --> 00:19:04.618 And that is the only thing which will actually let us understand quantum mechanics is to. 145 00:19:04.618 --> 00:19:07.709 Learn it mathematically all metaphors and analogies will lead you astray. 146 00:19:07.709 --> 00:19:11.699 There is a famous quote, here by a business name. 147 00:19:11.699 --> 00:19:14.939 Uh, David, which is shop and calculate. 148 00:19:14.939 --> 00:19:20.939 I forgot to quantum mechanics, so some grad students, maybe they lean back on their chair and they're like, oh, what does this all mean and chop and calculate. 149 00:19:20.939 --> 00:19:23.999 Just trust the math and that is the only thing which will not lead you astray. 150 00:19:23.999 --> 00:19:29.189 Okay, here's how this presentation will break down. 1st, we're going to learn. 151 00:19:29.189 --> 00:19:33.269 How do you represent computation using Super basically your algebra. 152 00:19:33.269 --> 00:19:37.318 Major vectors matrix multiplication then we will. 153 00:19:37.318 --> 00:19:42.028 Generalize that to learn about superposition in quantum logic. 154 00:19:42.028 --> 00:19:53.098 And finally, we will use all those tools we've developed to go over these simplest problem where quantum computer outperforms the classical computer. This is called the Oracle problem. I mean, there are a bunch of problems, but this is the simplest. 155 00:19:54.659 --> 00:20:04.019 And finally, it's been almost negligent to let you get out of here without learning about quantum entanglement and teleportation just because with the tools we'll develop. They're like, so easy to understand. 156 00:20:04.019 --> 00:20:08.638 So all of our topics, and then we'll have some demos. All right, let's get to it. 157 00:20:08.638 --> 00:20:11.669 You will become very, very. 158 00:20:11.669 --> 00:20:16.169 Familiar with these 2 vectors. This is how we represent a 0 and a 1. 159 00:20:16.169 --> 00:20:19.469 The 2 values of 1 that in classical terms. 160 00:20:19.469 --> 00:20:24.628 So the top 1 is just 1 over 0 and the bottom 1 is just you're over 1. 161 00:20:24.628 --> 00:20:31.528 You can also write them in that weird like, angle bracket thing that's called direct vector notation. 162 00:20:31.528 --> 00:20:40.528 Do you have a 0, and they asked me this is the value of that factor if you have trouble remembering this just think of it like an array index from 0. 163 00:20:40.528 --> 00:20:44.338 And so there's a 1 in the 0 index. 164 00:20:44.338 --> 00:20:47.368 I mean, this is 0, there's a 1 in the 1st index. That means it's a 1. 165 00:20:47.368 --> 00:20:51.898 Okay, everyone's fine with this. You'll be very familiar with this by the end of the. 166 00:20:51.898 --> 00:20:55.259 Presentation, we're going to go over. 167 00:20:55.259 --> 00:20:59.189 Matrix multiplication really quickly. 168 00:20:59.189 --> 00:21:02.429 So, average that okay. 169 00:21:02.429 --> 00:21:09.659 The way to think about major, small application, just reviewed for you. Most of you will have gone over this in your linear recourses like the 1st or 2nd day. 170 00:21:09.659 --> 00:21:15.179 So, if we have a matrix multiplied by vector, you kind of like, take this horizontal row and then flip it. 171 00:21:15.179 --> 00:21:22.439 And multiply point wise with these vector values. So a Times X times Y, and let's say you get the top 1. 172 00:21:22.439 --> 00:21:26.608 Need the same thing in the bottom row C a Times X. Y, you get a 2 vector. 173 00:21:26.608 --> 00:21:33.358 The same thing happens with 3 by 3 matrix a Times B times Y, see times that. 174 00:21:33.358 --> 00:21:40.439 And you get this matrix, you can also buy sorry this vector. You can also multiply matrices together. We won't really be doing much of that. 175 00:21:40.439 --> 00:21:44.818 And there's a very special matrix called the identity matrix. 176 00:21:44.818 --> 00:21:48.239 Which has zeroes everywhere. 177 00:21:48.239 --> 00:21:52.229 Except for ones along the main diagonal and anything. 178 00:21:52.229 --> 00:21:59.338 Multiplied by the identity matrix is itself. It's just like, multiplying by 1. in fact, usually refer to the identity matrix by the simple 1. 179 00:21:59.338 --> 00:22:05.729 Now, thankfully, for simplicity, a lot of them matrices will be using look a lot like this. There are multi zeros with just some ones. 180 00:22:05.729 --> 00:22:10.048 And you can notice is a nice rule of thumb that relative to the identity matrix. 181 00:22:10.048 --> 00:22:13.798 This Matrix has the middle 2 roads flipped. Okay. 182 00:22:13.798 --> 00:22:17.638 And that's actually the action it has on anything and multiplies. 183 00:22:17.638 --> 00:22:21.868 So, it flips the middle of the routes. It's a nice sort of rule of thumb. 184 00:22:21.868 --> 00:22:28.769 To remember how matrix application works does anyone have any trouble with this? Because you really need to know this. 185 00:22:28.769 --> 00:22:32.068 In order to go ahead. You got to know. Okay. 186 00:22:32.068 --> 00:22:41.578 Great operations. Actually, I'm I'd like to make this a quiz. Can anyone tell me what are the 4 operations on 1 bit. 187 00:22:41.578 --> 00:22:47.608 There's 4 of them 1. 188 00:22:47.608 --> 00:22:51.179 And or those are on 2 minutes. 189 00:22:51.179 --> 00:22:56.278 Identity identity, not such a 07071. there we go. 190 00:22:56.278 --> 00:23:02.338 Good yes, so we have identity this equals X. 191 00:23:02.338 --> 00:23:08.548 Effects equals not X constant. 0. it's always up to 0 console 1. it's always have the 1. you can see with the diagram. 192 00:23:08.548 --> 00:23:14.038 How it kind of looks okay and we can write. This is matrices. 193 00:23:14.038 --> 00:23:17.068 So, the identity is just the matrix. 194 00:23:17.068 --> 00:23:27.538 This remember is our symbol 40 identity about 12 00 and you multiply by 1 is 1. here's negation. You flip it from 0 to 1. 195 00:23:27.538 --> 00:23:32.128 And 1 to 0, same thing for constant, constant 1. 196 00:23:32.128 --> 00:23:37.858 So, I hope I convinced you, we can represent very simple functions, at least with major c's. 197 00:23:37.858 --> 00:23:42.058 Multiplied by our 01 doctors. Okay. 198 00:23:43.259 --> 00:23:47.548 And remember these 4 functions, because I will quiz you later on them, they'll come back up. 199 00:23:49.019 --> 00:23:57.328 So, let's talk about reversible computing. Reversal. Computing is a need sort of buzzword. Have you ever read the popular? It? 200 00:23:57.328 --> 00:24:00.929 Talks about rehearsal computing and basically it means that. 201 00:24:00.929 --> 00:24:04.979 If I tell you, the operation I did, and the output of that operation. 202 00:24:04.979 --> 00:24:09.509 You can always tell the input of that operation if the operations reversible. 203 00:24:09.509 --> 00:24:19.979 So intuitively operations that just shuffle bits around, like, permit them they are reversible, but operations that, like, a race fits and then overwrite them are not reversible. 204 00:24:19.979 --> 00:24:25.318 And in our previous slide, identity indication are both 1st of all. 205 00:24:25.318 --> 00:24:31.739 Because I say the output is 1, and I put it through the indication gate. That means the info was 0. you can always tell that. 206 00:24:31.739 --> 00:24:35.548 But if I say the output, it was 1, and I put it through the constant 1 operation. 207 00:24:35.548 --> 00:24:39.148 You can't tell what the input was. It's not reversible erases information. 208 00:24:39.148 --> 00:24:43.618 And the nice thing about quantum computing, what you can remember is that. 209 00:24:44.669 --> 00:24:53.519 Chronometers only use reversible operations. Those are the only ones we care about. So the only operation we actually care on both on the slide is really indication, like, also identity, but it's the same is doing nothing. 210 00:24:53.519 --> 00:24:56.939 A further. 211 00:24:56.939 --> 00:25:00.388 Fun fact of a clinical Gators is they actually. 212 00:25:00.388 --> 00:25:08.429 Only use operations are their own versus so not only are they reversible but if you apply them twice, you'll just get back the same input value. 213 00:25:08.429 --> 00:25:12.778 And we're going to go in a little this is kind of like going into pop science, but whatever. 214 00:25:12.778 --> 00:25:17.398 There's something called the land, our limit. 215 00:25:17.398 --> 00:25:20.818 Which is the smallest amount of energy. 216 00:25:20.818 --> 00:25:25.318 Physicists have calculated if necessary for the simplest possible calculation. 217 00:25:25.318 --> 00:25:29.608 Which erases is non reversible and. 218 00:25:29.608 --> 00:25:38.878 Reversible computing might eventually let us go beyond that limit more efficient than that theoretical limit. Now currently processors or millions of times these millions of times our energy. 219 00:25:38.878 --> 00:25:41.878 In the phenomena and our limit, but, you know, 5,100 years from now. 220 00:25:41.878 --> 00:25:49.229 You know, it's pretty nifty. All right, this is something you probably did not go over in your rough, but don't worry. 221 00:25:49.229 --> 00:25:55.169 It's still pretty simple. It's called the tenser product so if you have the product of 2 factors. 222 00:25:55.169 --> 00:25:59.368 It's kind of like you take the 2nd vector and you tile it. 223 00:25:59.368 --> 00:26:02.459 For each element in the 1st factor so. 224 00:26:02.459 --> 00:26:05.548 X0 gets its own copy and X1 gets its own copy. 225 00:26:05.548 --> 00:26:09.148 I mean, you multiply it out, so you get eventually. 226 00:26:09.148 --> 00:26:15.118 Why why is your extra y1, et cetera? It might be easier to look at this with real numbers. 227 00:26:15.118 --> 00:26:18.538 Here we have 1 times, 3, 1 times 4. 228 00:26:18.538 --> 00:26:21.808 2 times 3, and 2004, they got this. 229 00:26:21.808 --> 00:26:25.259 Here's how it looks like with 3. 230 00:26:25.259 --> 00:26:31.858 Factors you get this giant array. Don't worry about it too much. And I'd like to point out that. 231 00:26:31.858 --> 00:26:35.608 If we use our 0 and 1 values, you end up with a. 232 00:26:35.608 --> 00:26:45.118 Vector which has a 1 and only a single position. Okay. Does anyone have any real trouble with this concept? If I gave you 2 vectors you could probably calculate the tenser product. 233 00:26:45.118 --> 00:26:48.479 It's not entirely difficult. All right. 234 00:26:48.479 --> 00:26:53.249 But the, the final final it leads into, how do we represent multiple. 235 00:26:53.249 --> 00:26:56.398 Hospital beds, and we represent them by their tenser product. 236 00:26:56.398 --> 00:27:00.298 So, it's 00, remember we cancer. This is our 0 symbol. 237 00:27:00.298 --> 00:27:04.679 10 0 is this product state we call it. 238 00:27:04.679 --> 00:27:12.659 01, is this product state? 10 is this product state like 0 times 10000. 239 00:27:12.659 --> 00:27:16.769 1 times 111000. okay. 240 00:27:16.769 --> 00:27:23.038 And 11 is this product state and similar to just the single bit you can think of this is. 241 00:27:23.038 --> 00:27:26.128 An index in this vector so this is 0. 242 00:27:26.128 --> 00:27:29.818 There's a 1 at the 0 position. This is 1 there's value with the. 243 00:27:29.818 --> 00:27:36.328 1st position can be from 0 and this is to evaluate the 2nd position 3 value of the 3rd position. 244 00:27:36.328 --> 00:27:40.048 Okay, this is the only time I will 10. 0, 3. 245 00:27:40.048 --> 00:27:44.398 It's together, thankfully, because they use yours, they grow exponentially in size. 246 00:27:44.398 --> 00:27:50.009 But it works for that. So, 4 is 100. you have 01234. 247 00:27:50.009 --> 00:27:53.999 Is where the 1? Goodness. Okay. Yeah so we call this the product state. 248 00:27:53.999 --> 00:27:59.219 We can always factor it out into its individual states and. 249 00:27:59.219 --> 00:28:07.739 Product state of nbis is a vector size. 2 to the end. Here. We see a sort of whisper of the power of quantum computing this exponential turn, but hold that thought we'll get back to it. 250 00:28:07.739 --> 00:28:11.159 So, we're going to learn a very. 251 00:28:11.159 --> 00:28:18.118 Important operation very fundamental to reversible computing. And quantum computing is called C not conditional. Not. 252 00:28:18.118 --> 00:28:23.999 And operates on a, you designate 1 bit the control bit. The other bit is the target bit. 253 00:28:23.999 --> 00:28:27.179 So the control, and if it's 1, it flips the target fit. 254 00:28:27.179 --> 00:28:30.598 If it's 0, it leaves the target bit alone and the control of it. 255 00:28:30.598 --> 00:28:37.919 Is always left alone, so if we have the most significant bit of a 2 minute system is control. The least significant bit is target. 256 00:28:37.919 --> 00:28:44.338 Then 00 since this is the most that we can fit, it's the target. It just goes to 0. 0. 257 00:28:44.338 --> 00:28:48.989 01 goes to 01, now, 10 since the control is 1. 258 00:28:48.989 --> 00:28:52.019 Let's do 1 on 1, right? Because it flips the other. 259 00:28:52.019 --> 00:28:56.669 And 11, since it again, flips the target, it goes to 1 0. 260 00:28:56.669 --> 00:29:05.729 Okay, and this matrix, you can actually kind of write it. You see, there's like a sort of correspondence here we flip the bottom 2 rows versus the identity matrix. 261 00:29:05.729 --> 00:29:11.308 On this, I claim which I will show on the next couple slides, we'll change our product States. 262 00:29:11.308 --> 00:29:18.538 According to the semantics of C. not. So, does everyone get the semantics here? We have a control button to target a bit trouble. It's 1 we flip the other bit. 263 00:29:18.538 --> 00:29:23.219 Pretty good. Okay now, this is a lot of math. 264 00:29:23.219 --> 00:29:27.358 But don't panic we're going to go over step by step if we apply. 265 00:29:27.358 --> 00:29:33.538 Let's see not gate to 10. remember that. This is the control. That's 1. so we're going to flip the other bit. So we expect it to go to 1. 1. 266 00:29:33.538 --> 00:29:38.909 Now, let's expand this out, step by step so, 10 corresponds to this tenser product. 267 00:29:38.909 --> 00:29:43.318 This is the matrix from the previous slide and we're applying it to our product state. 268 00:29:43.318 --> 00:29:48.028 Remember just going to flip the bottom 2 rows so we get to this product state. 269 00:29:48.028 --> 00:29:51.209 Which when we factor it out, indeed gets us 1 1. 270 00:29:53.038 --> 00:29:56.669 Okay, now let's do with 11 same thing. 271 00:29:56.669 --> 00:30:02.608 We have our familiar matrix multiply by products date. We go to here, which we factor into 1. 0. 272 00:30:02.608 --> 00:30:10.318 So, I hope I've convinced you, at this point that we can use major CS to represent kind of more interesting, logical operations than just a bit. 273 00:30:10.318 --> 00:30:17.009 Does anyone have any trouble? Yes. Factoring always possible or practical for larger vectors. 274 00:30:17.009 --> 00:30:20.699 It is practical. Yes. 275 00:30:20.699 --> 00:30:26.638 Not always possible, which kind of we'll get into that when we can do quantum entanglement basically. 276 00:30:26.638 --> 00:30:30.209 My question. Oh, okay. 277 00:30:30.209 --> 00:30:37.798 So this is this is fine for everyone. Everyone, nobody's lost in the gigantic equations here. 278 00:30:37.798 --> 00:30:42.868 Okay, we're going to go over 10000 so the control 0. 279 00:30:42.868 --> 00:30:50.489 That means we're gonna leave the other bit alone right? And we see that. Indeed. This is what we do and same with 4. 0 1. 280 00:30:50.489 --> 00:30:59.429 Okay, so everyone maybe you're maybe familiar with how classical computers are, like real. See if you use, they're all built on the sort of navigate. 281 00:30:59.429 --> 00:31:10.078 Like, everything is fundamental built on that and see now is kind of the analogy, analogous NAND for reversible computing. It's a really basic thing that is used to build up. 282 00:31:10.078 --> 00:31:17.578 Larger and more complicated, logical statements. So I'm sure if I give you a task, like, build some more complicated, logical. 283 00:31:17.578 --> 00:31:21.328 Formula was seen how you'd probably be able to do it at this point. You just stick them all together and. 284 00:31:21.328 --> 00:31:30.749 Eventually it comes out, I will note that it's not actually universal. You can't make every logical function with a gate. You need something called it to fully it. 285 00:31:30.749 --> 00:31:34.288 But we want to we want to see that in this presentation. 286 00:31:34.288 --> 00:31:39.088 And we've learned everything we need to know about how to use linear. 287 00:31:39.088 --> 00:31:44.578 Excuse me to represent classical computation, and we can now go on to quantum computing. So to recap. 288 00:31:44.578 --> 00:31:48.509 We learned that we can represent classical bets and vector form is 1 0 for 0. 289 00:31:48.509 --> 00:31:54.209 01, for 1, rather than operations on bits by multiplications. 290 00:31:54.209 --> 00:31:58.739 On their bit factors, we don't have the quantum computers only use reversible operations. 291 00:31:58.739 --> 00:32:06.538 And, in fact, the operations are their own versus and multi mistakes we write is the tenser product of our single States. Finally. 292 00:32:06.538 --> 00:32:09.898 Leaner about learned about, which is a very fundamental gate. 293 00:32:09.898 --> 00:32:13.259 In reversible computing, iconic repeating. 294 00:32:13.259 --> 00:32:19.858 So this isn't anyone everyone's following along so far. This is pretty, pretty simple, right? It's not that bad. 295 00:32:20.939 --> 00:32:28.378 Okay, let's make the jump 2 minutes and I will surprise you. We've been using all along actually. 296 00:32:28.378 --> 00:32:33.179 Uh, the events are just special cases of. 297 00:32:33.179 --> 00:32:37.439 And a rough time to keep it in general by a vector of 2 elements a be. 298 00:32:37.439 --> 00:32:44.068 Where, and B are complex numbers don't worry. We'll only use real numbers for this presentation. We're keeping it simple. 299 00:32:44.068 --> 00:32:49.108 All these in real numbers, and we have the constraint that a square plus B squared is 1. 300 00:32:49.108 --> 00:32:53.848 So, 1001 foot this definition because 1 squared plus your Square is 1. 301 00:32:53.848 --> 00:33:02.429 And your square plus 1 squared is 1 right now here are some example Cupid values. This is 1 with which you'll become very familiar. 302 00:33:02.429 --> 00:33:06.778 It is 1 of our, it to 1 of our 2. can anyone tell me what 1 of our 2 square. 303 00:33:08.278 --> 00:33:13.138 1 app. Exactly. 1. apple's 1. half is 1. so this is a Cuban. What about. 304 00:33:13.138 --> 00:33:16.169 1, half and 3 over 2. what's 1 square? 305 00:33:16.169 --> 00:33:23.969 1 quarter, so 1 quarter plus 3 quarters is 1. Here's an interesting thing. We can actually use negative numbers now. 306 00:33:23.969 --> 00:33:28.739 As a negative number squared is a positive number. So negative 1 squared is 1 1 plus 1. 307 00:33:28.739 --> 00:33:31.888 And the same thing here, it's just the same is this 1st matrix. 308 00:33:31.888 --> 00:33:36.239 But 1 of values is negative. Pretty neat. 309 00:33:36.239 --> 00:33:41.398 Not bad so far. Right. Okay. But you might say this is. 310 00:33:41.398 --> 00:33:46.588 Kind of like saying that acumen has a value, which is not 01, but actually. 311 00:33:46.588 --> 00:33:50.909 Both 0, and 1, at the same time, this is called superposition. 312 00:33:50.909 --> 00:33:55.288 Everyone's probably heard of the being both alive and dead. 313 00:33:55.288 --> 00:33:58.679 That's just like a pop sciencey way of articulating this concept sort of. 314 00:33:58.679 --> 00:34:05.398 And the interesting thing about a human being and superposition it means we can kind of. 315 00:34:05.398 --> 00:34:11.188 Really qualify this kind of compute with the values of both 0 and 1 at the same time. 316 00:34:11.188 --> 00:34:15.148 Which again is the sort of whisper of some quantum speed up power. 317 00:34:15.148 --> 00:34:20.728 Okay, now, how this works is when we measure this cube, it. 318 00:34:20.728 --> 00:34:26.548 It collapses to our familiar values of 0 and 1 with some probability. 319 00:34:26.548 --> 00:34:34.018 And this is what we usually do at the end of a quantum computation to get the final result. We send it through the measurement gate. We see what the result is. 320 00:34:34.018 --> 00:34:37.918 So, how the probability works is, if you have a value, a be. 321 00:34:37.918 --> 00:34:42.748 Process to 0, with probability a squared and 1 with probability B squared. 322 00:34:42.748 --> 00:34:48.568 So, if we look at our familiar cube at from previous slide 1 of our 2, 1 or 2. 323 00:34:48.568 --> 00:34:54.478 It has a 1 half chance of collapsing to 0 and 1, half chance of collapsing to 1. it's like a coin flip. 324 00:34:54.478 --> 00:35:03.028 Now, if we just measure our classical Cuban or hospitals, see, that's the 10 has a 100% chance of collapsing to 0. 325 00:35:03.028 --> 00:35:08.309 And 01 has 100% chance of plugged into 1 so sort of item potent in that way. 326 00:35:08.309 --> 00:35:13.679 And I found it useful here to give people a sort of physical intuition about what we're doing. 327 00:35:13.679 --> 00:35:22.228 What Steven is or keep it is if you've heard of polarization, you've probably got fancy sunglasses. It says that they're polarized. 328 00:35:22.228 --> 00:35:27.208 So, polarization is actually sort of quantum phenomenon. You can be polarized like this way. 329 00:35:27.208 --> 00:35:37.469 Or this way, and your glasses will only be polarize a certain way. So the only led light polarized in this way through. Basically, it turns out that Cubans you can think of, like. 330 00:35:37.469 --> 00:35:41.548 This yeah, for time being horrible this way is 0. 331 00:35:41.548 --> 00:35:46.619 Fault on being polarized. This way is a 1 superposition means it's actually like both. Right? 332 00:35:46.619 --> 00:35:54.869 So, it could actually go through both grading and then if you measure it and a way to measure it is just to, like, fire it through a grading and see whether it goes through. 333 00:35:54.869 --> 00:35:58.739 Then you can collapse it to 0 or 1. so that's our sort of intuition. 334 00:35:58.739 --> 00:36:02.548 Pinning this to a physical concept. Of course modern. 335 00:36:02.548 --> 00:36:09.298 We found that photo was actually making very, very poor Cubans. They tend to collapse too easily so we don't use those, but you can maybe think of that. 336 00:36:10.438 --> 00:36:17.489 Does anyone have any confusion here really about measurement process? Collapse? Probably bullet probability works. 337 00:36:17.489 --> 00:36:26.159 Okay, great. We will learn about multiple events. The same is multiple. See bits we represent them by the Tensor product. 338 00:36:26.159 --> 00:36:29.548 It actually, why is that? Okay. 339 00:36:30.778 --> 00:36:33.958 Hello. 340 00:36:33.958 --> 00:36:38.369 Corner okay there we go. That's not. 341 00:36:41.099 --> 00:36:48.478 Okay, so if we have 2 units and we multiply them the sum of squares identity actually. 342 00:36:48.478 --> 00:36:53.429 It is maintained and so if we consider our. 343 00:36:53.429 --> 00:36:58.259 Perfectly balanced 50% coin, flip cube and we transfer with another corn flip Cupid. 344 00:36:58.259 --> 00:37:02.068 We get this 4 vector with 1 half and every single. 345 00:37:02.068 --> 00:37:09.449 Call them and what that squared is, of course 1 quarter. So 1 quarter plus 1 quarter 1 cordless 1 quarter is 1 this maintains the property. 346 00:37:09.449 --> 00:37:13.228 And the way to read, this is when you measure these 2, 2 events. 347 00:37:13.228 --> 00:37:17.789 It has a 1 quarter chance of collapsing to 000110 on 1. 1. 348 00:37:17.789 --> 00:37:25.108 This is sort of the probability of when you measure it finding a 1 there and 0, everywhere else. 349 00:37:25.108 --> 00:37:33.688 So, we could find that there's a 1 quarter probability of finding a 1 in this position and all zeros everywhere else. Does that kind of make sense for multiple Cubans? 350 00:37:33.688 --> 00:37:37.858 Okay, great. You guys are just trucking along and no problem. 351 00:37:37.858 --> 00:37:43.559 Super easy yes, absolutely. So only, there's only going to be 1. 352 00:37:43.559 --> 00:37:51.509 11 left in the vector. Yes, there could be 23rd not be 2 ones because I would actually violate our B squared. 353 00:37:51.509 --> 00:37:54.929 Equals 1 restraint. 354 00:37:54.929 --> 00:37:58.048 Yeah, after measurement they will only be 1 1 here. 355 00:37:58.048 --> 00:38:02.338 In this in this 4 vector, which we can then factor into 2 bits actually. 356 00:38:02.338 --> 00:38:06.809 Okay, yes, you can also find collapsing, Florida. 357 00:38:06.809 --> 00:38:13.920 Longer yeah, yeah. So if you enter these 2 events at once you can. 358 00:38:13.920 --> 00:38:17.820 Think of this product state factor is collapsing and. 359 00:38:17.820 --> 00:38:21.929 And then that's just the probability of finding a 1 here and a 0, everywhere else. 360 00:38:21.929 --> 00:38:27.150 It'll become useful in the future when we can't actually factor out the product state, but that's. 361 00:38:27.150 --> 00:38:35.969 Anyway, okay. How about operations on campus? We have these Cubans. How do we actually manipulate them? The same way we do with the classical bits? 362 00:38:35.969 --> 00:38:40.739 And all the matrix operators, we've learned also work with. 363 00:38:40.739 --> 00:38:44.969 So, for example, negation, if we have our cube that as. 364 00:38:44.969 --> 00:38:49.619 A 25% chance of collapse into 075% chance clubs into 1 if we run it through the flip. 365 00:38:49.619 --> 00:38:55.710 Operator now it has a 75% chance of collapse into 0 and a 25% chance of collapsing to 1. so we can manipulate. 366 00:38:55.710 --> 00:39:09.960 These probability is actually called aptitudes with Gates, and you can think of like a Gates correspond to some device scientists have created, which is sort of, like, manipulate the key with with a gentle touch, not enough to collapse them, but enough to change their state. 367 00:39:09.960 --> 00:39:13.440 So pretty neat and. 368 00:39:13.440 --> 00:39:16.559 There are also a couple important matrix operators. 369 00:39:16.559 --> 00:39:21.090 Which we haven't learned yet, because they only really make sense in quantum context of the. 370 00:39:21.090 --> 00:39:25.170 What is okay sorry okay. All right. We're going to. 371 00:39:25.170 --> 00:39:31.500 The most important 1 is the, how to market and what it does is, it takes a 0 or 1 cube. It. 372 00:39:31.500 --> 00:39:36.239 And it takes it to the corn flip state where it's exactly equal superposition. 373 00:39:36.239 --> 00:39:42.690 The Matrix looks like this, it has 1 over to in every single entry, except for a negative in the bottom right? Corner. 374 00:39:42.690 --> 00:39:46.110 So, multiply that by this, you get 1 of her 2, whatever 2. 375 00:39:46.110 --> 00:39:50.130 If you modify this by 1, we get 1 over 2 negative 1 of 2. 376 00:39:50.130 --> 00:39:55.139 So this is how we actually get into superposition. We use the device, which. 377 00:39:55.139 --> 00:39:58.949 Scientists have develops and. 378 00:39:58.949 --> 00:40:03.239 We can make much use of so. 379 00:40:03.239 --> 00:40:08.309 Kind of quiz time. Can anyone tell me why we need a negative 1 in the bottom right? Corner? 380 00:40:09.360 --> 00:40:15.510 Why do we need a negative sign here? It's like it. 381 00:40:15.510 --> 00:40:19.170 That's a matrix. 382 00:40:19.170 --> 00:40:23.340 Probation, it's like a catches there. 383 00:40:23.340 --> 00:40:26.670 And I know exactly why it seems like it's. 384 00:40:26.670 --> 00:40:29.849 Well, that's the answer is, it has to be reversible. 385 00:40:29.849 --> 00:40:37.860 So, if we didn't have this negative number here, then both 0, and 1 would map to the same value. It would not be reversible. So we need this negative. Yes. 386 00:40:37.860 --> 00:40:42.960 Correct okay. Question yes. 387 00:40:42.960 --> 00:40:50.489 Review of Super position. What does that mean? Exactly the superposition. Yeah. So, supervision means that you bet. 388 00:40:50.489 --> 00:40:54.030 Is it kind of in in both state 0 and 1 at the same time? 389 00:40:54.030 --> 00:40:57.659 And I want to be some, I guess, really clear about something here. 390 00:40:57.659 --> 00:41:02.039 This doesn't mean that Cuba is secretly 0 or secretly 1 and we don't know. 391 00:41:02.039 --> 00:41:06.840 It means it's actually in both the same time this is like, just quantum weirdness. 392 00:41:06.840 --> 00:41:10.829 This is just like, what happens at the very, very small level of our world. 393 00:41:10.829 --> 00:41:13.889 Which is that things don't have definitely values until we measure them. 394 00:41:13.889 --> 00:41:18.510 Here you have 01 and Super position. 395 00:41:18.510 --> 00:41:26.429 1, over 21, over to 1 and superposition of warmer 2 minus 1. so we seem to have different. 396 00:41:26.429 --> 00:41:31.710 Values, but they're representing conceptually the same thing. 397 00:41:31.710 --> 00:41:37.800 Right. Which is so this kind of comes down to the distinction between the quantum and classical world. 398 00:41:37.800 --> 00:41:44.909 Which is that from a classical perspective? These 2 Cubans are kind of the same. They both have 50% probability of collapsing or 1. 399 00:41:44.909 --> 00:41:53.519 But if we stay within the quantum realm, this sign information is preserved as long as we don't measure it and it can affect our computations. 400 00:41:53.519 --> 00:41:57.869 So, kind of like, it's got a 50, 50 chance, but it. 401 00:41:57.869 --> 00:42:02.010 It has a sense that it needs to be a 1. 402 00:42:03.300 --> 00:42:11.940 As a sense that he used to be, what not really, it's just, it ends up in this state, which is different within the quantum realm than this state. These are 2 different states. 403 00:42:11.940 --> 00:42:17.489 Yes, yes, exactly. Both when you measure it and convert it into the classical world. 404 00:42:17.489 --> 00:42:21.630 Come down with a 50, 50, 50 chance. Yes, but they are different. 405 00:42:21.630 --> 00:42:25.440 And they have different combinations properties. Okay. 406 00:42:25.440 --> 00:42:29.849 Now, I'm going to show you something really cool about the how to market. 407 00:42:29.849 --> 00:42:32.969 Which it actually takes us out of superposition. 408 00:42:32.969 --> 00:42:37.469 Into the classical, this should be unsurprising everything's its own inverse right? 409 00:42:37.469 --> 00:42:41.159 If we just apply the, how to market to our. 410 00:42:41.159 --> 00:42:44.849 Value here are our value. We get 0. 411 00:42:44.849 --> 00:42:49.710 And for this 1, we get 1, so this suggests to us a sort of. 412 00:42:49.710 --> 00:42:56.250 Quantum computation structure we can start with our classical bit values. We put them in a superposition. 413 00:42:56.250 --> 00:43:00.929 Do a bunch of quantum stuff to them, and at the end of our clever we can transition them. 414 00:43:00.929 --> 00:43:04.679 To 01, so that our whole computation was deterministic. 415 00:43:04.679 --> 00:43:08.190 Pretty neat, right. Okay. 416 00:43:08.190 --> 00:43:12.690 Now, I should say that not all algorithms work this way. There are algorithms like, shores algorithms. 417 00:43:12.690 --> 00:43:16.800 Which only gives the right answer 50% of the time. You can't be clever and. 418 00:43:16.800 --> 00:43:24.119 I always get the right answers just not doable, but the 1, the problem we'll look at does give this sort of deterministic property. 419 00:43:25.889 --> 00:43:29.969 So, does anyone anyone have trouble with this concept? 420 00:43:29.969 --> 00:43:33.150 Can transition out a superposition without measurement? Pretty interesting. 421 00:43:33.150 --> 00:43:37.079 Both, what would this transition from that classical kind of computation. 422 00:43:37.079 --> 00:43:42.869 To quantum and then converting it back. Yes, that's correct. Let's say 2 plus 2 out that. 423 00:43:42.869 --> 00:43:45.989 The, how it just the simplest. 424 00:43:45.989 --> 00:43:49.050 Of math, that's a math. 425 00:43:49.050 --> 00:43:53.010 What we're going to go over that? Basically we're going to go over a problem. 426 00:43:53.010 --> 00:43:56.309 There are certain problems where quantum computers are better. 427 00:43:56.309 --> 00:44:00.780 For that, then classical, like, edition is not 1 of the. 428 00:44:00.780 --> 00:44:05.489 It works the exact same way in both. It should also be said that all. 429 00:44:05.489 --> 00:44:11.760 Classical communication can be done on a quantum computer. All you do is just restrict all the values as your own 1 and it works. 430 00:44:11.760 --> 00:44:17.460 But it's kind of a waste because you can also use all the extra values that we, we found. 431 00:44:18.690 --> 00:44:27.059 Okay, and we've actually sort of defined the state machine here, which I'm sure many of you will appreciate if your software engineers are computer scientists. 432 00:44:27.059 --> 00:44:32.940 Here is the action. Okay so this isn't the unit circle which many of you will remember from. 433 00:44:32.940 --> 00:44:38.070 High School trigonometry right? It's a circle of radius 1 around the origin. 434 00:44:38.070 --> 00:44:44.940 Here's the X access. This is the why access you can think of the top value is the. 435 00:44:44.940 --> 00:44:49.619 Coordinate on the value of the X value and the bottom 1 is the Y value. So. 436 00:44:49.619 --> 00:44:52.860 Our 0 value is that position 1 0. 437 00:44:52.860 --> 00:44:57.449 Which is along the X axis are 1 values of position. 01, which is the on the Y, axis. 438 00:44:57.449 --> 00:45:01.289 And this is the flip operator it defines transitions. 439 00:45:01.289 --> 00:45:05.730 Around this unit circle, so 0 goes to 1 when it goes to 0. 440 00:45:05.730 --> 00:45:11.369 0, negative 1 goes to negative 10, which when you clap them. 441 00:45:11.369 --> 00:45:19.110 This is 0, this is 1 and vice versa. Excuse me? And if we look at the superposition values, Here's something interesting. 442 00:45:19.110 --> 00:45:25.650 And flip operator really kind of has no effect on them for this 1. if you flip it upside down, it's just. 443 00:45:25.650 --> 00:45:30.539 It stays itself staying with this over here. This was kind of interesting. And then it. 444 00:45:30.539 --> 00:45:34.559 Transitions you all the way across the unit circle, and we'll make use of that. 445 00:45:34.559 --> 00:45:42.630 But, yeah, okay, this is pretty much the full action of the flip operator on the unit circle. I should mention here that if we were using complex numbers. 446 00:45:42.630 --> 00:45:51.780 This would actually be a sphere much more difficult to sort of intuitively represent are graphically represent. So we're sticking with real numbers as soon as just fine. 447 00:45:53.159 --> 00:45:57.960 What's nice to know that additional dimension is always available to us. If we want to make our competition even more powerful. 448 00:45:57.960 --> 00:46:01.110 And here's the action of the, our gate, which we saw the past couple slides. 449 00:46:01.110 --> 00:46:04.650 It takes us from 0 to whatever 2, whatever 2. 450 00:46:04.650 --> 00:46:10.440 1, to 1, over 2, negative 1 of our 2, and kind of symmetrically it also works in this way over here. 451 00:46:10.440 --> 00:46:19.050 So we have a nice day machine that we've developed and now we can start running things on the state machine. Very nice, intuitive sort of representation. You don't need to do much matrix multiplication all the time. 452 00:46:19.050 --> 00:46:25.320 Everyone kind of gets this. Okay great. 453 00:46:25.320 --> 00:46:30.269 So here's an example of something we could run through the state machine here. I'm introducing something called. 454 00:46:30.269 --> 00:46:35.010 Quantum circuit notation, you're going to think of the Cuban traveling along this line. 455 00:46:35.010 --> 00:46:38.760 Has these operations applied to it is it goes through the boxes. 456 00:46:38.760 --> 00:46:44.730 So, access bet, flip it just how to mark and then so we go to our and 5th of how to Martin. 457 00:46:44.730 --> 00:46:48.300 And we, if we start out here, we go bit flip. 458 00:46:48.300 --> 00:46:53.639 How to mark fit? Flip? So we start at 1. 0. 459 00:46:53.639 --> 00:46:59.610 I get to 01, we got across the circle. I know this is reversible. So we can also go this way and get back. 460 00:46:59.610 --> 00:47:03.690 So this is kind of how we'll be. 461 00:47:03.690 --> 00:47:08.550 We'll be doing all of our conversations like this basically just. 462 00:47:08.550 --> 00:47:15.329 Hopping around the unit circle pretty simple. Pretty intuitive right? It's just a state machine. We've seen we use this in classical computing all the time. 463 00:47:15.329 --> 00:47:21.210 It just hasn't been a weird rules with claps or whatever, but don't worry. Okay. 464 00:47:21.210 --> 00:47:30.059 Now, we have everything we need to learn the simplest problem where a quantum computer performs a classical computer. We learn that events are just a special case of Cuba. 465 00:47:30.059 --> 00:47:33.989 And humans are 2 vectors of complex numbers, such that 8 squared plus B squared. It goes 1. 466 00:47:33.989 --> 00:47:39.809 Within that team is going to be in superposition. They're probably politically collapsed to see if it's by measurement. 467 00:47:39.809 --> 00:47:44.579 Multi human systems, they learned our tens of products of single Cuban systems. Same as what. 468 00:47:44.579 --> 00:47:50.579 Matrices again represent operations on Cubics and we learn about the head of our gate. Very important. 469 00:47:50.579 --> 00:47:53.909 Which takes are your own 1 bits to superposition and back. 470 00:47:53.909 --> 00:48:03.030 And finally, we learn that we can think of cubes in their operations as forming the state machine on the unit circle our unit sphere. If we're using complex numbers, it's actually called the block sphere. 471 00:48:03.030 --> 00:48:09.989 The sphere, if you want to look that up for the full complex numbers, but unit circle just fine for us. 472 00:48:09.989 --> 00:48:15.780 Okay, so we're going to learn about something called the Oracle. 473 00:48:15.780 --> 00:48:25.409 Okay, so that was. 474 00:48:25.409 --> 00:48:34.289 Half an hour on very nice introduction from Microsoft and quantum computing you can watch the rest, or I might even show more of it later. 475 00:48:34.289 --> 00:48:38.849 But back to, let's see, here. 476 00:48:39.954 --> 00:48:49.195 That was that I'd like to do some getting a little less technical now for the rest of the thing. Well, 3 minutes on what? 477 00:48:49.224 --> 00:48:55.224 Quantum algorithms from somebody at IBM, and then a managerial update of the state of the technology. 478 00:48:59.130 --> 00:49:05.519 In the future quantum computers have the potential to solve certain problems exponentially faster than classical computers. 479 00:49:05.519 --> 00:49:08.699 But which problems can they solve exponentially faster. 480 00:49:08.699 --> 00:49:17.250 And which problems would we do just as well to solve using the classical computer? So, these are the questions that we're really interested in as quantum algorithms researchers. 481 00:49:17.250 --> 00:49:24.510 Now, to set the stage, it makes sense to talk a little bit about the relationship between algorithms and computer hardware and more generally. 482 00:49:24.510 --> 00:49:27.659 A computer is a programmable machine. 483 00:49:27.659 --> 00:49:32.429 Which will base the laws of physics an algorithm is a recipe for solving a problem. 484 00:49:32.429 --> 00:49:35.969 So, an algorithm is a sequence of steps that lead you to the solution. 485 00:49:35.969 --> 00:49:42.570 Of course, if you design an algorithm, you have to make sure that the machine you plan to run it on can execute each of those steps. 486 00:49:42.570 --> 00:49:46.980 Generally speaking, this is not much of a problem because. 487 00:49:46.980 --> 00:49:52.739 Algorithms designers like to abstract away the machine and we're using a basic construction set. 488 00:49:52.739 --> 00:49:59.969 That can then be mapped to the machine afterwards and 1 of the central principals of theoretical computer science is that. 489 00:49:59.969 --> 00:50:04.710 If you use the right basic construction set, then any real world machine, you can build. 490 00:50:04.710 --> 00:50:08.579 You can run that algorithm on it. Um. 491 00:50:08.579 --> 00:50:15.960 This central tenet of theoretical computer science held out for about 50 years, but we now understand that. 492 00:50:15.960 --> 00:50:20.909 There's 1 exception to this rule and the exception is quantum computers. 493 00:50:20.909 --> 00:50:30.690 Quantum computers use a fundamentally different instruction set than classical computers as a result. They can run different algorithms algorithms. You could not run on classical computers. 494 00:50:30.690 --> 00:50:34.769 Those algorithms make use of quantum effects, like non locality. 495 00:50:34.769 --> 00:50:42.000 Entanglement superposition and interference and so when you're designing a quantum algorithm, your aim is to exploit those quantum effects. 496 00:50:42.000 --> 00:50:49.800 To make you get the solution faster generally speaking an algorithm might start by taking your classical input to the problem. 497 00:50:49.800 --> 00:50:56.550 Processing that into a quantum state, which is a superposition of an exponential number of classical States. 498 00:50:56.550 --> 00:51:01.289 You then to transform that quantum state, which encodes the problem. 499 00:51:01.289 --> 00:51:05.760 Into a quantum state, which includes the solution and from that quantum say, you measure. 500 00:51:05.760 --> 00:51:12.360 And you get the solution now, of course, to make this work, it's a bit of an art, and you have to design an algorithm. 501 00:51:12.360 --> 00:51:16.829 Which follows these these steps for a given problem and some of the areas where. 502 00:51:16.829 --> 00:51:22.500 We have such algorithms where we have quantum speed ups include the simulation of quantum systems. 503 00:51:22.500 --> 00:51:25.769 Number theoretical problems, like factorization. 504 00:51:25.769 --> 00:51:29.940 And certain types of search problems now, 1 of the caveats is that. 505 00:51:29.940 --> 00:51:34.409 Most of these algorithms are designed for future quantum computers, which have many, many Cubans. 506 00:51:34.409 --> 00:51:45.119 And right now we're at a moment where people are building, small and modest sized quantum computers and so it was an exciting research area to understand what can we do with those devices before we have. 507 00:51:45.119 --> 00:51:53.340 Basically, an unlimited number of okay. 508 00:51:53.340 --> 00:51:56.670 And, like, I see here. 509 00:52:09.570 --> 00:52:16.769 Welcome to another video explaining computers. 510 00:52:16.769 --> 00:52:23.969 Dot com, this time, it's my 4th annual update on the state of play in quantum computing. 511 00:52:23.969 --> 00:52:31.769 This video is mainly going to focus on what the pioneers in the area have been achieving in the past 12 months. 512 00:52:31.769 --> 00:52:37.769 However, as usual, I thought we should start out with a brief summary of what quantum computing. 513 00:52:37.769 --> 00:52:42.719 Is all about. 514 00:52:44.219 --> 00:52:54.989 Micro processes in conventional or classical computers are built from billions of transistors that are turned on or off to represent a value of either 1 or 0. 515 00:52:54.989 --> 00:53:02.429 In turn this allows classical computers to store and process data, using binary digits or bits. 516 00:53:02.429 --> 00:53:09.360 In contrast, quantum computers process, information using quantum beds or Cubics. 517 00:53:09.360 --> 00:53:20.460 These can be created in many different ways, for example, using superconducting electronic circuits or by tapping ionized atoms as I'll discuss later in the video. 518 00:53:22.050 --> 00:53:29.699 Unlike classical bits, Cubics can exist in more than 1 state or superposition, but exactly. The same point in time. 519 00:53:29.699 --> 00:53:36.690 This allows a cubic to assume a value of 1 or 0 or both of these numbers simultaneously. 520 00:53:36.690 --> 00:53:44.730 In turn is potentially enables a quantum computer to process a far high number of data possibilities than a classical computer. 521 00:53:44.730 --> 00:53:52.949 Indeed, in theory, quantum computers will be able to accomplish tasks, which are impossible to perform using conventional hardware. 522 00:53:56.070 --> 00:54:06.780 Today, quantum computing remains in the research phase with all quantum hardware created for experimental purposes. 523 00:54:06.780 --> 00:54:13.079 This said many traditional computing companies and now developing and operating quantum computers. 524 00:54:13.079 --> 00:54:20.130 With key players, including IBM, Google, and they bought Microsoft, Intel and Honeywell. 525 00:54:20.130 --> 00:54:25.650 There are also a whole host of pure play hardware and software startups, including. 526 00:54:25.650 --> 00:54:30.329 Cupid D wave systems. I. Q. 527 00:54:30.329 --> 00:54:33.840 Juicy where he was simulate quantum circuits. 528 00:54:33.840 --> 00:54:37.829 Rocco with Getty and. 529 00:54:37.829 --> 00:54:45.389 You can find out more about what these companies are doing on the quantum computing page on expanding computers dot com. 530 00:54:46.469 --> 00:54:55.710 Quantum computers will only become a viable commercial technology when they can perform useful calculations that cannot be executed on a classical computer. 531 00:54:55.710 --> 00:55:05.730 Back in 2012 professor, John presco coins to term quantum supremacy to define this concept writing. But in the era of quantum supremacy. 532 00:55:05.730 --> 00:55:13.380 We will be able to perform tasks with controlled quantum systems going beyond what can be achieved with ordinary digital computers. 533 00:55:15.300 --> 00:55:20.340 In October, 2019, Google claim to have achieved quantum supremacy. 534 00:55:20.340 --> 00:55:29.280 As it reported a team of Google scientists used a quantum processor called sycamore, the sample, the output of a pseudo random quantum circuit. 535 00:55:29.280 --> 00:55:35.699 Sycamore took about 200 seconds to sample 1 instance of a circuit a 1Million times. 536 00:55:35.699 --> 00:55:44.639 In comparison, the Google team estimated classical supercomputer would take about 10,000 years to perform the same calculations. 537 00:55:44.639 --> 00:55:55.739 As a team 1 contract conclude quantum processes based on superconducting Cubics, and now perform computation beyond the reach of the fastest classical supercomputers available today. 538 00:55:55.739 --> 00:56:02.579 To our knowledge with experiment 1st computation that can be performed only on a quantum processor. 539 00:56:02.579 --> 00:56:07.320 Quantum processes have reached the regime of quantum supremacy. 540 00:56:08.969 --> 00:56:12.960 The Google announcement was big news, but soon controversial. 541 00:56:12.960 --> 00:56:25.019 Not least IBM published a blog post in which they stated that the computations in google's experiment could be undertaken on a classical computer in 2 and a half days rather than 10,000 years. 542 00:56:25.019 --> 00:56:30.570 As IBM went on to content, because the original meaning of the term quantum supremacy. 543 00:56:30.570 --> 00:56:40.860 Was proposed by John practical in 2012, which described a point where quantum computers can do things that classical computers can't. This threshold has not been met. 544 00:56:42.750 --> 00:56:51.059 Ibm and Google arrivals who have created quantum computers based on the same kind of superconducting Joseph some junction Cubics. 545 00:56:51.059 --> 00:56:55.829 It's therefore perhaps not surprising that IBM disputed google's achievement. 546 00:56:55.829 --> 00:57:05.550 However, and more broadly IBM and some other quantum computing specialists, increasingly question the importance of the quantum supremacy concept. 547 00:57:05.550 --> 00:57:11.340 But least they are concerned that future experiments, which demonstrate quantum supremacy. 548 00:57:11.340 --> 00:57:15.150 May involve tasks with knowing useful or commercial application. 549 00:57:18.269 --> 00:57:28.590 Whilst the quantum supremacy 2 basis raged the last 12 months I've seen a notable expansion of cloud based quantum computing. 550 00:57:28.590 --> 00:57:36.239 Or quantum computing as a service, the birth of Q. C. A. S dates back to May 2016. 551 00:57:36.239 --> 00:57:42.750 When IBM launched the IBM queue experience to make a quantum computer available over the Internet. 552 00:57:42.750 --> 00:57:51.119 Just under 2 years later, Chinese E, commerce, giant Ali Barb followed suit with its superconducting quantum computing cloud. 553 00:57:51.119 --> 00:58:01.320 This was developed with the Chinese Academy of sciences, and, like, IBM to allow users to run quantum programs in the cloud and download the results. 554 00:58:02.460 --> 00:58:13.079 October 2018, Canadian, quantum computing, Pioneer D wave systems also made available a service called leap to provide cloud access to each quantum computing hardware. 555 00:58:13.079 --> 00:58:22.050 And in January 2019, rival, quantum computing, pure play with Getty, launched its content cloud services all platform. 556 00:58:24.059 --> 00:58:31.949 The big quantum cloud news of the past 12 months has been the entry of Amazon and Microsoft as new market players. 557 00:58:31.949 --> 00:58:37.650 amazon's offerings were announced on December. 2nd, 2019. 558 00:58:37.650 --> 00:58:47.909 And include an, a W s solution called Amazon bracket, but allows scientist researchers and developers to begin experimenting with quantum computers from multiple hardware providers. 559 00:58:47.909 --> 00:58:52.500 Specifically customers could access hardware from D, Wave systems. 560 00:58:52.500 --> 00:59:00.420 On 2, and we're getting which means they can experiment with quantum computers based on 3 different cubic technologies. 561 00:59:02.010 --> 00:59:07.349 In addition to bracket, Amazon also launched the Amazon solution to lab. 562 00:59:07.349 --> 00:59:14.909 This is intended to help companies to get ready for quantum computing by allowing them to work with leading pioneers. 563 00:59:14.909 --> 00:59:21.869 So, the thing that Allison is doing was its quantum computing offerings is to act as a broker in the cloud. 564 00:59:23.849 --> 00:59:27.929 On maybe 19th 2020, Microsoft entered the fray. 565 00:59:27.929 --> 00:59:35.489 Well, it's announced as your quantum boss, Microsoft is working with several universities to develop a tone quantum hardware. 566 00:59:35.489 --> 00:59:43.980 Approach current cloud offering, it's adopted the same strategy as Amazon to partner with those who already have quantum hardware and software solutions available. 567 00:59:44.394 --> 00:59:47.755 This allows Microsoft to offer what is your quantum stack, 568 00:59:48.025 --> 00:59:51.175 which it describes as an open cloud ecosystem, 569 00:59:51.235 --> 00:59:54.594 offering access to a diverse set of quantum resources, 570 00:59:54.744 --> 00:59:55.525 including Pre, 571 00:59:55.525 --> 00:59:56.574 built solutions, 572 00:59:56.755 --> 00:59:58.135 quantum development tools, 573 00:59:58.135 --> 00:59:59.155 such as simulators, 574 00:59:59.244 --> 01:00:00.954 resource estimation tools, 575 01:00:01.135 --> 01:00:05.545 accelerated classical hardware and a variety of quantum hardware. 576 01:00:09.389 --> 01:00:19.829 1 of the developments that's most captured my own attention in the past 12 months has been progress in trapped ion quantum computing. 577 01:00:19.829 --> 01:00:30.389 Keep players here include the University of Oxford as well as a company called iron cube, but in October 2019 secure and 55Million dollars of funding. 578 01:00:30.389 --> 01:00:40.289 The technology page of the Q, a website explains what quantum computing is all about and really is an excellent learning resource. 579 01:00:41.400 --> 01:00:52.199 In simple terms, chapter on quantum computing, create teach Cupid by removing an electron from an atom in order to turn it into an ion with a positive electrical charge. 580 01:00:52.199 --> 01:01:02.909 The chip known as a linear ion trap is then used to help change that ion cube it in 3 D space where they can be manipulated and red using lasers. 581 01:01:02.909 --> 01:01:15.690 All of this is extremely difficult to achieve and control, but least because the cubital need to be super cold to almost absolutely. 0 as well as being isolated in a very high quality vacuum chamber. 582 01:01:17.789 --> 01:01:21.659 In addition to, I'm Q, another key player is Honeywell. 583 01:01:21.659 --> 01:01:29.400 Go in June 2020 announced we didn't use trapped iron technology to create the world's highest performance quantum computer. 584 01:01:29.400 --> 01:01:32.940 Clearly, others may take a different view. 585 01:01:32.940 --> 01:01:39.295 But regardless, this is another important development, particularly as Honeywell report that J. 586 01:01:39.295 --> 01:01:49.315 P Morgan chase is already experimenting with its system to develop quantum financial services software, including for detection and trading applications. 587 01:01:52.650 --> 01:02:02.309 Always when I make a video about quantum computing, I get questions about how quantum computers are programmed. 588 01:02:02.309 --> 01:02:10.349 Present this takes place using a classical computer as a controlled device where the language is used, including Python. 589 01:02:10.349 --> 01:02:15.449 An explanation of exactly what is involved is well, beyond the scope of this video. 590 01:02:15.449 --> 01:02:23.880 But if you want to learn more and actually experiment, go to the IBM quantum computing Web site, and click on for developers. 591 01:02:23.880 --> 01:02:30.989 Next select try kiss kit, which will allow you to experiment with this open source development environment. 592 01:02:32.579 --> 01:02:41.519 1, great, but still quite complex example shows how it is possible to estimate the value of pie using a quantum phase estimation algorithm. 593 01:02:41.519 --> 01:02:47.010 As you can see the Python code begins by importing relevant libraries. 594 01:02:47.010 --> 01:02:52.349 Before progressing using commands that will be pretty familiar with some programming background. 595 01:02:52.349 --> 01:02:57.389 Absolutely, this is complex and something you need to get your head around. 596 01:02:57.389 --> 01:03:05.369 But what I hope it makes clear is that quantum computing is not quite so removed from traditional computing as many people believe. 597 01:03:08.639 --> 01:03:15.510 Quantum computing is yet to become a commercial reality. 598 01:03:15.510 --> 01:03:21.150 This said, as we've seen in this video, significant progress continues to be made. 599 01:03:21.150 --> 01:03:28.139 And I remain confident that some kind of quantum hardware will find useful application in the 2nd, half. 600 01:03:28.139 --> 01:03:36.000 Of this decade, but now that it for another video, if it's job, you see the light button. 601 01:03:36.000 --> 01:03:39.030 If you have a subscribe subscribe. 602 01:03:39.030 --> 01:03:42.329 And I hope to talk to you again so. 603 01:03:43.829 --> 01:03:57.239 Okay, so the earlier, so the, he does 1 of these every year, the earlier ones are also interesting because they talk about. 604 01:03:57.239 --> 01:04:01.289 I'll stop the earlier, but still relevant to quantum computing. 605 01:04:01.289 --> 01:04:07.110 Okay, that's a good point. I think I'll just stop today. 606 01:04:07.110 --> 01:04:14.550 And what we'll continue on Monday well, to chance for you to get. 607 01:04:15.750 --> 01:04:24.719 Just a quick say teaser about what you're going to do for your term projects and I'll continue on with my summary, but algorithm details and I'll. 608 01:04:24.719 --> 01:04:31.860 Had it out with a lot of videos because the videos, I think talk about it better than I do and they have the visual assist. 609 01:04:31.860 --> 01:04:37.710 So, if there's any questions, other than that, have a good weekend. 610 01:04:37.710 --> 01:04:41.159 Questions no. 611 01:04:42.570 --> 01:04:47.039 See, you Monday.