WEBVTT 1 00:00:12.689 --> 00:00:16.768 Hello. 2 00:00:16.768 --> 00:00:27.600 Okay, good afternoon class. This is quantum computer programming class 18 if I'm counting right? 3 00:00:27.600 --> 00:00:34.890 So, I think I'm going to assume that the audio works, if not. 4 00:00:34.890 --> 00:00:39.719 2nd here. Okay. I have a chat window open. 5 00:00:39.719 --> 00:00:53.399 So, this concept is that you can send me chat messages, and also in an emergency, if you want to get hold of me, I have my mobile phone off here. 6 00:00:53.399 --> 00:01:00.000 With her on even so we'll see how many siding calls I get during the talk. 7 00:01:00.000 --> 00:01:03.840 Okay, so let us see. 8 00:01:05.430 --> 00:01:12.180 Silence. 9 00:01:17.459 --> 00:01:20.670 Actually work. 10 00:01:22.829 --> 00:01:33.209 Cory, so, what's happening in this course I just want some general stops and get more to IBM and then talk a little about Microsoft and Google. 11 00:01:33.209 --> 00:01:38.819 And so on 1st, ask a question for you. 12 00:01:38.819 --> 00:01:52.799 Do you like the pace of this course? Do you feel you're getting value for the money you're paying? Or am I working you to death? So I'm trying to keep things relaxed here, introduce you to fun stuff that you can pursue. 13 00:01:52.799 --> 00:02:03.930 And it occurred to me, maybe you want deeper math maybe you want to go into every little detail of algorithms and so on. 14 00:02:03.930 --> 00:02:17.455 Do you want me I could give you required readings and viewings videos before class that you have to watch on your own and then we discuss them in class that's with some classes. Do I could require you to turn your cameras on? I'm the only person never has my camera on. 15 00:02:17.455 --> 00:02:19.705 I've noticed and then I could start. 16 00:02:19.949 --> 00:02:25.110 Calling on questions. So what do you think you are you. 17 00:02:25.110 --> 00:02:31.259 Wouldn't trade well, thank you. Anyone else have any comments you can you said, use the chat window. 18 00:02:32.125 --> 00:02:47.064 Again, I like to introduce you to stop, you know, I mean, I asked to create this. Course, I asked the department head, you know, could I do this and he said yes, so I have fun. Fun, learning this stuff and I enjoy teaching stuff to other people. 19 00:02:47.094 --> 00:02:48.414 But, okay. 20 00:02:48.689 --> 00:02:55.110 So, again, you know, you think of something later on and you can do that and. 21 00:02:55.110 --> 00:03:08.039 Also, so good various stuff here for you. I got the wire stuff I've been browsing around so I thought you might like, also now, the word guide, it's not that deep, but. 22 00:03:08.039 --> 00:03:13.469 I have it up here basically because of the timetable here. 23 00:03:13.469 --> 00:03:16.860 Which I think is cool. So. 24 00:03:16.860 --> 00:03:25.349 You know, they, it was 40 years ago when they the physicist's 1st, thought of this. 25 00:03:25.349 --> 00:03:39.599 Then you often find men named it you know, the di chat algorithm is 35 years ago. Magnification showers qualification algorithm is 15 year so, 25 years ago. 26 00:03:39.599 --> 00:03:44.219 D, wave is more than it's a dozen years old now and so. 27 00:03:45.985 --> 00:03:52.824 And we still don't have it actually being useful frankly, it's at the point where everyone thinks it's going to be useful, real. 28 00:03:52.824 --> 00:04:01.014 So now I agree with them but it's still at the prototype stage says all these different companies competing to do it, but. 29 00:04:01.409 --> 00:04:16.230 You know, it hasn't quantum supremacy. Google says they've got it. There's some debate about well, IBM says quantum supremacy is meaningless and then there's some question of what they've gotten there, whether or not it's meaningful. 30 00:04:16.795 --> 00:04:30.535 So, I, I personally think the genesis of and again, they did it with atomic energy, twenties, thirties and forties. And we know how that turned out. My view is the physicists have gone and done it again. They and atomic energy again. 31 00:04:30.564 --> 00:04:32.875 So theory was 1st, and. 32 00:04:33.178 --> 00:04:45.149 Yeah, thank you. Thanks guys, it's, we're going to see an enormous breakthrough. Maybe I'll be wrong. You know, there was a joke some said. 33 00:04:45.149 --> 00:04:48.569 More time and more depth. Okay. 34 00:04:48.569 --> 00:04:52.048 Think about that then thank you. 35 00:04:52.048 --> 00:04:55.709 You know, so I think we're about to have a big breakthrough, but. 36 00:04:55.709 --> 00:04:59.399 Who knows any case. 37 00:04:59.399 --> 00:05:12.988 That was just a nice summary. I have here an update, which I'll, I'll play the whole thing so it starts off a little review. Reviews aren't so bad and then it gets into what's happening in industry. 38 00:05:12.988 --> 00:05:18.869 Now. 39 00:05:25.439 --> 00:05:32.459 Welcome to another video explaining computers. 40 00:05:32.459 --> 00:05:39.689 Dot com, this time, it's my 4th annual update on the state of pay in quantum computing. 41 00:05:39.689 --> 00:05:47.519 This video is mainly going to focus on what the partners in the area have been achieving in the past 12 months. 42 00:05:47.519 --> 00:05:53.459 However, as usual, I thought we should start out with a brief summary of what quantum computing. 43 00:05:53.459 --> 00:05:58.139 Is all about. 44 00:05:59.788 --> 00:06:10.738 The processors 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. 45 00:06:10.738 --> 00:06:17.968 In turn this allows classical computers to store the protest data, using binary digits or bits. 46 00:06:17.968 --> 00:06:24.988 In contrast, quantum computers process, information using quantum beds, or. 47 00:06:24.988 --> 00:06:36.178 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. 48 00:06:37.439 --> 00:06:45.418 Unlike classical bits, Cubics can exist in more than 1 state or superposition at exactly. The same point in time. 49 00:06:45.418 --> 00:06:52.379 This allows a Cupid to assume a value of 1 or 0T or both of these numbers simultaneously. 50 00:06:52.379 --> 00:07:00.418 In turn, this potentially enables a quantum computer to process a far high, a number of data possibilities in a classical computer. 51 00:07:00.418 --> 00:07:08.699 Indeed in Siri, quantum computers will be able to accomplish tasks, which are impossible to perform using conventional hardware. 52 00:07:14.309 --> 00:07:22.379 Today, quantum computing remains in the research phase with all quantum hardware created for experimental purposes. 53 00:07:22.379 --> 00:07:28.649 This said many traditional computing companies and now developing and operating quantum computers. 54 00:07:28.649 --> 00:07:35.819 With key players, including IBM Barb, Microsoft, Intel and Honeywell. 55 00:07:35.819 --> 00:07:41.309 Are also a whole host of pure play hardware and software startups, including. 56 00:07:41.309 --> 00:07:45.959 Cupid C. D. wave systems. I. Q. 57 00:07:45.959 --> 00:07:49.559 Juicy where you assimilate quantum circuits. 58 00:07:49.559 --> 00:07:53.579 Rocco with Getty and. 59 00:07:53.579 --> 00:07:59.668 You can find out more about what these companies are doing on the quantum computing page on expanding computers. 60 00:07:59.668 --> 00:08:06.149 Dot com, quantum computers will become a viable commercial technology. 61 00:08:06.149 --> 00:08:11.369 When they can perform useful calculations, they cannot be executed on a classical computer. 62 00:08:11.369 --> 00:08:21.389 Back in 2012 professor, John presco coins to term quantum supremacy to define this concept writing in the era of quantum supremacy. 63 00:08:21.389 --> 00:08:27.028 We will be able to perform tasks with controlled content systems going beyond what can be achieved. 64 00:08:27.028 --> 00:08:32.969 Ordinary computers in October 2019. 65 00:08:32.969 --> 00:08:35.999 Google claim to have achieved quantum supremacy. 66 00:08:35.999 --> 00:08:45.359 As it reported a team of user content processor called sycamore, the sample, the output of a pseudo random quantum circuit. 67 00:08:45.359 --> 00:08:51.599 Sycamore took about 200 seconds to sample 1 instance of a circuit a 1M times. 68 00:08:51.599 --> 00:09:00.629 In comparison, the Google team estimated classical supercomputer would take about 10000 years to perform the same calculations. 69 00:09:00.629 --> 00:09:11.428 As a team went on to conclude quantum processes based on superconducting Cubics and now perform computations beyond the reach of the fastest classical supercomputers available today. 70 00:09:11.428 --> 00:09:18.239 To our knowledge experiment box, the 1st computation that can be performed only on a quantum processor. 71 00:09:18.239 --> 00:09:23.009 Quantum processes have reached the regime of quantum supremacy. 72 00:09:23.009 --> 00:09:30.778 If I can just pause for a minute response to that, is that the particular job they were doing was sort of silly. 73 00:09:30.778 --> 00:09:34.168 And not a realistically useful job. 74 00:09:34.168 --> 00:09:37.408 Get that response for Thursday. 75 00:09:37.408 --> 00:09:41.548 The Google announcement was big news, but soon controversial. 76 00:09:41.548 --> 00:09:53.668 Not least IBM published a blog post in which they stated that the computations in google's experiment could be undertaken a classical computer in 2 and a half days rather than 10000 years. 77 00:09:53.668 --> 00:09:59.219 As IBM went on to content, because the original meaning of the term quantum supremacy. 78 00:09:59.219 --> 00:10:07.408 Was proposed by John presco in 2012 was described a point where quantum computers can do things that classical computers can't. 79 00:10:07.408 --> 00:10:19.798 This threshold has not been that IBM and Google a liable who have created quantum computers based on the same kind of superconducting Joseph some junction Cuba. 80 00:10:19.798 --> 00:10:24.479 It's therefore perhaps not surprising that IBM disputed google's achievement. 81 00:10:24.479 --> 00:10:34.229 However, and more broadly IBM and some other quantum computing specialists, increasingly question the important of the quantum supremacy concept. 82 00:10:34.229 --> 00:10:39.989 But at least they are concerned that future experiments, which demonstrate quantum supremacy. 83 00:10:39.989 --> 00:10:43.828 Involve tasks with no use for or commercial application. 84 00:10:49.589 --> 00:10:57.269 Whilst a quantum supremacy to basis raged the last 12 months I've seen a notable expansion of cloud based quantum computing. 85 00:10:57.269 --> 00:11:04.889 Or quantum computing as a service, the birth of s dates back to May 2016. 86 00:11:04.889 --> 00:11:11.399 When IBM launch the IBM Q experience to make a quantum computer available over the Internet. 87 00:11:11.399 --> 00:11:19.769 Just under 2 years later, Chinese ecommerce, giant Ali barber followed suit with its superconducting quantum computing cloud. 88 00:11:19.769 --> 00:11:29.969 This was developed with a Chinese Academy of sciences, and, like, IBM to allows users to run quantum programs in the cloud and download the results. 89 00:11:31.078 --> 00:11:41.788 October 2018, Canadian computing, Pioneer D wave system also made available a service called leap to provide cloud access to each quantum computing hardware. 90 00:11:41.788 --> 00:11:50.759 And in January, 2019, rival, quantum computing, pure play, we Getty launched its quantum cloud services all platform. 91 00:11:52.678 --> 00:12:00.629 The big quantum cloud news of the past 12 months has been the entry of outlets and Microsoft as new market players. 92 00:12:00.629 --> 00:12:06.298 amazon's offerings were announced on December. 2nd, 2019. 93 00:12:06.298 --> 00:12:16.678 And include an, a W s solution called Allison bracket, but allows scientists researches developers to begin experimenting with quantum computers from multiple hardware providers. 94 00:12:16.678 --> 00:12:21.178 Specifically customers could access hardware from D, Wave systems. 95 00:12:21.178 --> 00:12:29.129 I'm 2, and we're getting, which means they can experiment with quantum computers based on 3 different cubic technologies. 96 00:12:30.688 --> 00:12:35.938 In addition to bracket, Amazon also launched the Amazon quantum solutions. 97 00:12:35.938 --> 00:12:43.589 This is intended to help companies to get ready quantum computing by allowing them to work with leading pioneers. 98 00:12:43.589 --> 00:12:50.548 So, a key thing that Allison is doing was its quantum computing offerings is to act as a broker in the cloud. 99 00:12:50.548 --> 00:12:54.688 On May 19th, 2020. 100 00:12:54.688 --> 00:12:58.859 Microsoft enter the fray when it's announced as your quantum. 101 00:12:58.859 --> 00:13:04.139 Microsoft is working with several of universities to develop the term quantum hardware. 102 00:13:04.139 --> 00:13:12.658 Broach current cloud offering, it's adopted the same strategy as Amazon to partner with those who already have quantum hardware and software solutions available. 103 00:13:12.658 --> 00:13:16.708 This allows Microsoft, the offer it to 0T quantum stack. 104 00:13:16.708 --> 00:13:19.979 Which it describes as an open cloud ecosystem. 105 00:13:19.979 --> 00:13:25.558 Offering access to a diverse set of quantum resources, including Pre, built solutions. 106 00:13:25.558 --> 00:13:34.678 Content development tool, such as simulators and resource estimation, tools, accelerated, classical hardware and a variety of quantum hardware. 107 00:13:37.979 --> 00:13:48.479 1 of the developments that's most captured. My owner in the past 12 months has been progress in trapped ion quantum computing. 108 00:13:48.479 --> 00:13:59.038 Is here include the University of Oxford as well as a company called? Ion cube between October 2019 secured 55M dollars of funding. 109 00:13:59.038 --> 00:14:08.938 The technology page of the on your website explains what quantum computing is all about and really is an excellent learning resource. 110 00:14:10.078 --> 00:14:20.969 In simple terms, John quantum computing created cubic by removing an electron from an atom in order to turn it into an ion with a positive electrical charge. 111 00:14:20.969 --> 00:14:24.178 The chip known as a linear ion trap. 112 00:14:24.178 --> 00:14:31.649 Is then used to help change that ion cubic in 3 D, space where they can be manipulated and red using lasers. 113 00:14:31.649 --> 00:14:40.109 All of this is extremely difficult to achieve and control, but least because the Cubans need to be super cold to all those. Absolutely. 0. 114 00:14:40.109 --> 00:14:44.519 As well, as being isolated in a very high quality vacuum chamber. 115 00:14:46.379 --> 00:14:50.369 In addition to iron to another key player is Honeywell. 116 00:14:50.369 --> 00:14:57.418 In June 2020 announce we didn't use trapped ion technology to create the world's highest performance quality. 117 00:14:57.418 --> 00:15:01.589 Computer clearly, others may take a different view. 118 00:15:01.589 --> 00:15:07.974 But regardless, this is another important development, particularly as Honeywell report that J. 119 00:15:07.974 --> 00:15:17.994 P Morgan chase is already experimenting with its system to develop quantum financial services software, including for detection and trading applications. 120 00:15:23.969 --> 00:15:27.269 Always when I make a video about quantum computing. 121 00:15:27.269 --> 00:15:30.989 I get questions about how quantum computers are programmed. 122 00:15:30.989 --> 00:15:38.668 At present, this takes place using a classical computer as a controlled advice where the language is used, including Python. 123 00:15:38.668 --> 00:15:44.099 An explanation of exactly what is involved is well, beyond the scope of this video. 124 00:15:44.099 --> 00:15:52.589 But if you want to learn more and actually experiment, go to the IBM quantum computing Web site, and click on for developers. 125 00:15:52.589 --> 00:15:59.639 Next select try casket, which will allow you to experiment with this open source development environment. 126 00:16:01.288 --> 00:16:10.109 1, great, but still quite complex example shows how it is possible to estimate the value of pie using a quantum phase estimation algorithm. 127 00:16:10.109 --> 00:16:15.688 As you can see the Python code begins by importing relevant libraries. 128 00:16:15.688 --> 00:16:20.908 Before progressing using command that will be pretty familiar with some programming background. 129 00:16:20.908 --> 00:16:26.038 Absolutely, this is complex and something you need to get your head around. 130 00:16:26.038 --> 00:16:34.078 But what I hope it makes clear is that quantum computing is not quite so removed from traditional computing as many people believe. 131 00:16:37.318 --> 00:16:44.158 Quantum computing is to get to become a commercial reality. 132 00:16:44.158 --> 00:16:49.739 This said is we've seen in this video significant progress, continues to be made. 133 00:16:49.739 --> 00:16:56.818 And I remain confident that some kind of quantum hardware will find useful application in the 2nd, half. 134 00:16:56.818 --> 00:17:02.188 Of this decade, but now that it for a lot of video. 135 00:17:02.188 --> 00:17:07.709 It doesn't job, you see the best type like button. You'll have a subscribes please subscribe. 136 00:17:07.709 --> 00:17:10.979 And I hope to talk to you again Thank you, sir. 137 00:17:19.288 --> 00:17:28.169 Okay, that was the 2020 update. Now. I'd like to show you. This is a nice page here. 138 00:17:28.169 --> 00:17:33.778 From yet another quantum report, these are open source. 139 00:17:33.778 --> 00:17:39.148 Development projects that you saw kissed kit. Well, there's many I Microsoft. 140 00:17:39.148 --> 00:17:44.489 As it, excuse me, has their development kit, or there's IBM saying. 141 00:17:44.489 --> 00:17:49.288 Um, and there's already others, so it's quite nice. Is. 142 00:17:49.288 --> 00:17:56.669 I believe what these companies are doing is they see a marketing thing to get people to use their product is to provide the software tools. 143 00:17:56.669 --> 00:18:01.078 Which is nice for us in universities. So there's all of these tools that are. 144 00:18:01.078 --> 00:18:05.489 That are available for free and so you can. 145 00:18:05.489 --> 00:18:14.759 Browse through this, and actually this is a chance actually to talk about the next homework, which will be to pick a tool and then to. 146 00:18:14.759 --> 00:18:18.989 Write up something to summarize it for us. So. 147 00:18:18.989 --> 00:18:25.288 Let me actually divert myself and I'll come back to that, but 1st. 148 00:18:25.288 --> 00:18:28.979 So, pick a software project, write a page on it. 149 00:18:28.979 --> 00:18:35.608 And you could start from that page I just showed you and then also to write another page summarizing Microsoft. 150 00:18:35.608 --> 00:18:38.788 So, because I'll come up to them in a minute. 151 00:18:38.788 --> 00:18:43.048 Oh, okay so can I was open source quantum software projects. 152 00:18:43.048 --> 00:18:47.699 I'd like to show you a little video. 153 00:18:47.699 --> 00:18:58.229 It's an introductory thing, but this introduces you to 1 of main people, Abraham asphalt. So he's down at Yorktown Heights and I just thought. 154 00:18:58.229 --> 00:19:03.598 It'd be good for you to see him. I like to show you videos of leading people in the field. 155 00:19:03.598 --> 00:19:18.509 So so, as you can see, here, I'm losing pretty badly. Oh, that's funny. I can not beat the classical computer. That's not a thing we should keep from the video. 156 00:19:20.243 --> 00:19:20.814 Very hard. 157 00:19:34.794 --> 00:19:36.084 Hello my name is. 158 00:19:37.229 --> 00:19:46.763 And you can call me if you'd like, I'm originally from, and today I live in New York City programming and learning how to develop on quantum computers. 159 00:19:46.854 --> 00:19:54.324 The point of this YouTube series is to take you through the journey of learning and discovering quantum computers and programming quantum algorithms on them. 160 00:19:55.463 --> 00:20:08.753 So, this is how you program IBM, quantum computers, the best thing about open source and freely available. And what it means is that you can use it, not only to build quantum algorithms, but also in real world applications. 161 00:20:09.624 --> 00:20:21.354 So I started graduate school in 2012 and wanted to study experimental quantum computing back. Then, in order to do these experiments, you actually needed access to a research lab that works on quantum computing. 162 00:20:21.354 --> 00:20:27.834 So the time it took for you to go from, have an idea to okay, I cannot do. The experiment was several days. 163 00:20:29.788 --> 00:20:35.548 But now using his kit, you can do all of the work with quantum computing from your laptop. 164 00:20:35.548 --> 00:20:39.179 Once, you know how to program quantum computers using. 165 00:20:39.179 --> 00:20:52.943 Now, you can focus on various application areas 1, particular area that you can focus on is quantum chemistry. So, for example, calculating the bond length of molecules. So, another area that you can focus on is the development of quantum algorithms. 166 00:20:53.153 --> 00:21:08.094 1 of my favorite quantum algorithms is what's called the on the algorithm. This is a really interesting algorithm. So imagine, you have a box with a number inside that you don't know, you can find out what secret number is inside that box in 1 shot. 167 00:21:08.574 --> 00:21:23.364 And 1 of the really cool things that we can also do is program games based on quantum computing. So, on this screen, what we have is game called quantum punk or a coupon for short. And the idea is to create a quantum circuit at the very bottom of the. 168 00:21:23.608 --> 00:21:34.288 Uh, screen here as you can see which moves the paddle based on the outcome of the quantum computation from that circuit. So, let me try and meet the classical computer. 169 00:21:35.548 --> 00:21:42.959 Oh, there we go. So, watch me create a superposition now between those 2, but it doesn't matter because I'm going to lose anyway. 170 00:21:42.959 --> 00:21:46.769 However, now we have an interesting situation, look about. 171 00:21:46.769 --> 00:21:49.828 Just lost. 172 00:21:49.828 --> 00:21:55.588 I went through because the measurement force, the superposition to collapse into the bottom. 173 00:21:55.588 --> 00:22:03.479 A paddle and not. This is a very good way to teach quantum computing and generally how to create quantum algorithms. 174 00:22:03.479 --> 00:22:18.298 Even though I'm losing pretty hard, and as you can see, you can develop not only quantum algorithms, but also quantum applications and excusing. So, throughout this whole YouTube series, the goal will be to explore this different range of things that you can do. 175 00:22:18.298 --> 00:22:27.148 And eventually to learn how to program a quantum computer in the next episode, we'll be covering in detail how to install kit and get ready to start programming with it. 176 00:22:27.148 --> 00:22:32.818 And then will show you how to use kit. So, for those of you have started on your journey with quantum computing. 177 00:22:32.818 --> 00:22:45.743 Or, even those are people I've generally been curious about the field. What are the kinds of things that you will want to know more about? What questions can we answer? Please leave those in the comments thanks for watching and we'll see you in the next episode. 178 00:22:50.429 --> 00:22:53.818 Okay, so now. 179 00:22:53.818 --> 00:23:00.173 I'm getting a little tougher and more mathematical in the course. Okay so I mentioned last time. 180 00:23:00.324 --> 00:23:09.144 I mean, today the algorithm, those initials of the 3 authors to solve linear system, may X equals B and. 181 00:23:09.449 --> 00:23:17.909 Quantum terms, it's just if it's a very sparse system, then on a quantum computer, it's, it's exponentially faster than on the classical. 182 00:23:17.909 --> 00:23:22.378 Computer, so what I have here is an introduction in 6 minutes. 183 00:23:22.378 --> 00:23:28.108 And then a much longer thing, I'll probably show you about half of it or something and that. 184 00:23:28.108 --> 00:23:31.378 And, like, you watch the rest on your own. 185 00:23:32.638 --> 00:23:35.638 And we've got some pages, we can look at it so. 186 00:23:37.229 --> 00:23:43.979 Yeah, it's 36 minutes long and I'll show you half an hour, 20 minutes or something of that. 2nd 1. 187 00:23:48.058 --> 00:23:51.479 Let's continue with other very important. 188 00:23:51.479 --> 00:23:57.298 The hero hazard employed algorithm, which is also called matrix inversion. 189 00:23:57.298 --> 00:24:02.909 This protocol is used in many, many quantum machine learning algorithm. 190 00:24:02.909 --> 00:24:10.199 As a fundamental so it is really important that we understand the logic behind it, but it's tricky algorithm. 191 00:24:10.199 --> 00:24:14.759 So, 1st, let's go through a quick overview of how it works. 192 00:24:14.759 --> 00:24:18.898 So, the problem that we want to solve is given. 193 00:24:18.898 --> 00:24:24.148 A system of linear equations we want to and assuming that. 194 00:24:24.148 --> 00:24:31.769 The coefficient matrix is impossible, then we want to calculate the solution X as the universe of 8 times. 195 00:24:31.769 --> 00:24:39.719 And we would like to find a quantum algorithm for this. Classically this would take some pulling on your time. Probably. 196 00:24:39.719 --> 00:24:44.368 Cubic in time, or maybe a bit better and. 197 00:24:44.368 --> 00:24:51.719 There is, uh, this algorithm gives you an exponentially faster quantum protocol for performing something very similar. 198 00:24:51.719 --> 00:24:55.499 So 1st, we have to consider state preparation. 199 00:24:55.499 --> 00:25:01.078 So you have to prepare the be the state, the state, the vector. 200 00:25:01.078 --> 00:25:04.499 And this is in amplitude and coding. 201 00:25:06.719 --> 00:25:10.888 So, you might have to use a fairly involved circuit. 202 00:25:10.888 --> 00:25:15.929 To pick that, so that for the increases, your circuit them and then. 203 00:25:15.929 --> 00:25:21.179 Here would be corresponding to Tony and including. 204 00:25:23.788 --> 00:25:27.298 Says you can think of this. 205 00:25:27.298 --> 00:25:31.648 As a, as a habit, Antonio, so if your a is not permission. 206 00:25:31.648 --> 00:25:38.338 Then you have to make it permission by applying a simple transformation and then you would be able to include it as a hammered attorney and. 207 00:25:38.338 --> 00:25:44.818 And then eventually as a unitary, the next stage is to perform the quantum phase estimation. 208 00:25:44.818 --> 00:25:48.719 Because we are going to estimate the values of this, a operator. 209 00:25:48.719 --> 00:25:59.459 And by estimating diagonal values of this operator, what we are going to get is a very simple description of the structure of a. 210 00:25:59.459 --> 00:26:03.328 And that allows us to easily invert. 211 00:26:03.328 --> 00:26:08.608 The icon value, so of the quantum phase estimation. 212 00:26:08.608 --> 00:26:16.409 We have this state, roughly the States so it's some state and what we are going to do is we introduce and silver register. 213 00:26:16.409 --> 00:26:20.729 1 more and we applied a control rotation. 214 00:26:20.729 --> 00:26:25.679 On, uh, on the value estimates. 215 00:26:25.679 --> 00:26:31.798 I was going to give us is, this is now the answer register is additional 1 Cuban. 216 00:26:34.739 --> 00:26:38.368 For the time being, let's forget about what the rest of the state is here, because. 217 00:26:38.368 --> 00:26:41.459 This is what happens this is why the important thing that happens. 218 00:26:41.459 --> 00:26:46.138 So, by creating this condition are additional Dan. 219 00:26:46.138 --> 00:26:51.868 You can think code some costs them times the inverse of dragon value. 220 00:26:51.868 --> 00:26:59.068 In the amplitude of the 1 state of your ansolar register, and that's what we are going to use. 221 00:27:00.239 --> 00:27:04.259 But before we can use that, plus you have to compute. 222 00:27:04.259 --> 00:27:11.308 I can value register, which means that everything that we did for the final phase estimation we have to reverse. 223 00:27:11.308 --> 00:27:16.949 so this will be a four point of his estimation reverse in this circuit . 224 00:27:16.949 --> 00:27:23.278 And the reason we have to do that is because if you start doing any kind of measurement here. 225 00:27:23.278 --> 00:27:31.648 That these registers are entangled and if you remember the very beginning of this course, you know, that if you, if you just trace out or get rid of. 226 00:27:31.648 --> 00:27:38.999 A part of an entangled system, then you're going to end up with some kind of from next day you want to avoid. 227 00:27:38.999 --> 00:27:42.628 So, we have to pump computer everything in these registers. 228 00:27:42.628 --> 00:27:46.348 So we have this state, so this is just essentially. 229 00:27:46.348 --> 00:27:51.148 Itself expanded in it's in an item basis. 230 00:27:51.148 --> 00:27:56.368 And then this would be the vacuum state of the I can register. 231 00:27:56.368 --> 00:28:01.019 And then your answer lies on touched by this inverse operation. 232 00:28:01.019 --> 00:28:05.578 So, once we do it is, we can confident that your rejection sampling, which means that. 233 00:28:05.578 --> 00:28:10.739 Measure them, and if we get 0T, we discard everything and be restarted calculation. 234 00:28:10.739 --> 00:28:14.308 And if we get the 1, that means that now. 235 00:28:14.308 --> 00:28:24.929 Uh, whatever whatever I'll put you get is in probate proportional to the inverse of London and that's what we are going to use. 236 00:28:24.929 --> 00:28:29.548 To estimate, observable, or some measurable. 237 00:28:29.548 --> 00:28:36.118 Think why we needed the solution of the nurse system of equations. 238 00:28:36.118 --> 00:28:41.249 So, the resource requirements of this protocol are steep. 239 00:28:41.249 --> 00:28:46.288 So, if you watch, Roger matters guest lecture, he talked about shores. 240 00:28:46.288 --> 00:28:53.338 If you look at how short algorithm works, it's actually has a similar structure. It's a very similar start. The planning phase estimation. 241 00:28:53.338 --> 00:28:59.429 So, that gives us a lower amount of how much resource we need to to run is protocol. 242 00:28:59.429 --> 00:29:03.358 Software 48 bits encryption. 243 00:29:03.358 --> 00:29:07.318 You need 4000 logical habits. 244 00:29:07.318 --> 00:29:13.439 And that will take us in a range of millions of physical attributes on which we have to apply error correction. 245 00:29:13.439 --> 00:29:20.308 So, if we look at the number of Cubics, we have today on gateway, the computers, they are less than a 100. 246 00:29:20.308 --> 00:29:26.548 So, we are very, very far from being able to implement this algorithm in a practical manner. 247 00:29:26.548 --> 00:29:33.028 Well, this algorithm is definitely very important as we move forward and the better and better and computers. 248 00:29:33.028 --> 00:29:41.009 Silence. 249 00:29:41.009 --> 00:29:44.159 And this is the 2nd care. 250 00:29:44.159 --> 00:29:47.398 So, I didn't really understand it either, but, um. 251 00:29:47.398 --> 00:29:50.999 The thing was to give you a flavor of it. 252 00:29:50.999 --> 00:29:56.429 So, the next video. 253 00:30:00.179 --> 00:30:05.638 Is more relaxed and you'll be able to understand more. 254 00:30:07.858 --> 00:30:11.038 Okay. 255 00:30:18.509 --> 00:30:22.739 So. 256 00:30:22.739 --> 00:30:25.949 The intuition behind this problem. 257 00:30:25.949 --> 00:30:31.019 So solving linear equations, why might you think that quantum mechanics. 258 00:30:31.019 --> 00:30:34.858 So, a is no. 259 00:30:34.858 --> 00:30:40.318 There is no X is unknown. 260 00:30:40.318 --> 00:30:45.989 I won't say that there are variables. 261 00:30:45.989 --> 00:30:50.519 The motivation for solving this 1. well, this is a very common. 262 00:30:50.519 --> 00:30:53.878 Problem you want to find. 263 00:30:53.878 --> 00:30:57.538 Max is equal to a. 264 00:30:57.538 --> 00:31:03.868 Times B, this solving linear equations is something that happens a lot on its own. 265 00:31:03.868 --> 00:31:07.229 My daughters who are in in junior high school. 266 00:31:07.229 --> 00:31:18.509 Have to do this in their class, and they would definitely like to have a quantum computer for solving their linear equations. And it grows very frequently as a software team for. 267 00:31:18.509 --> 00:31:20.183 Some larger problems, 268 00:31:21.023 --> 00:31:24.114 and 1 thing that's happened recently, 269 00:31:24.114 --> 00:31:29.513 is that because with the huge amount of data that's being collected in the universe, 270 00:31:29.513 --> 00:31:29.723 or, 271 00:31:29.844 --> 00:31:31.794 at least on earth by human beings, 272 00:31:32.634 --> 00:31:33.023 this, 273 00:31:33.054 --> 00:31:37.433 this number and so the number of variables involved in such equations. 274 00:31:37.679 --> 00:31:42.479 Can get very, very large. It's perfectly reasonable. 275 00:31:42.479 --> 00:31:47.278 To have things where you have tend to the 9th. 276 00:31:47.278 --> 00:31:51.929 Variables or 10 of 12 variables. And the problem is that the best. 277 00:31:53.669 --> 00:31:58.558 The best classical algorithms. 278 00:32:01.019 --> 00:32:04.618 Brian, in time. 279 00:32:04.618 --> 00:32:11.278 Well, you're probably familiar with the 1 that you probably everybody here learned in school the 1 where, you know, you gradually. 280 00:32:11.278 --> 00:32:19.048 Eliminate variables along the way and then you bring down the number of variables each time and. 281 00:32:19.048 --> 00:32:25.648 There are you have and variables and it takes and square steps so you definitely it, it takes time. 282 00:32:25.648 --> 00:32:29.548 Order and squared were and could be. 283 00:32:29.548 --> 00:32:32.848 A big number so for instance. 284 00:32:32.848 --> 00:32:42.239 Nobody is going to invert a set of equations over 10 to the 12 variable soon, but it's perfectly reasonable to have systems where you have terabytes of data and might want to do that. 285 00:32:42.239 --> 00:32:47.818 If a is sparse, so this is the, this is called a. 286 00:32:47.818 --> 00:32:51.479 Kelsey and elimination. 287 00:32:51.479 --> 00:33:04.618 I mean, I didn't know that until I didn't know that until I started looking at this slide, but it's just, you know, just eliminate variables, express them in terms of the other variables, et cetera. 288 00:33:04.618 --> 00:33:08.159 And if a is sparse. 289 00:33:10.919 --> 00:33:15.808 So that means. 290 00:33:15.808 --> 00:33:18.838 No, more than. 291 00:33:18.838 --> 00:33:22.019 S, which is say, less as a lot smaller than an. 292 00:33:22.019 --> 00:33:27.419 Entries per. 293 00:33:27.419 --> 00:33:30.568 To them. 294 00:33:30.568 --> 00:33:35.999 It takes time order and log. 295 00:33:35.999 --> 00:33:41.999 Which is pretty good, actually, because, you know, there are and rose. 296 00:33:41.999 --> 00:33:50.848 And so, by the way, this kind of problem occurs all the time when, for instance, you have a version of some partial differential equation. 297 00:33:50.848 --> 00:33:56.098 Right then so each variable is only related to a few other variables. 298 00:33:56.098 --> 00:34:01.558 And this is this is done via what's called. 299 00:34:01.558 --> 00:34:09.568 Conjugate gradient. Excuse me? Gradient. 300 00:34:12.599 --> 00:34:23.039 Decide what all that means is a fancy way of saying, you try out a potential solution for X and you say oh, it's really not good. Okay. 301 00:34:23.039 --> 00:34:33.568 It's not very good and you say, well, let's look at solutions in the surrounding area. We'll take a set of solutions in the surrounding area. 302 00:34:33.568 --> 00:34:38.009 And we'll see which ones look better and you just move in the directions of the 1 that look better. 303 00:34:38.009 --> 00:34:44.489 And then you try take your new guess, and you try out a bunch of solutions in that area. And then again you move in the direction. 304 00:34:44.489 --> 00:34:53.068 That things are better, that's the gradient South Park. And the conjugate part simply means that you keep track of where you've been and you don't go backwards. 305 00:34:53.068 --> 00:34:56.278 So, because it's a linear problem. 306 00:34:56.278 --> 00:35:01.048 Actually, you're trying to minimize what you're trying to you minimize. 307 00:35:01.048 --> 00:35:04.469 A quadratic functional, which is, you know. 308 00:35:04.469 --> 00:35:07.768 A X minus. 309 00:35:07.768 --> 00:35:11.068 Squared because you're minimizing a quadratic functional. 310 00:35:11.068 --> 00:35:16.858 There is a unique minimum for this and so gradient set will actually get a good job. 311 00:35:16.858 --> 00:35:25.349 Okay, I mean, I'm not here to talk about the classical algorithms, but they are easy to describe and I just tell you what the. 312 00:35:25.349 --> 00:35:29.699 What the best what the best ones are right? 313 00:35:29.699 --> 00:35:38.128 Now, why might you think that quantum mechanics could do better? Well, they're really stupid. 314 00:35:38.128 --> 00:35:41.429 Because argument is, Gee. 315 00:35:41.429 --> 00:35:56.188 Let me, I going to use red for quantum mechanics of the blue as classical here. Red is for quantum mechanics. I actually, I, once I normally draw electrons as being blue, and then a famous. 316 00:35:56.188 --> 00:36:10.469 Italian physicist who has been blind from birth told me no, Seth you're wrong. Electrons are yellow. Hey. Yeah. It works for him. So. 317 00:36:11.548 --> 00:36:15.179 So, why am I quantum mechanics? Do better. 318 00:36:15.179 --> 00:36:22.679 Well, quantum mechanics. 319 00:36:22.679 --> 00:36:32.579 Might do better. They're really stupid. Intuition is whoa. Quantum mechanics. It's linear and Gee, these are linear equations. So hey, maybe we can solve them. 320 00:36:32.579 --> 00:36:42.568 Right. Where does it really stupid argument but it actually turns out to be correct? So so and I'll tell you how that works and the quantum mechanical version. 321 00:36:42.568 --> 00:36:47.909 As we have some Matrix, a, which is going to be a quantum mechanical operator. 322 00:36:47.909 --> 00:36:54.028 Now, and we're going to multiply it by and quantum mechanical vector. 323 00:36:54.028 --> 00:36:58.648 And is going to be equal to another quantum, mechanical vector. 324 00:36:58.648 --> 00:37:02.309 And we were going to assume that. 325 00:37:02.309 --> 00:37:11.699 Somebody gives us this B, and somebody gives us this a, in some suitable quantum, mechanical fashion and we want to find. 326 00:37:12.958 --> 00:37:19.588 X is equal to a inverse B or I'm going to say. 327 00:37:19.588 --> 00:37:24.088 Rather than find, I think we should really say construct. 328 00:37:24.088 --> 00:37:30.628 Because this is going to be a very physical process. We're going to start with inspector B and then. 329 00:37:30.628 --> 00:37:37.289 We're going to act on it by a dynamically process, which Institutes some, which is related to this a. 330 00:37:37.289 --> 00:37:47.878 For instance, if a is permission, then we're going to think of a, as being the Hamel Tony. 331 00:37:47.878 --> 00:37:51.208 For a system. 332 00:37:51.208 --> 00:37:55.739 Okay. 333 00:37:55.739 --> 00:38:01.978 Each of the minus, I, a T, acting on B. so that's the kind of thing we can do. 334 00:38:01.978 --> 00:38:06.809 All right, so we're going to actually construct this axe. 335 00:38:06.809 --> 00:38:11.099 And then this will give us our answer. So, the question. 336 00:38:11.099 --> 00:38:14.519 In the quantum mechanical version is, can we construct. 337 00:38:14.519 --> 00:38:17.789 Such an X, so. 338 00:38:17.789 --> 00:38:21.389 Let's just look at the. 339 00:38:21.389 --> 00:38:29.280 At the, at the accounting for this, how how how hard is is to do. 340 00:38:29.280 --> 00:38:35.010 Well, so if this is an, if this is an end dimensional problem. 341 00:38:35.010 --> 00:38:38.849 Then we need log the base 2. 342 00:38:38.849 --> 00:38:44.070 I'll call this little end is log the base 2 and Cubics. 343 00:38:44.070 --> 00:38:47.760 And similarly we need to roughly that for this. 344 00:38:47.760 --> 00:38:51.539 Actually, it, it doesn't have to be square here. 345 00:38:51.539 --> 00:38:56.010 For or even her mission, but we'll show you in a moment how to deal with that. 346 00:38:56.010 --> 00:38:59.760 So, okay, so we can actually encode the problem. 347 00:38:59.760 --> 00:39:03.840 In blog and space log. 348 00:39:03.840 --> 00:39:12.239 A is sparse, so we're going to look at the sparse case. I'll tell you about the non case later then. 349 00:39:12.239 --> 00:39:16.230 This takes order. 350 00:39:18.059 --> 00:39:21.480 Hopefully. 351 00:39:21.480 --> 00:39:25.170 Then this exponentiates takes order log. 352 00:39:25.170 --> 00:39:28.559 The base 2 events steps on a quantum computer. 353 00:39:28.559 --> 00:39:35.789 So, actually, we have, we really have a fighting chance here. I mean, I'd like to point out how. 354 00:39:35.789 --> 00:39:39.539 How the bizarre it is that we could, in fact. 355 00:39:39.539 --> 00:39:47.940 Solve this problem using only logarithmic resources right? I mean, this says if I have a terabyte of data. 356 00:39:47.940 --> 00:39:52.829 So that I could code everything that's so 10 to 12. 357 00:39:52.829 --> 00:39:57.659 By it, so, let's say a tear a bit of data, so I can calculate the numbers easily. So. 358 00:39:57.659 --> 00:40:02.849 That's 10 to the 12 bits because 10 to the. 359 00:40:02.849 --> 00:40:07.530 2 to the 10 this tend to the 3rd I need 40 Cubans. 360 00:40:07.530 --> 00:40:11.369 To encode this information and so. 361 00:40:11.369 --> 00:40:14.610 I can only need 40 and. 362 00:40:14.610 --> 00:40:18.570 Order, you know, a 100 or so steps. 363 00:40:18.570 --> 00:40:22.920 To invert test matrix over 10 to the 12. 364 00:40:22.920 --> 00:40:30.059 Variables so it's pretty shocking. And again, I think this is why people didn't think of this before. 365 00:40:30.059 --> 00:40:33.449 But I'm going to show you, I can do this. 366 00:40:33.449 --> 00:40:37.050 I got to be careful not to use up too much time here. 367 00:40:37.050 --> 00:40:40.829 This doesn't take that long. So, I mean, it's pretty simple algorithms. 368 00:40:40.829 --> 00:40:44.849 I should say that when we, when we publish this. 369 00:40:44.849 --> 00:40:53.010 It was, it was 1 every now and then, you know, you probably something that catches the attention of some community and. 370 00:40:53.010 --> 00:40:57.869 And this called the attention of slash doctor who, who, who knows what slash status. 371 00:40:57.869 --> 00:41:01.500 Who doesn't know who doesn't know what slashed out is. 372 00:41:01.500 --> 00:41:05.909 Oh, yeah, so slash dot is 1 of these, you know, Internet. 373 00:41:06.175 --> 00:41:17.815 Discussion farms it habit primarily by really annoying geeks. So I think that a lot of them are probably like, you know, write compilers for a living and stuff like that. 374 00:41:17.815 --> 00:41:25.795 So, they are very irreverent and they say mean things to each other, and they discuss things at great technical depth. It's a pretty funny. 375 00:41:26.010 --> 00:41:30.449 It's a pretty funny forum it's worth checking out anyway, so. 376 00:41:30.449 --> 00:41:38.219 When we posted this paper on the archive, slash dot thread, started up with people discussing it and somewhere. 377 00:41:38.219 --> 00:41:50.909 Down somewhere down, like, you know, person, number 14 or something. This, this is ridiculous. You can't possibly do this because just to read in the input. 378 00:41:50.909 --> 00:42:04.284 Takes time for our app and the next person on the thread and said, you moron you idiot if you read more than log and of the paper, then you were to find out how they do it. 379 00:42:08.369 --> 00:42:13.079 Of course, this brings up an issue, which is, you also can't read out. 380 00:42:13.079 --> 00:42:17.309 The output, right because it has and variables for what you can do. 381 00:42:17.309 --> 00:42:24.389 Is in the end, you can you can take some kind of expectation value of the following form. 382 00:42:24.389 --> 00:42:28.679 X where some operator, so you can figure out. 383 00:42:28.679 --> 00:42:35.219 You can't figure out the exact entries of X in time log in, but you can evaluate some. 384 00:42:35.219 --> 00:42:39.090 You know, some expectations on your global properties. 385 00:42:39.090 --> 00:42:42.239 Of this of this operator. 386 00:42:42.239 --> 00:42:45.960 Okay. 387 00:42:45.960 --> 00:42:49.079 So. 388 00:42:49.079 --> 00:42:58.260 That's how the this operation works. Let me just switch to green to show you what's going on here. So, this is. 389 00:42:58.260 --> 00:43:04.349 And coding step so this is right here. 390 00:43:04.349 --> 00:43:11.610 Yes, this is our goal and this is the kind of thing we're going to do. 391 00:43:11.610 --> 00:43:19.739 In order to find the answer, and this is evaluate the answer. 392 00:43:19.739 --> 00:43:25.230 So these are the steps. 393 00:43:25.230 --> 00:43:28.469 Okay, everybody happy. 394 00:43:28.469 --> 00:43:33.900 People questions before I tell you how this works, anybody want to object. 395 00:43:33.900 --> 00:43:39.360 Strongly or weekly, or start their own slash dot thread. 396 00:43:39.360 --> 00:43:42.840 Okay. 397 00:43:42.840 --> 00:43:47.639 All right, so all right, then I'll just tell you how it works so the 1st step. 398 00:43:47.639 --> 00:43:51.449 We're going to assume. 399 00:43:51.449 --> 00:44:05.550 A is equal to a dagger this actually this says, oh, you're cheating. You're only using Square permission matrices, but it's very easy to. 400 00:44:05.550 --> 00:44:08.760 To solve the problem in the following context. 401 00:44:08.760 --> 00:44:19.289 0T a, a dagger 0T acting on the 0T X is equal to the 0. 402 00:44:19.289 --> 00:44:23.159 Right. And then and then actually, I can. 403 00:44:23.159 --> 00:44:29.190 I can if I solve this, this implies that a X is equal to be. 404 00:44:29.190 --> 00:44:33.630 For today for any. 405 00:44:33.630 --> 00:44:44.309 Including a is that are not square they could have, you know, there could be more. There could be more. The problem could be either over constrained or under constrained. 406 00:44:44.309 --> 00:44:51.119 Okay, so everybody happy with that. So now I'm going to assume the day is. 407 00:44:51.119 --> 00:44:57.449 Is permission all right a sparse. 408 00:44:58.650 --> 00:45:06.420 Much less than entries per per row. 409 00:45:11.880 --> 00:45:24.690 We can this implies that again, as I mentioned there that we can perform, we're given B, we can do each of the minus a T, acting on B. and this now becomes essentially just the simulating. 410 00:45:24.690 --> 00:45:32.250 The action of this local camel, Tony, this means that a equals a dagger is local. 411 00:45:33.900 --> 00:45:36.900 So, it corresponds to a Hamel, Tony and where. 412 00:45:36.900 --> 00:45:46.079 Things are only where the variables are only only interacting with local variables and there's, it's been known for quite a while. Now. 413 00:45:46.079 --> 00:45:57.269 That if you, if there's a well, actually, the 1st is a local Hamel telling him, I prove many years ago. Now, like, in 1996 that you can do this time log in. 414 00:45:57.269 --> 00:46:03.090 And it's been known that for maybe 8 or 10 years, sort of as a sparse. 415 00:46:03.090 --> 00:46:06.420 Then you can do this for any sparse day. 416 00:46:09.420 --> 00:46:13.110 As I mentioned over here okay. 417 00:46:13.110 --> 00:46:19.500 So, now, let's come up with we come up with a a trek. 418 00:46:19.500 --> 00:46:24.659 So, 1 way to suppose that. 419 00:46:24.659 --> 00:46:29.639 Uh, we knew supposed to how to diagonal as a. 420 00:46:29.639 --> 00:46:33.750 If you know how to analyze the matrix, there's a very easy way. 421 00:46:33.750 --> 00:46:43.559 To invert the matrix because once, you know, the icon values, the inverse matrix is just the 1 where the values are inverted. So if we could. 422 00:46:43.559 --> 00:46:47.550 To idolize. 423 00:46:47.550 --> 00:46:50.730 Exactly. 424 00:46:50.730 --> 00:46:55.019 Okay, so a is equal to. 425 00:46:55.019 --> 00:47:00.329 So, what's the rate of the following way? You a U dagger. 426 00:47:01.739 --> 00:47:05.250 Is equal to land on 1. 427 00:47:05.250 --> 00:47:09.630 Up to Lambda capital n0 0. 428 00:47:09.630 --> 00:47:13.769 Then this implies that a inverse. 429 00:47:13.769 --> 00:47:21.389 Is equal to you dagger Landa 1 inverse up to plan to an inverse. 430 00:47:21.389 --> 00:47:26.010 0T as you can easily see by. 431 00:47:26.010 --> 00:47:40.284 Multiplying these 2 a, and a inverse together and this fits in this diagonal form. I'm sorry if you can't see them in the back anyway, you can't see the back. You just get the values, you invert them and that gives you the inverse matrix now. 432 00:47:40.284 --> 00:47:41.094 Classically. 433 00:47:41.309 --> 00:47:51.239 This would be, you know, this is certainly 1 way to solve the problem, but it's not a very good way to solve the problem classically because classically. 434 00:47:51.239 --> 00:47:55.800 You know, I think this is actually an end cube, not square. Sorry? 435 00:47:55.800 --> 00:48:00.539 Classically to invert the matrix basically. 436 00:48:00.539 --> 00:48:14.010 The way that I remember from high school about how grand bird matrices is doing calcium elimination. So it started to find the to the matrix is as hard as doing this calcium elimination. 437 00:48:14.010 --> 00:48:20.010 So this doesn't help you classically, but now quantum mechanically. 438 00:48:20.010 --> 00:48:26.550 We actually have an algorithm for doing this the so called. 439 00:48:26.550 --> 00:48:33.389 Thing is algorithm. 440 00:48:33.715 --> 00:48:41.724 This was invented by a tie in around 90, 95 or 90, 96, forgive alternative way of solving shore's algorithm. 441 00:48:41.724 --> 00:48:53.184 But my graduate student, Dan Abrams, and I showed that this is an algorithm that allows you to find the values. This is what. 442 00:48:54.179 --> 00:49:03.030 Time use it for, but also very importantly, the icon vectors of. 443 00:49:03.030 --> 00:49:15.900 Okay, it takes the ability to exponentially in exponentially the matrix and find it valuable and I can vectors and I actually claim. 444 00:49:15.900 --> 00:49:23.610 That this has been known for a very long time, much longer than quantum information because all it really is. 445 00:49:23.610 --> 00:49:28.079 Is fungi minds, quantum, mechanical model for measurements. 446 00:49:28.079 --> 00:49:33.869 Right because when you make a measurement, what you do is, you, you know, have a measurement, takes the system. 447 00:49:33.869 --> 00:49:41.849 You make an operation of measurement that corresponds to a permission operator a, and what you get is 1 register. 448 00:49:41.849 --> 00:49:44.940 You get 1 register that gives you the. 449 00:49:44.940 --> 00:49:50.730 Value of a, and then in the in the system is left in the state. 450 00:49:50.730 --> 00:49:56.670 Right. And so actually the way that this works, the way that fungi mine described it. 451 00:49:56.670 --> 00:50:00.269 Let me just remind you as that here at the state. 452 00:50:00.269 --> 00:50:11.159 Fee and sorry B and B is going to be equal to the sum over. We don't know what these and values are. Let's just call them. 453 00:50:11.159 --> 00:50:14.190 Size, so, Jay, these are the eye conductors. 454 00:50:14.190 --> 00:50:19.409 Where this is the icon vector associated with the icon value Landa. 455 00:50:19.409 --> 00:50:24.000 So, Jay, and you had you take a. 456 00:50:24.000 --> 00:50:31.739 Another register, which basically you can think of as a particle or a point of variable, which is we're visually localized at. 457 00:50:31.739 --> 00:50:36.840 From the point 0T and then you apply the following operator. 458 00:50:36.840 --> 00:50:42.659 So, the, we apply a, and we Tensor the momentum operator. 459 00:50:42.659 --> 00:50:51.329 For this this point or variable, and what the momentum operator does, the momentum takes position and it moves it. 460 00:50:51.329 --> 00:51:03.360 Right. This is true. Classically it's also to your quantum mechanically and it moves at buying them out that's proportional to the value of a. so this is actually equal to the sum. 461 00:51:03.360 --> 00:51:08.190 Over this Jay beta sub J side Jay. 462 00:51:08.190 --> 00:51:12.570 And then we have basically some land to Jay times chief. 463 00:51:14.460 --> 00:51:28.110 This is on this is in, like, I forget which page on page? 233 of 5, 9 months, 1930 book on the foundations of quantum mechanics. And that's really all this quantum phase algorithm is. 464 00:51:28.110 --> 00:51:34.440 Except that there is the quantum phase algorithm tells you explicitly how to do this once you're given. 465 00:51:34.440 --> 00:51:39.059 Those are people okay about this. This is some. 466 00:51:39.059 --> 00:51:44.190 This is very antique quantum mechanics in some sense, but a. 467 00:51:44.190 --> 00:51:48.210 Uh, I'm just reminding you that you can do this quantum mechanical. 468 00:51:48.210 --> 00:51:55.289 Okay, and I'll call this or call this use of 5. this is a quantum phase. 469 00:51:55.289 --> 00:52:00.360 Algorithm okay. Are people right? This is kind of the key. 470 00:52:00.360 --> 00:52:05.070 Staff, actually there are several key steps, but this is a key step. 471 00:52:05.070 --> 00:52:08.699 Sorry for everybody. Okay anybody not. Okay. 472 00:52:08.699 --> 00:52:16.710 All right, so now, once you have this, you see, of course, you have to be you have to be careful because. 473 00:52:16.710 --> 00:52:19.800 All along the way. 474 00:52:19.800 --> 00:52:27.960 You have to be careful because you've been coded things into a finite number of Cubics. You don't get this. Exactly. You only get it to a particular degree. 475 00:52:27.960 --> 00:52:32.159 Of accuracy, but I'm going to ignore all that stuff now. 476 00:52:32.159 --> 00:52:36.539 And at the end, I'll tell you how the, all the accuracies work out. 477 00:52:36.539 --> 00:52:41.039 But now you see, we're in a really good position. 478 00:52:41.039 --> 00:52:45.630 To do this inversion because we actually have. 479 00:52:45.630 --> 00:52:54.750 This vector sitting here, so we have an estimate here and so in quantum parallel, we can now do the following step. 480 00:52:54.750 --> 00:52:59.280 We can multiply this times a phase. 481 00:52:59.280 --> 00:53:05.730 So, I'll call 2 and I'll put a little delta in here. It's some small number. 482 00:53:05.730 --> 00:53:09.449 I sure could even be this cheap for that are all those today that as a team. 483 00:53:09.449 --> 00:53:14.489 Could be the same team? No, no I better make it a don't sorry. 484 00:53:14.489 --> 00:53:21.989 My quantum computer, it looks what's in this register says a ha now I know the, I can value. 485 00:53:21.989 --> 00:53:27.090 So, let me turn it into a face because you can always turn the face. You invert it. 486 00:53:27.090 --> 00:53:31.139 And we say, or what's 1 over the icon you turn it into a face. 487 00:53:35.219 --> 00:53:39.179 Bye. 488 00:53:40.199 --> 00:53:47.730 This is, of course, the key stuff, because actually, once you realize you can do this, you'll see the rest of it follows almost directly. 489 00:53:48.869 --> 00:53:57.690 And now, let me go, I'm sorry to kind of wander around like this. This is maybe bad blackboard technique. Now I undo. 490 00:53:57.690 --> 00:54:02.099 The phase algorithm just kind of measurement. 491 00:54:02.099 --> 00:54:05.940 Algorithm and I get some of over Jay. 492 00:54:05.940 --> 00:54:11.730 Need to the delta Glam to inverse. J. 493 00:54:11.730 --> 00:54:26.039 j0T, but this now is equal to. 494 00:54:26.039 --> 00:54:30.179 To the delta a inverse. 495 00:54:31.349 --> 00:54:38.639 Acting on the 0. 496 00:54:41.909 --> 00:54:48.360 So, now we're almost there because the only step last is if we make Delta small enough. 497 00:54:48.360 --> 00:54:52.409 This is equal to 1, approximately equal to 1. 498 00:54:52.409 --> 00:54:56.340 Plus, I Delta a inverse. 499 00:54:57.960 --> 00:55:05.730 Acting on B was 0T, but you can, it's actually turns out to be not very hard to. 500 00:55:05.730 --> 00:55:08.730 Extracts just this part. 501 00:55:08.730 --> 00:55:16.409 So some Delta is small, but some reasonable fraction of the time you're actually going to get. 502 00:55:16.409 --> 00:55:23.579 Your answer and so the net result, and we get. 503 00:55:23.579 --> 00:55:26.940 A inverse acting on B. 504 00:55:26.940 --> 00:55:31.980 And the number of in a number of steps. 505 00:55:33.329 --> 00:55:38.039 That goes as let me see if I can get this. Right? 506 00:55:38.039 --> 00:55:44.219 I think it goes as s squared. That's the sparseness. 507 00:55:44.219 --> 00:55:51.449 Divided by Epsilon, so we get it to accuracy. Epsilon. Excuse me standing of the way. 508 00:55:51.449 --> 00:55:55.139 I know it's bad. 509 00:55:55.139 --> 00:55:58.380 That board technique and time. 510 00:55:58.380 --> 00:56:01.769 Order squared over on. 511 00:56:01.769 --> 00:56:09.480 Logs the best of app. Oh. And then actually, there is another number here. 512 00:56:09.480 --> 00:56:15.929 Which is a campus where I go to this number Kappa. 513 00:56:15.929 --> 00:56:22.289 Is equal to it's called the condition number. It's equal to Lambda. Max overlap the map. 514 00:56:22.289 --> 00:56:27.840 So, you see if some of these values are really small. 515 00:56:27.840 --> 00:56:31.079 Then the inversion becomes tricky. 516 00:56:31.079 --> 00:56:34.559 If 1 of them is 0T, for instance, then. 517 00:56:34.559 --> 00:56:42.869 You can have trouble, so it's going to take longer because the small icon values, you need to take longer to find them and then you have to be careful. 518 00:56:42.869 --> 00:56:51.989 When you invert, so the condition number of a matrix, and if everybody all the classical people who solve these problems, the condition number of the matrix is 1 of the key. 519 00:56:51.989 --> 00:56:55.530 Questions here for how how long it actually takes. 520 00:56:55.855 --> 00:57:10.494 To do this problem, I should mention that even if some of the lamb does are 0T. So the problem is not solvable. That's actually still. Okay, because at this step right here you can have your quantum computer. 521 00:57:10.739 --> 00:57:18.210 Associate say, oh, this land is 0T. I'm not going to invert it and it raises a flag saying danger danger. 522 00:57:18.210 --> 00:57:26.789 Small land does and then so what you can actually do is you can even do better. You can get a inverse together with. 523 00:57:26.789 --> 00:57:31.619 Go on the part that is in vertebral together with information about. 524 00:57:31.619 --> 00:57:36.090 The part that's not vertical so it actually, it's not really a bug. It's a feature. 525 00:57:36.090 --> 00:57:48.929 Love this and let me see the best classical algorithm. Let me see the best classical algorithm over here. I seem to recall. 526 00:57:48.929 --> 00:57:52.559 It takes time. 527 00:57:52.559 --> 00:57:56.579 I'll give you the full for this order. 528 00:57:56.579 --> 00:58:01.349 Tab. 529 00:58:01.349 --> 00:58:04.440 Squared s. 530 00:58:04.440 --> 00:58:09.300 Maybe just Kappa asks, I don't remember exactly over log. 531 00:58:09.300 --> 00:58:16.920 Of Epsilon, so minus log of Epsilon times and. 532 00:58:16.920 --> 00:58:22.349 Can I please. 533 00:58:22.349 --> 00:58:27.989 Can you show the. 534 00:58:27.989 --> 00:58:32.070 Welcome to talks about. Can you. 535 00:58:33.570 --> 00:58:40.500 This so this, these steps to get this. 536 00:58:40.500 --> 00:58:45.420 Yeah, yeah, so so. 537 00:58:46.405 --> 00:59:00.534 I can, but, you know, I can, but I don't have time so it's this quantum phase algorithm. It's a very standard algorithm. If you look at Nielsen strong, it's there, it just simply involves doing conditional for transforms. 538 00:59:02.880 --> 00:59:10.050 But all it really is is, you can think of it is another way of thinking of it is doing just quantum simulation. 539 00:59:10.050 --> 00:59:14.849 Of this Hamilton, which I think is maybe even easier. 540 00:59:14.849 --> 00:59:21.090 Perfect. Okay, so you see, I mean, there are some issues here. 541 00:59:21.090 --> 00:59:34.045 From there are unimportant 1, is that you can't read out every entry to the answer. So that's a restriction. I mean, of course, you can't because it will take time order and read out every entry. 542 00:59:34.735 --> 00:59:37.704 If you compare these things, the, the. 543 00:59:40.554 --> 00:59:52.704 The dependence on condition number as far since those are other similar. I don't even know if I've gotten the right 1 over there. I'd have to think about it for just a 2nd, since I gave you a set of options, because I didn't actually go and prepare the talk individually. 544 00:59:52.974 --> 00:59:59.155 I don't remember off hand the exact dependents there, but I do remember. So, here we have a 1 over Epsilon. 545 00:59:59.460 --> 01:00:04.260 And then we have a 1 of our log Epsilon. So the classical 1 is. 546 01:00:04.260 --> 01:00:11.789 Is has a better dependence on the error on the other hand the most important thing. And the big thing is, this is and. 547 01:00:11.789 --> 01:00:19.469 Emphasis log app and in fact, it says log in, it would actually be quite bizarre. It would be very strange if you had a better. 548 01:00:19.469 --> 01:00:23.460 A better dependence on the air and in fact. 549 01:00:23.460 --> 01:00:28.349 You can show that since you can actually you can rephrase computational problems. 550 01:00:28.349 --> 01:00:31.349 In this way, you would actually show that if you actually had. 551 01:00:31.349 --> 01:00:36.840 They log Epsilon down here, then you'd be able to solve keep complete problems and. 552 01:00:36.840 --> 01:00:40.440 Time log in on a, on a quantum computer. 553 01:00:40.440 --> 01:00:46.860 And that seems unlikely. So do you. 554 01:00:46.860 --> 01:00:57.900 Right. So so right here right here you see that you have some amplitude of finding a, and verse B and so you just repeat the process until you get. 555 01:00:57.900 --> 01:01:02.579 You know, this is actually B, plus adverse pay. 556 01:01:03.780 --> 01:01:15.150 Yeah, that's right there are all kinds of sneaky things. If you look at the longer version of our paper disappeared and physical Review letters a couple of years ago, we did a longer version of the paper. 557 01:01:15.505 --> 01:01:23.994 On that there are all kinds of sneaky things that you can do along the way in order to get this kind of this dependence that we actually get. 558 01:01:24.295 --> 01:01:34.195 But the short answer is, yeah, you just, you know, this has some probably a probably ordered Delta square to find this. You can actually turn that if your probability of order, Delta, vice Nikki tricks. 559 01:01:35.304 --> 01:01:39.804 But you but the point is, a Delta is only goes as. 560 01:01:40.050 --> 01:01:45.210 Log at so you're actually so you're okay. 561 01:01:45.210 --> 01:01:53.789 Well, you have to check carefully that all these steps work. I mean, you're trying to fit something really huge into our little little, little, tiny box. 562 01:01:53.789 --> 01:02:02.400 And and so it's got to be folded up can really treat sneaky way to get it in there. So you have to check each stage along the way careful. 563 01:02:02.400 --> 01:02:06.570 What's the, what's the experimental situations. 564 01:02:06.570 --> 01:02:14.880 Yeah, so actually, so together with Dave, Corey, and 1 of his graduate students, we, we looked about at a. 565 01:02:14.880 --> 01:02:20.849 Implementing this on an Amar quantum computer, but then this kind of fell apart when he moved up to a. 566 01:02:20.849 --> 01:02:26.340 To Waterloo, because the graduate students stayed at MIT, doing something else. So we came up. 567 01:02:26.340 --> 01:02:30.119 You actually need. 568 01:02:30.119 --> 01:02:35.159 So, you have this be register. 569 01:02:35.159 --> 01:02:41.250 You have the register which stores the, the States. 570 01:02:41.250 --> 01:02:47.760 To some degree of accuracy and then actually it turns out that you need you need to actually need another register here. 571 01:02:47.760 --> 01:02:54.869 To tell you when you've got the answer, so you end up doing something that gives you, you have another letter that says, hey, did we get the answer or not? 572 01:02:54.869 --> 01:03:00.000 So we figured we could invert. 573 01:03:00.000 --> 01:03:08.969 8 by 8 matrices where 3 qbr tier 3 to vocera plus 7 cubax you put in Jordan a, by Matrix. 574 01:03:08.969 --> 01:03:19.710 We're just something hurting 8 by matrices is something I don't like to do by hand. So we figured that was pretty good. If you wanted to do a 4 by 4, you could have 2 with 5. 575 01:03:19.710 --> 01:03:24.599 You can adapt and by the the. 576 01:03:24.599 --> 01:03:30.480 Yeah, yeah, and then tomorrow, I mean, now they're, they're like. 577 01:03:30.480 --> 01:03:42.809 12 to 15 cubic molecules. So, I mean, you know, there are technical issues that make some of the partner and some of them easier. So you could easily use easily could not easily. I should say that's a fearless talking. You could readily use. 578 01:03:42.809 --> 01:03:56.155 Room temperature and Omar to perform this algorithm and there's some group in China that's trying to do it right now, though, from my correspondence with them. I'm not actually hopeful that they really understand what they're doing. 579 01:03:59.094 --> 01:04:01.344 Some of the things they say are, like, whoa. 580 01:04:04.380 --> 01:04:12.750 Yeah, so I mean, it's nice. It's well, I should say minimum number can work, but. 581 01:04:12.750 --> 01:04:17.969 2, 4 by 4, 4 by 5th of 5 QB 4 by 4. 582 01:04:17.969 --> 01:04:23.280 I think that's that's the minimum number. That's I think well, that's what they're trying to do in China. 583 01:04:23.280 --> 01:04:36.809 But, yeah, so and they can do, and I think they're not really at that stage yet. Actually both ion traps and superconducting to pets. I mean, it traps. 584 01:04:36.809 --> 01:04:42.869 They have really have a limited ability right now to address individual. 585 01:04:42.869 --> 01:04:50.880 So, I mean, people can do it. There are, there are experiments that do it, but I think 5 Cubans with individual cubic addressability. 586 01:04:50.880 --> 01:05:01.199 For iron traps would be tough and the same was super good. I think they're, they're trying, you know, they've done 2 pubic algorithms, but they're not doing 5 Cuban algorithms yet. 587 01:05:01.199 --> 01:05:07.409 So certainly the easiest way to do this, even though it would require a bit of effort would be at room temperature and Amar. 588 01:05:07.409 --> 01:05:14.219 But I'm, I'm hoping that the trap folks and the Super electing folks may soon be able to do. 589 01:05:14.219 --> 01:05:17.880 Uh, 5, 5, Cupid algorithm it'd be a good test for them. 590 01:05:19.349 --> 01:05:24.630 Okay, so as I told you, it was, it's actually, you know, it's not. 591 01:05:24.630 --> 01:05:28.920 Not that it's not that complicated an algorithm and. 592 01:05:28.920 --> 01:05:32.010 I mean, it relies on kind of tried and true. 593 01:05:32.010 --> 01:05:35.550 Techniques together with 1 sneaky trick. 594 01:05:35.550 --> 01:05:48.179 Right and the overall sneaky trick is just, I think, as I said, the main issue reason that people didn't discover this before is because it's kind of unbelievable. 595 01:05:48.179 --> 01:05:55.289 Right. It's kind of unbelievable. I mean, it seems like way too much that you should be able to do this. And in fact. 596 01:05:55.289 --> 01:05:59.369 When i1st started working on this, that's the suggestion. It was it the suggestion of a. 597 01:05:59.369 --> 01:06:03.510 Classical computational complexity guy said, hey, why don't you. 598 01:06:03.510 --> 01:06:07.289 Try to solve some of these linear equations because they actually, you know. 599 01:06:07.289 --> 01:06:11.070 They take a long time and they're often the sticking point from any algorithms. 600 01:06:11.070 --> 01:06:15.420 And I thought, oh, well, maybe we'll get you like a square root of and speed up or something like that. 601 01:06:15.420 --> 01:06:18.630 But as it turned out, it wasn't log, it was like. 602 01:06:18.630 --> 01:06:23.340 Exponential speed up. 603 01:06:23.340 --> 01:06:26.699 Okay, more questions about topic number 1. 604 01:06:26.699 --> 01:06:32.940 I still pretty well, I think it's 40. it's a little more than I started 45 minutes ago. 605 01:06:32.940 --> 01:06:36.539 Silence. 606 01:06:36.539 --> 01:06:42.840 All right, so let's vote again on topic number 2. 607 01:06:42.840 --> 01:06:46.590 Okay, so I decide to play the whole thing because it was interesting, but. 608 01:06:46.590 --> 01:06:51.750 It's fun to see this. Okay. 609 01:06:54.150 --> 01:06:57.599 Where is. 610 01:07:01.619 --> 01:07:09.239 So that was this again to the page. 611 01:07:09.239 --> 01:07:15.630 I had this other thing here. It's something called a quantum summary. Symposium has stuff. 612 01:07:15.630 --> 01:07:22.320 You watch it if you wish are tactical and again, I showed you this last time. I. 613 01:07:23.340 --> 01:07:28.199 Guess as examples and stuff. 614 01:07:28.199 --> 01:07:34.199 And might even perhaps look at that Thursday attach perhaps. 615 01:07:35.639 --> 01:07:45.000 And also on the wall away page, so you got lots of different ways. You can help you understand that if you would like. 616 01:07:45.000 --> 01:07:48.119 Wikipedia pages. 617 01:07:49.230 --> 01:07:55.860 It's very long and detailed, but if you want. 618 01:07:55.860 --> 01:08:01.050 Can browse through this and so between 1, another if you wish you could learn. 619 01:08:01.050 --> 01:08:08.070 A lot of information about solving the sparser near systems, quantum computer. So. 620 01:08:09.960 --> 01:08:13.949 Okay. 621 01:08:17.310 --> 01:08:23.819 Um, it's getting late now. Let me just introduce you to Microsoft so they're also a player in this. 622 01:08:23.819 --> 01:08:30.989 And they have a lot of stuff. I joking, it's low signal to noise ratio, but. 623 01:08:30.989 --> 01:08:35.130 This is our homework question so. 624 01:08:35.130 --> 01:08:39.840 Again, no words, not the most technical magazine, but it has some nice summaries. 625 01:08:39.840 --> 01:08:47.189 So so they're talking about the provide cloud computing services on. 626 01:08:47.189 --> 01:08:57.239 Various some different quantum computers, so they're not building hardware. They're providing access to 3 competing types of hardware. So they would see, like, hey, would have. 627 01:08:57.239 --> 01:09:03.210 Develop software and provide a service and so you so you cannot. 628 01:09:03.210 --> 01:09:06.720 A cloud provider and so on. So. 629 01:09:08.340 --> 01:09:12.630 They don't do their own hardware doing the software. Well, that's microsoft's. 630 01:09:12.630 --> 01:09:17.970 Or most of the time, so you can check that. 631 01:09:17.970 --> 01:09:26.189 Um, Microsoft has this Gateway page here and a lot of stuff. 632 01:09:27.720 --> 01:09:31.649 Can browse through they have an API, they have something. 633 01:09:33.149 --> 01:09:38.399 A development kit and stuff like that and. 634 01:09:39.420 --> 01:09:45.300 I think their description here send out the descriptions I showed you earlier. I'd like better, but they have a lot of stops there. 635 01:09:47.399 --> 01:09:51.329 A blog you can look at. 636 01:09:56.399 --> 01:10:03.300 Kind of developer and so on. 637 01:10:05.850 --> 01:10:09.029 Yeah, they got sharp, I guess. 638 01:10:09.029 --> 01:10:17.010 I prefer C plus plus C sharp and if you want more information, they had a workshop here. 639 01:10:17.010 --> 01:10:23.819 Which is 5 hours long you're ready to start. The recording is 5 hours long. It's. 640 01:10:23.819 --> 01:10:26.880 Um, it's a full day workshops, so you can. 641 01:10:26.880 --> 01:10:30.149 Register watch it on your registration is free. 642 01:10:32.399 --> 01:10:36.359 I seen it. 643 01:10:38.520 --> 01:10:42.989 From last year. 644 01:10:42.989 --> 01:10:48.750 So, you can get a nice summary of what's going on here. 645 01:10:52.050 --> 01:10:59.069 And I think I didn't think I think this before. 646 01:10:59.069 --> 01:11:06.000 Okay, so piles and piles of documentation that companies spending some resources to make it easy for people to. 647 01:11:06.000 --> 01:11:09.869 You know, do quantum computing with them learning modules. 648 01:11:09.869 --> 01:11:15.779 References and stuff like that and different ways to describe things. So, in any case. 649 01:11:17.609 --> 01:11:22.859 So, tutorials documentation and their development. 650 01:11:22.859 --> 01:11:28.079 Libraries for different things, so you can have, um. 651 01:11:28.079 --> 01:11:32.039 Fun playing quantum computing with Microsoft if you like. 652 01:11:33.090 --> 01:11:36.510 Machine learning I should talk about at some point. 653 01:11:38.850 --> 01:11:48.960 Yeah oh, okay. Do talk about Google next time perhaps. 654 01:11:48.960 --> 01:11:52.109 You can read ahead if you would like. 655 01:11:53.250 --> 01:12:00.750 And I'll look a little at some of these other companies and may circle back at some point and get into some. 656 01:12:01.949 --> 01:12:06.180 Some of these algorithms in greater depth. Perhaps if I can figure them out. 657 01:12:06.180 --> 01:12:10.079 Oh, it's just review what I showed you today some. 658 01:12:10.079 --> 01:12:18.060 On general stuff about quantum computing, that's not technically all that deep, but it can be interesting. Current. I thought the current. 659 01:12:18.060 --> 01:12:23.310 2020 update was quite nice to historical timeline and wired. 660 01:12:23.310 --> 01:12:28.170 And a blog of a lot of quantum software projects, how many there are. 661 01:12:28.170 --> 01:12:31.560 Engineer should Abraham at IBM. 662 01:12:31.560 --> 01:12:35.609 And the algorithm. 663 01:12:35.609 --> 01:12:39.960 For, and since it's a very deep algorithm. 664 01:12:39.960 --> 01:12:43.949 I hit you various ways as a quick deep in trials and this longer. 665 01:12:43.949 --> 01:12:49.890 Thing which I didn't understand the details either, but it gives you a feeling for it. 666 01:12:49.890 --> 01:12:54.930 And, but lots of other sources to help, you. 667 01:12:54.930 --> 01:12:58.529 So these are as easy as anything it is a typical algorithm, I guess. 668 01:12:58.529 --> 01:13:01.829 And Microsoft another 1 of the company. So Microsoft. 669 01:13:01.829 --> 01:13:07.229 They're not trying to do quantum hardware. They're trying to be ambitions of providing a service. 670 01:13:07.229 --> 01:13:15.960 So this is wants to provide a hardware and a service. Microsoft is providing just a service. 671 01:13:15.960 --> 01:13:20.550 And there's all these other small companies starting up. So we'll see who wins. 672 01:13:20.550 --> 01:13:25.529 So that was enough stuff for today. 673 01:13:25.529 --> 01:13:29.460 I'll stay around to see if there's any questions other than that. 674 01:13:29.460 --> 01:13:33.149 Have a good week and see you Thursday. 675 01:13:39.510 --> 01:13:43.680 You're welcome, Amanda. 676 01:13:47.789 --> 01:13:52.770 Silence. 677 01:13:56.699 --> 01:14:02.279 Silence. 678 01:14:08.220 --> 01:14:11.430 Silence. 679 01:14:40.739 --> 01:14:44.729 Silence. 680 01:14:44.729 --> 01:14:48.479 Silence.