WEBVTT 1 00:00:51.295 --> 00:00:55.524 Okay, this is why I have two laptops in front of me. My other laptop. 2 00:02:28.139 --> 00:02:29.275 This is crazy. 3 00:02:29.694 --> 00:03:03.354 Okay. 4 00:03:12.569 --> 00:03:22.164 Cool, what I did is I switched to my laptop because the first laptop for some reason Tom asked me why. 5 00:03:23.094 --> 00:03:29.935 And so finally. 6 00:03:34.020 --> 00:03:40.645 We have a blog. Okay. 7 00:03:40.949 --> 00:03:50.844 So if people can see this, I can see this up here. 8 00:03:52.435 --> 00:04:03.324 Last seven. Oh, good. And just to make certain that people can still hear me. 9 00:04:04.860 --> 00:04:08.814 We can see the browser and the webcam. Thanks. 10 00:04:10.884 --> 00:04:20.214 And me, can you hear me now? 11 00:04:21.834 --> 00:04:25.824 Well, if you can, we can't let me know. Okay. 12 00:04:26.759 --> 00:04:37.800 Thank you. Cool. So, what be happening today is I put homework. Three up homework. 13 00:04:37.795 --> 00:04:52.314 Three will be in two weeks, not one week and that will give you a chance for you to present an idea. I will take two days for. It will be the whole week. 14 00:04:52.855 --> 00:05:01.615 And so you can do it in teams of up to three people, pick some idea relating to quantum computing and presented in class for a few minutes. 15 00:05:01.615 --> 00:05:16.495 If it's one person, six or seven minutes, two people, two or three people, ten to twelve minutes. And I'll have to be strict about the time because we want to get, you know, we want to class to end on time. 16 00:05:17.035 --> 00:05:17.785 So that. 17 00:05:19.165 --> 00:05:33.654 So, you can form teams if you like, I recommend using teammates and so, and I will set up an online form for people to pick today and to pick topics and it will be first come first serve. 18 00:05:34.795 --> 00:05:41.310 A good example of a big list of topics is in the appendix E textbook. So. 19 00:05:47.790 --> 00:05:49.644 Forward to appendix. 20 00:06:07.920 --> 00:06:08.610 Okay, 21 00:06:09.475 --> 00:06:13.615 now the first couple of topics will not be allowable, 22 00:06:13.615 --> 00:06:16.404 like complex numbers, 23 00:06:16.404 --> 00:06:18.654 unless you to do something very sophisticated, 24 00:06:19.194 --> 00:06:22.524 but if you want to talk about these transformations, 25 00:06:23.095 --> 00:06:26.125 get into perhaps affect your spaces in more detail now, 26 00:06:26.125 --> 00:06:37.884 we can start getting it classical to quantum transfer visit transition from against principal interference that side of thing talked about the double slit experiment, 27 00:06:41.154 --> 00:06:46.464 other other topics relating to the double double slit experiment digging to more recent. 28 00:06:46.464 --> 00:06:54.324 Now we're getting to twenty th, century topics. Some. So, quantum mechanics has worked out, basically give or take nineteen twenties. 29 00:06:54.925 --> 00:06:56.605 And well, 30 00:06:57.024 --> 00:07:00.925 people were explaining things like earlier, 31 00:07:00.925 --> 00:07:07.285 like brownie in motion and with quantum tip, 32 00:07:07.704 --> 00:07:08.845 quantum for light, 33 00:07:09.240 --> 00:07:10.435 it's electric effect. 34 00:07:10.824 --> 00:07:12.745 But then the twenty people started. 35 00:07:13.740 --> 00:07:25.915 You know, I'm trying to think about quantum mechanics in more detail, so you can talk about some of that if you want. The paradox is about not getting. 36 00:07:25.944 --> 00:07:35.514 We're now we're getting into entanglement and so on are hidden variables a possibility. And if you're going to have locality, you cannot have hidden variable so that you can. 37 00:07:35.694 --> 00:07:48.204 So you could talk for a few minutes about that and bell's near, which was proving the problem with that back here. So, bells serum subject to some reasonable assumptions. 38 00:07:48.564 --> 00:08:02.694 Then subject to some reasonable assumptions, then there are no hidden variables. So you can you can see that. Somebody might talk about that and getting into more. 39 00:08:04.345 --> 00:08:17.665 The state collapsed by cost by measurement and talk about that. And just as a famous one, probably most of you have heard of it, but cat in the box. 40 00:08:17.665 --> 00:08:32.664 And if you, and maybe a quantum causes it, perhaps to be killed. But so it's interesting from both life and bed state until you observe. It thought experiment could Duncan experiment and then section five gets possible topics relating to quantum architecture. Here. 41 00:08:33.205 --> 00:08:44.304 And well, first classical, reversible complication, you can make classical Gates reversible. And if they're reversible, they're not increasing entropy. And the machines could perhaps be faster. 42 00:08:45.600 --> 00:08:57.504 So, in smaller groups that we can talk about that we talked about the universe, the quantum Gates, but somebody could present more information. Probably there are a probabilistic algorithm. 43 00:08:57.504 --> 00:09:09.835 So what's happening here is that in classical computation you can have algorithms, which are probabilistic. A classical example is prime allergy testing algorithm. 44 00:09:09.835 --> 00:09:22.649 We're given the thousand digit number, and we want to know is it primer or composite and a probabilistic method? It's not quite this simple, but you can imagine you pick a random number and see if it divides into your input. 45 00:09:22.644 --> 00:09:33.865 And if it does, it's approved that the numbers compensate. If it does not, then you still don't know and you pick another test factor. Well, that's a little simple that doesn't actually work very efficiently, but there are extension to that the, to work. 46 00:09:34.434 --> 00:09:41.784 And this is actually some of the best prime casting algorithms like, this is the probabilistic algorithm here. 47 00:10:02.815 --> 00:10:15.745 It's possible some algorithms, they will prove that a number is conscious that, but they do not produce the factors. The sound sort of counterintuitive you might think so. Let me just give you an example here. 48 00:10:15.745 --> 00:10:23.934 Actually, I'll go over to my other tab, come on hover cam. We have the screen. 49 00:10:27.774 --> 00:10:28.134 No. 50 00:10:35.004 --> 00:10:44.154 I have a screen and my hover can good. 51 00:10:50.184 --> 00:10:57.235 Let me try it. One more thing, log into a different output here. 52 00:11:07.644 --> 00:11:09.955 Well, maybe I won't be showing you until the hundred. 53 00:11:12.210 --> 00:11:16.825 Besides the other one second here. 54 00:11:31.615 --> 00:11:32.274 Okay. 55 00:11:38.754 --> 00:11:40.195 we do not have 56 00:11:51.414 --> 00:12:04.254 No, I'm restart next attempt. 57 00:12:04.559 --> 00:12:17.575 We actually have so seven. 58 00:12:21.059 --> 00:12:31.465 So, I just want to motivate how counterintuitive some of these things are. And let me bring up my web page. My book again just to show you what I'm talking about. 59 00:12:32.965 --> 00:12:33.745 Okay. 60 00:12:40.164 --> 00:12:40.764 Okay. 61 00:13:15.899 --> 00:13:24.325 The number is conscious, it is. Okay. Giving the factors. 62 00:13:28.945 --> 00:13:31.554 Something do something called Wilson serum. 63 00:13:35.215 --> 00:13:42.384 And I'll show you what that is. Let me just in case. Somebody is coming up. 64 00:13:43.465 --> 00:13:54.085 Do we need to so, let me just answer the questions. Yes. Okay so I'm showing you. I'll give an example of how counterintuitive this is. 65 00:13:54.565 --> 00:14:05.365 Wilson serum says visit P is prime. If, and only if. 66 00:14:09.504 --> 00:14:10.644 See, if I get this right? 67 00:14:20.335 --> 00:14:24.445 If you minus one factorial plus one. 68 00:14:25.884 --> 00:14:26.245 Oh. 69 00:14:34.679 --> 00:14:38.815 Is divisible by P. 70 00:14:40.200 --> 00:14:43.195 Make sure you example, the clothes. 71 00:14:44.815 --> 00:14:52.705 To one factorial, plus one equals to three plus one equals three. 72 00:14:53.580 --> 00:14:59.695 For free factorial plus one because seven? No, so that's okay. 73 00:15:00.235 --> 00:15:00.745 Five, 74 00:15:00.804 --> 00:15:03.445 four factorial plus one equals twenty five, 75 00:15:03.750 --> 00:15:12.240 the visible six by factor plus one equals one hundred and twenty one does not divide and so on seven, 76 00:15:12.235 --> 00:15:18.804 six secretarial plus one is exactly seven twenty seven. 77 00:15:18.835 --> 00:15:21.264 Twenty one and so on. 78 00:15:23.125 --> 00:15:36.384 So, we're engineers, we can say, approved it for six cases. Therefore. It's true. No. Okay. But this is the point that I'm making here is this sort of stuff. It can be counter intuitive. Okay. 79 00:15:36.384 --> 00:15:42.355 Getting back to classical factoring algorithms. You know, science is weird. What can we say. 80 00:15:47.125 --> 00:15:49.434 this column here let me just say that's divisible 81 00:15:51.120 --> 00:15:54.715 Okay, so that's for transforms. Well. 82 00:15:57.534 --> 00:16:12.294 This is getting a little far from the course you wanted to pick that topic, make good programming languages since we're going to be looking. At IBM thing. I would say, perhaps not pick one of these topics. I'll write this down getting back to theoretical things. 83 00:16:12.294 --> 00:16:14.754 I mentioned primarily testing and so on. 84 00:16:15.809 --> 00:16:22.375 Calling them a chance to do some research and some photography stuff. 85 00:16:23.514 --> 00:16:36.024 So, let me mention here, the people are not into this. The classical type of photography was that there was a key. The same key was used to encrypt and decrypt a message. 86 00:16:36.269 --> 00:16:47.784 The secret key, the big advance here. The same method was that you could have two separate keys one key to encrypt a message and a second key to decrypt a message. 87 00:16:48.955 --> 00:16:57.445 So you could publish your encryption key on your web page. Let's say anyone could use it to encrypt messages to you and you'd use your secret key to decrypts. That was a big advantage. 88 00:17:00.174 --> 00:17:03.174 And so all encryption methods are based on some. 89 00:17:04.230 --> 00:17:17.424 Type up some mathematical theorem, and many of these are based on the idea that it's difficult to factor large numbers. So if you could tack large numbers fast, this whole large class of keys would be breakable. Talk about that. 90 00:17:18.505 --> 00:17:29.994 Here's the cool thing. Quantum with medication. I may talk, I'm gonna talk about it later probably, but somebody can give an intro on this. This is sort of like quantum cryptography so that I could send you a message and that you would. 91 00:17:31.704 --> 00:17:35.275 Know that I sent it no one else other things here. 92 00:17:37.619 --> 00:17:52.194 Error correcting, so so these are possible topics if if none of these topics are if you like, none of the topics you can browse around the web and pick your own topic and so on but in any case. 93 00:17:52.194 --> 00:17:58.674 So, this this will be and I'm giving you two weeks to think about this. So okay. 94 00:18:06.835 --> 00:18:07.375 Come on. 95 00:18:09.480 --> 00:18:12.234 Good I'll come back to that. So that's this. 96 00:18:13.464 --> 00:18:21.414 Next thing what I did is I put up a large class of videos here. Don't have to watch them all, but sample them, you know, the first few perhaps. 97 00:18:23.694 --> 00:18:30.835 And you can look at some of them, they have different levels here of different, different levels of. 98 00:18:32.424 --> 00:18:37.734 Ease in depth and some of the last thing I got to fix the link and then go for his algorithm, 99 00:18:37.734 --> 00:18:41.065 there are a lot of videos and grover's algorithm because it's fairly hard in fact, 100 00:18:41.335 --> 00:18:46.434 I could spend a whole class here just free running video grover's algorithm videos for you, 101 00:18:46.825 --> 00:18:49.795 so you can look at this somewhat if you like, 102 00:18:51.984 --> 00:18:53.065 so again grope resell, 103 00:18:53.065 --> 00:18:53.335 Chris, 104 00:18:53.335 --> 00:18:57.535 and we have a function on and inputs and one input makes the output. 105 00:18:57.535 --> 00:19:09.684 True the others do not make it. True. So, find this and classically, you've got a test every input one by one takes it and tests grover's algorithm, take square root event tests. 106 00:19:10.289 --> 00:19:21.805 Okay, the next one I got some videos here on shore's algorithm, which factors? A large integer. And so I'm going to be presenting it, but you could also look at this. 107 00:19:22.950 --> 00:19:28.855 Okay, other things there is another set of pages here. 108 00:19:29.214 --> 00:19:44.035 The textbook has a good description of things on computing computer scientist, but sometimes it's nice to see a separate description. Different words and so on. So I'm giving you stuff from three different sources here. 109 00:19:44.460 --> 00:19:59.424 There's many things on the web, but these three are tolerable varying quality. There's the quantity of quantum computing things, which has some nice stuff here. The problem is that I don't know that the quantity has been maintained that much late. 110 00:19:59.575 --> 00:20:03.775 For example, a lot of the links are dead, but you can go to something like this here. 111 00:20:05.724 --> 00:20:17.815 And I think this is a very nice, you know, description here and some of this stuff, which I showed you before, but you can go through. 112 00:20:19.019 --> 00:20:33.355 So, you can have fun looking at that and feel like later on the thing gets a little harder and then some of the image links are dead. So, this is not perfect, but this isn't bad for some stuff. Okay. 113 00:20:34.914 --> 00:20:49.134 And so the algorithm I was talking with the last few times, one of them was Josh, the algorithm and just as a refresher, you're given a function on and inputs and Cupid inputs and the function. 114 00:20:49.855 --> 00:21:04.795 It's a black box. You don't know what it's doing. All you can do is give it inputs and watch the output and the function is in one of two types. It either is constant. It gives the same output all the time or it's not constant. 115 00:21:04.795 --> 00:21:16.704 And if it's not constant, it is required to be balanced, which means half the time, its output is one and the other half the time and stuff, but it's zero. So, the book has a nice a nice description. But there are some other ones. 116 00:21:17.035 --> 00:21:21.384 I'm like this one here, for example, nice description of what the problem is. 117 00:21:22.525 --> 00:21:26.755 And, yeah, so. 118 00:21:28.105 --> 00:21:38.095 In bits in, and now it says what's happening. It doesn't say how it works here. Button. Any cases is a nice executive introduction on what the problem is. 119 00:21:38.910 --> 00:21:43.045 So, it takes one evaluation is function. There is constant balance. 120 00:21:47.515 --> 00:21:48.144 Okay. 121 00:21:49.950 --> 00:22:01.434 The nice a nice detailed description here, which I may walk through is the Wikipedia one I'll come back to kiss, get later. So I just for fun. Spend a few minutes on this. 122 00:22:04.375 --> 00:22:11.335 Large this a little here motivation and whatever. 123 00:22:13.285 --> 00:22:15.984 You can go through this, but the idea is that. 124 00:22:18.894 --> 00:22:31.644 We have the black box here, so what we did is so it takes and inputs and is constant. Does the same output roll inputs or else? It's this half and half and again, just to remind you. 125 00:22:32.964 --> 00:22:45.535 The technique is a general technique as we had a controller bit. Why? And this because this makes the Blackbox reversible. So, what happens here? The X inputs that are the real inputs. 126 00:22:45.535 --> 00:22:58.944 We just pass them straight through to the output and the last output and plus first output. We take the we take that FX. It's a serious part of this box and resort with the control. 127 00:22:58.944 --> 00:23:11.634 But why, and because of this little trick, you might say, if we apply, you have twice the second application canceled out the first application. So which means it's reversible. So any case. 128 00:23:11.634 --> 00:23:19.404 So, we have the end inputs here, and we mix them up each input with a matrix. 129 00:23:19.404 --> 00:23:30.865 So, what happens here where I've got the we got to a point here is so we have to, to the end States here, quantum to, to the end and inputs cost to the States. 130 00:23:31.134 --> 00:23:36.204 And after those matrices, all of the to, to the end stage are equally probable. 131 00:23:37.315 --> 00:23:40.585 And so this function app is going to groups. 132 00:23:41.365 --> 00:23:54.505 I'm sorry about that one the function app is gonna act in parallel on all of those to the end possible input States and we also spread out the control bit, but it's a one. 133 00:23:54.505 --> 00:23:58.464 So it's going to spread out in an opposite phase things so on with it's had in mind. 134 00:23:58.464 --> 00:23:58.644 But, 135 00:23:59.755 --> 00:24:00.654 and now, 136 00:24:03.775 --> 00:24:05.785 because everything's it's equally probable, 137 00:24:05.815 --> 00:24:09.565 waving my hands and so on we take the X output bits, 138 00:24:09.565 --> 00:24:15.595 and we had to mark them again and now we can look at the top one and we can tell us the function is. 139 00:24:47.250 --> 00:25:00.535 Okay, so IBM is putting a lot of money in the quantum computation. This will be the last part of the course, and they have something called, which are a couple of things here. 140 00:25:01.194 --> 00:25:09.384 They have a log of tutorial material available online for free, or they said, okay, you could spend a few dollars on. 141 00:25:09.625 --> 00:25:19.765 So they're trying to make it easy for people to learn quantum computing, learn their version of quantum computing. And they also provide a simulator. 142 00:25:19.765 --> 00:25:31.105 You can use with all be showing you later and of course, they have the real quantum computers and the public you are allowed to access their older, real quantum computers. 143 00:25:31.105 --> 00:25:44.845 There batch computing is an old idea coming back and not interactive. You prepare a program and you submitted to a queue to the processing queue and when it gets processed, you get a, you can email back. 144 00:25:46.765 --> 00:25:57.924 Okay. So that okay so lots of things that IBM providing for you and again just occasionally guests expressed an opinion. 145 00:25:58.255 --> 00:26:09.505 My opinion is that this is the future of the company their cloud computing is not they're not a major player in the cloud computing market. They've got a couple percent of the market. 146 00:26:09.984 --> 00:26:16.555 Their mainframes are still interesting, but mainframes are sort of the past, not the future. 147 00:26:17.815 --> 00:26:22.825 And IBM has a precedent for doing this back around sixty years ago. 148 00:26:23.125 --> 00:26:37.255 They bet the whole company on on a new type of mainframe and their idea, which was revolutionary at the times they had a whole series of mainframes of different sizes. 149 00:26:37.855 --> 00:26:50.994 You could get cheap ones. That were small and slow or expensive ones that were big and fast, but they were all compatible with each other. You could move a program from one to the other and it would run. It would just run. You can run bigger programs, faster on expensive machines. 150 00:26:51.444 --> 00:27:04.555 But this was a major gamble inside the machine that totally different hardware. What they presented the same software to the user and going into this. Ibm was equal with several other computer companies largely coming out of this. 151 00:27:04.555 --> 00:27:19.464 Ibm was by far the dominant company. And it basically a monopoly. So the company has some, you might say some, some genealogy embedding the copy on it on a new technology. And my guess is, they're doing this with quantum computing. 152 00:27:19.765 --> 00:27:29.934 So, in any case, they have the tutorial material here. The various websites I'll give you links so kids get dot Org here and just go to it. 153 00:27:33.115 --> 00:27:44.904 Okay so the open source quantum development and you can, and you can just read the thing. Also I'll show you I'll click through some of that. 154 00:27:46.105 --> 00:27:47.994 I'll click on it now. Let me just. 155 00:27:50.789 --> 00:27:53.454 Okay, so you can get started and. 156 00:27:55.795 --> 00:28:04.375 And so is their open source emulator you can install it's all in Python and. 157 00:28:05.849 --> 00:28:20.694 You can have fun looking at that. And so what they're doing is they're providing tools to help you get started with quantum computing. And one thing is that they gotta drag and drop things. 158 00:28:20.694 --> 00:28:23.065 So you can do quantum circuits. I'll show you this on Thursday. 159 00:28:23.065 --> 00:28:36.115 I think I just kinda analyzing you, so you can drag and drop circuits like that up there or you can get access to the real machine and they've got tutorial materials, noise, mitigation. 160 00:28:38.184 --> 00:28:48.684 This is getting down into the electrical engineering and physics is a big problem with the quantum circuits is they're noisy or classical circuits to actually. 161 00:28:49.015 --> 00:29:03.835 And so there's a lot of money being spent a lot of research being spent on trying to make the actual hardware less noisy so you can do longer computations. The problem is that when eventually the noise builds up and you stop getting useful results. 162 00:29:04.134 --> 00:29:17.845 Now, this is separate from the issues that most quantum algorithms are probabilistic and grover's algorithm that we'll see is the drosha one is not the algorithm is and it produces it has a probability of an error. 163 00:29:18.355 --> 00:29:29.484 And what you do is you run the algorithm, you run the computation many times, and you get the error down small upsets a separate sort of noise thing. So we'll see more of that and quick star Tom. 164 00:29:29.484 --> 00:29:44.095 So, you can download the, you can lots of lots of platforms and so on Jupiter notebook or interactive Python notebooks. So there's lots of information and you cases one of my recommended books. 165 00:29:44.125 --> 00:29:57.115 So you can have fun going ahead of me and reading some of this. But that's not so much for today yet. Let me come back to the algorithm. So I'm still want to get you with algorithms a little more here. 166 00:29:58.404 --> 00:30:12.535 And on and hit you with a couple of ones, if I can get. Oh, okay. 167 00:30:13.045 --> 00:30:25.615 So, what's happening here is that the mathematics is being interpreted at the client. It's written in basically tech math and is a JavaScript client side interpreter. 168 00:30:25.615 --> 00:30:32.275 So, when I enable JavaScript, then I, I have JavaScript, basically disabled as much as I can. 169 00:30:33.119 --> 00:30:45.835 Yeah. Okay. Good. Okay. So I just wanted to show you their version of it. And again, I'm just gonna walk you through somebody. You can look at the details yourself and ask questions. Let me just. 170 00:30:49.224 --> 00:30:53.454 Okay, let me check here. Well, we know that. Yes. 171 00:30:54.954 --> 00:31:07.914 Okay, okay, I'll check every few minutes for questions. Okay so it's just going to the quantum solution again here spreading out their probability on all the input bits and then reversing. 172 00:31:07.914 --> 00:31:17.994 I'm not going to talk you through it again, but this also shows examples here for why? It works and and. 173 00:31:19.589 --> 00:31:32.664 So, you, you can read this yourself basically. Well, basically, the summary is if the machine is constant, this is the output. It's, it's not even. 174 00:31:32.664 --> 00:31:40.194 And if it's balance, the output is all even and they can separate them out and then they work through examples here, two and three cube bits. 175 00:32:03.299 --> 00:32:15.384 So so you can have fun, but this is, and then this gets into the implementation of this and if I hit try, it's not gonna work. So I'm not gonna try. 176 00:32:15.744 --> 00:32:29.275 So, basically, what's happening is you can inform edit in case you can either write a on program, or you can drag and drop circuits either way. So, and they can drag and drop Gates into a flowchart. 177 00:32:29.335 --> 00:32:40.615 So okay. So again, I'm not gonna try. But so you can have fun looking at that, get gates that look like, and then execute the thing. And so on. 178 00:32:42.204 --> 00:32:46.974 Okay, any case that would be awesome. 179 00:32:50.970 --> 00:33:04.674 So, grover's algorithm is probabilistic and that's the one that you have a function. One input is true. The other inputs are false and the textbook has a description. 180 00:33:04.674 --> 00:33:10.855 I want to walk you through a superficial view of what some of the other ones are doing here. 181 00:33:11.785 --> 00:33:25.734 And and the idea is, it does a linear search and event so this so nice application tells you why it's worth talking about here. 182 00:33:26.184 --> 00:33:33.414 You can use it for estimating statistics of a set of numbers on mean, and median and faster. 183 00:33:34.529 --> 00:33:38.184 Fantastic thing. Okay. 184 00:33:38.964 --> 00:33:51.295 Excuse me now what we're going to do here with grover's algorithm. 185 00:33:53.365 --> 00:33:58.105 Is that on. 186 00:34:54.570 --> 00:35:08.724 Yeah, it's a good description here and it's all it's all it's all fairly complicated, which is why I'm hitting a different directions. But the concept is that. 187 00:35:11.184 --> 00:35:16.704 We have okay, we have an operator called you several Mega. 188 00:35:17.755 --> 00:35:23.695 And if it's applied to X, which is the X, which causes the output to be true. 189 00:35:25.585 --> 00:35:40.554 Then it invert the, it puts a phase change on the on X otherwise it does not put a page change on next to this minus sign. Here is multiplying X by by minus one. Now, here's the thing here. 190 00:35:41.219 --> 00:35:55.885 You could not if you put a measurement right? Here you could not detect that, because the magnitude of the output is still the same you've got a minus sign, you find the length of the factor, the link to the state factor didn't change. 191 00:35:56.724 --> 00:36:02.065 So this is not enough here to detect. 192 00:36:02.514 --> 00:36:16.164 Is this the X switches causes the output to be true but waiting my hands and so on what they're doing is that they are if actually the true thing. 193 00:36:16.164 --> 00:36:30.385 They're increasing its probability. You see, the thing is this that we have to the end possible exercise. Each has equal probability. We created them with our head of marred matrices. And what we're doing is some, some reflections and some operators. 194 00:36:31.465 --> 00:36:44.815 And and well, if I can walk through the flowchart here, so we have an mbit input, because their function has and and bit. So we mix it up with a header garden matrix. 195 00:36:44.815 --> 00:36:49.105 Then we got this control bed also. So, this is our thing. 196 00:36:49.135 --> 00:36:49.434 So, 197 00:36:49.434 --> 00:36:57.534 what we do here is we unmixed the output with again and now what this thing here that does it, 198 00:36:59.875 --> 00:37:06.144 it amplifies the probability of acts where axis is the input, 199 00:37:06.144 --> 00:37:07.405 the cause the output to be true. 200 00:37:07.735 --> 00:37:16.344 And the sex is different, because its direction had a minus one applied, but by the you here so this. 201 00:37:18.420 --> 00:37:33.085 What this thing here does is going into the probabilities at the to the end different states for all the same this one increases a probability of the state that causes the 202 00:37:33.085 --> 00:37:38.934 output to be one because it was May distinguish use of Omega operator. 203 00:37:39.355 --> 00:37:52.434 And then we apply this thing, we keep repeating this and repeating us and repeating this. And what happens is eventually this date that we want is now fairly high probability. 204 00:37:53.184 --> 00:38:05.844 And then if we measure, we're probably gonna find it not certainly. But and the more times that reiterate the defeat, what's called the diffusion operator we do, it's quite good event times. Then was. 205 00:38:07.344 --> 00:38:10.434 We have we, we can, we can get the correct decks. 206 00:38:13.224 --> 00:38:14.605 And that's what's happening here. 207 00:38:17.485 --> 00:38:21.054 So, and this is the, this is the operator there, Tom. 208 00:38:24.925 --> 00:38:29.905 These operators take on the amplify the ex, which causes the function if to be true. 209 00:38:31.585 --> 00:38:33.204 oh waving my hand somewhat 210 00:38:39.385 --> 00:38:42.445 Okay, let me just see any other question. 211 00:38:47.574 --> 00:39:02.335 Okay, good and so it's called an Oracle here what an Oracle operator is is oh, it's a function that's a black box. Actually, it does something in this if it gets an input, that's the solution to the search problem. 212 00:39:02.335 --> 00:39:10.945 It gives a negative sign. How does this happen is a separate issue? Well, they're talking about it down here so I'm just giving you an executive summary. 213 00:39:11.605 --> 00:39:22.255 And what they're doing is series aquatic operators that eventually start amplifying the probability of that. And I will skip through some of this. 214 00:39:24.114 --> 00:39:31.135 You can, you can handle functional So where more than one input causes the output to be two K of them you need for generations to find them. 215 00:39:32.280 --> 00:39:40.375 Okay, and some applications and so on and. 216 00:39:41.849 --> 00:39:42.510 Okay. 217 00:39:47.335 --> 00:39:59.394 And in the real world, you've got a database, you gotta read the database and which takes time. So this might work might not be useful to modifying things. Okay. 218 00:40:00.445 --> 00:40:04.764 And the one is perhaps the most detailed. 219 00:40:08.005 --> 00:40:13.554 Okay, take a touch bigger for people. 220 00:40:15.775 --> 00:40:19.344 Okay, so we've got two of the inputs and one of them. 221 00:40:20.400 --> 00:40:22.764 Causes the output to be true. The one that's purple. 222 00:40:24.119 --> 00:40:28.494 Okay, okay so. 223 00:40:31.614 --> 00:40:45.235 What we need to do is, so that's our function. We have to take that function and turn it into this. Oracle. The function is the one, which is true for one of the two of the inputs and falls for all the others. So, the article again. 224 00:40:46.914 --> 00:40:53.335 We create from that function, we create the article and again, it negates X effects is the one. 225 00:40:54.150 --> 00:41:06.085 Which will, which is the true answer so, in the articles created by someone who has the function. So that is the oracle's input to the to growers the grover's algorithm. 226 00:41:06.085 --> 00:41:11.425 It's created by the person that created the function is use the Oracle. Then you use the Oracle to find what. 227 00:41:12.599 --> 00:41:19.074 What's the input that makes it? True so and so what does the article look like? 228 00:41:20.699 --> 00:41:34.735 Could be something like this, for example. So, what this will do is again, our input to this is our state vector that is to to the elements and this is unitary. It's its own inverse and so on. 229 00:41:34.735 --> 00:41:49.164 And what it does is for this particular input that negates that. And again now, this is legal, because the same factor, it's not probabilities from the state factor. If you take the magnitude squared of the elements of the state vector are probabilities. 230 00:41:50.065 --> 00:41:54.355 So, let me write that down because this could get confusing here. 231 00:41:58.735 --> 00:41:59.094 So. 232 00:42:23.369 --> 00:42:33.505 That is today and it's not the probabilities. 233 00:42:37.949 --> 00:42:44.724 It's the entry where is the probability. 234 00:42:46.590 --> 00:42:49.735 Okay, probability of that entry. 235 00:42:54.480 --> 00:43:02.215 So, in other words, you know, this here say, one, half, one half will have one half. Okay that's legal. 236 00:43:02.215 --> 00:43:17.125 So, on the probabilities or one quarter well, if we do something like this, say, one, half, one, half, minus one. That's legal. 237 00:43:17.155 --> 00:43:17.695 Okay. 238 00:43:19.434 --> 00:43:23.994 And so what I did was I changed that one entry there now. 239 00:43:25.644 --> 00:43:35.454 So here's the thing is, we can't measure the difference. 240 00:43:44.784 --> 00:43:51.385 But you affect future. 241 00:43:53.010 --> 00:43:59.215 Block them operators. Okay. 242 00:44:02.875 --> 00:44:11.664 It affects future quantum operators and this is the key here. This is the thing here. So, I mean, it's crazy. 243 00:44:11.664 --> 00:44:25.675 It's, you know, it's a computational trick, which you use here stances any questions. Okay. Let's say the input is wrong. 244 00:44:25.675 --> 00:44:26.994 If you run it many times. 245 00:44:33.085 --> 00:44:41.724 Think what the answer is well, 246 00:44:41.755 --> 00:44:42.894 you're running this, 247 00:44:43.914 --> 00:44:49.045 you're computing on all the possible inputs and parallel. 248 00:44:49.045 --> 00:44:49.525 So. 249 00:46:44.400 --> 00:46:50.934 Okay, so. 250 00:46:53.454 --> 00:46:56.364 Here over. 251 00:47:03.054 --> 00:47:03.655 Okay. 252 00:47:11.309 --> 00:47:21.684 It's not going over off to the edge. Okay. 253 00:47:24.385 --> 00:47:25.224 In kids. 254 00:47:35.304 --> 00:47:42.085 I'm trying to put lots of things on the same screen simultaneously. This is not scrolling. 255 00:47:43.440 --> 00:47:57.150 The second here. Okay. Okay. 256 00:47:59.094 --> 00:47:59.454 So. 257 00:48:03.655 --> 00:48:17.394 Any case, so we have this Oracle operator for the function and just invert that. So again, this difference is not immediately measurable is one way you describe it here that it. 258 00:48:21.599 --> 00:48:36.355 For any given zero, then minus one zero is one. If axes one minus one is the one is fine. This one to that particular thing gets flipped here. Just the location gets a little if he changed. So we could. 259 00:48:38.190 --> 00:48:42.534 Something like that and. 260 00:48:44.400 --> 00:48:45.114 You could. 261 00:48:54.809 --> 00:48:55.530 Okay. 262 00:49:04.074 --> 00:49:14.695 Something like now, why does that do that? X goes right through? And if the function is true Yeah it flips. I don't necessarily see. 263 00:49:16.619 --> 00:49:31.014 Okay now, skip that for the moment. Okay so the thing is, like I said, so it's slipped. 264 00:49:33.360 --> 00:49:35.755 It clipped the amplitude of the. 265 00:49:38.635 --> 00:49:52.945 That the element of X, which gives you the true output, but we got no way to find it because as ahead, you can't just measure the thing. Measurement won't detect that because the magnitude it's the same. Okay. So. 266 00:49:55.079 --> 00:49:59.934 It just just saying that right here. Yeah. 267 00:50:01.380 --> 00:50:13.885 Okay, so this this is the thing coming up here, this current paragraph. So it's increase it's app it's called amplitude amplification. It's increasing the probability of that one state. 268 00:50:14.244 --> 00:50:20.545 The other two of the impossible in, which is increasing the aptitude of the one that we want. 269 00:50:21.385 --> 00:50:32.364 And and it's skipping to this a little what's happening is that. 270 00:50:36.000 --> 00:50:42.534 I was going to wave my hand somewhat W, is the vector, which has the one for the state that. 271 00:50:44.190 --> 00:50:49.375 Can see output being true and s is just a uniform one probably only want to recruit at the end and. 272 00:50:50.815 --> 00:50:55.315 Well, we can do with quantum operators is we can do a reflections and. 273 00:50:56.335 --> 00:51:08.275 I'm waiting my hand somewhat here, but the effect is that the winning factor gets longer. Basically you another call it, Tom, they call it rotations. 274 00:51:09.269 --> 00:51:11.364 It's not precisely that and so, 275 00:51:13.135 --> 00:51:20.394 and because of rotation or function change of state but the effect is that, 276 00:51:20.394 --> 00:51:21.085 after this, 277 00:51:21.085 --> 00:51:22.135 the probability, 278 00:51:23.394 --> 00:51:26.994 the probability of this date that we want got more. 279 00:51:28.405 --> 00:51:43.135 And and every time we do this, it gets relatively higher and you just get a little less than we do this enough times square root event times. This is now large enough that if we now do a measurement, which will collapse, let's take back to. 280 00:51:43.135 --> 00:51:46.074 We're probably going to collapse it to there. 281 00:51:47.875 --> 00:52:02.815 So that is, that's what's happening here and I'll let you read this on your own and then ask questions on Thursday. No point in handwriting. Various things. I'll just. 282 00:52:05.485 --> 00:52:18.474 Well, okay, yeah, let me I'll just walk through things quickly. I guess so this is a case we have to que bits, so for possible inputs and the function for the input one one. 283 00:52:19.375 --> 00:52:23.875 Will cause the output to be true for others? It's false. So. 284 00:52:26.065 --> 00:52:26.635 So, we've got. 285 00:52:29.760 --> 00:52:33.775 Such a Super physician on these four states here equal probability. 286 00:52:34.704 --> 00:52:35.304 And, 287 00:52:37.554 --> 00:52:40.105 and this is this operator here, 288 00:52:40.500 --> 00:52:49.795 which for the one one state will flip its lines and it will cause this thing down here to the plus one one went to the minus here. 289 00:52:50.550 --> 00:52:55.764 Okay. And to give any control to the state. 290 00:52:58.260 --> 00:52:58.860 Okay, 291 00:52:58.855 --> 00:53:09.054 now I'm okay now we are, 292 00:53:14.304 --> 00:53:15.744 we had way through this, 293 00:53:15.744 --> 00:53:18.414 we're going to and flipping magnitude States, 294 00:53:19.380 --> 00:53:28.045 cause reflections and and we end up with something like something like this, 295 00:53:29.905 --> 00:53:32.844 and the users if you up here. 296 00:53:33.204 --> 00:53:39.835 So, basically, this will be a circuit, which will increase the probability of one one. 297 00:53:41.130 --> 00:53:43.914 Stage in the X factor here. 298 00:53:47.244 --> 00:53:52.284 And we apply this a few times and increase the probability enough. 299 00:53:53.514 --> 00:54:07.105 And as an actual circuit, if I try, it's not gonna work, but demonstrate that to get a working for Thursday. Yeah. 300 00:54:07.105 --> 00:54:17.724 It's waiting for the actual machine to become available. Okay. Any case. So, I gave you a high level version of Grover, and you cannot look at this and work on this. 301 00:54:18.985 --> 00:54:24.744 And ask questions for Thursday, and you can run it in real device. Okay. 302 00:54:26.784 --> 00:54:31.855 And then they give you another example with three bits that you can have fun with up here. 303 00:54:33.000 --> 00:54:39.114 Okay, any case and in implementation and then we'll talk more. 304 00:54:41.215 --> 00:54:44.155 And simulated experiment. Okay. 305 00:54:45.030 --> 00:54:48.684 So this is a more handwaving about the Grover algorithm again, 306 00:54:48.684 --> 00:54:59.335 you got the function with which one input to the functional cause you have to be true and do it in square root of and time where the classical case with and time oh, 307 00:54:59.335 --> 00:55:07.045 they give another example here with the Oracle cool little thing into this adult thing, 308 00:55:07.045 --> 00:55:08.574 and the one input, 309 00:55:08.574 --> 00:55:13.525 which makes it true is where the various entries all of the Sonoco roles. 310 00:55:14.250 --> 00:55:24.894 So it's you can use brokers algorithm on it. And so they just have to take this rules and turn them into. 311 00:55:28.199 --> 00:55:42.804 Turn them into so, so you're not repeating a number and then you call them or any role. So it's the zero is not equal to B, one zero equal DV, two, etc etc. You get these things and. 312 00:55:45.235 --> 00:55:59.364 So, there's gotta be more than one possible answer. This is a case a more general search thing where you a couple of true answers, unless you add some more rules, but basically they take this and they apply use grover's algorithm on it. 313 00:55:59.364 --> 00:56:05.184 Now, it's a little silly, such a simple thing, but it gives you an idea. 314 00:56:06.474 --> 00:56:10.494 Okay, talk about it down here. 315 00:56:12.954 --> 00:56:17.815 Okay on the full algorithm. So you're gonna have fun with this. 316 00:56:22.920 --> 00:56:23.550 Okay, 317 00:56:25.585 --> 00:56:27.235 so that was the Josh, 318 00:56:27.235 --> 00:56:37.105 the algorithm and the Grover algorithm and another major one is shore's factoring algorithm get in the book as an explanation we can walk through, 319 00:56:37.105 --> 00:56:39.925 but just just to see other explanations. 320 00:56:41.094 --> 00:56:42.264 We've got things here. 321 00:56:43.949 --> 00:56:57.655 Okay, and the thing is that it's probabilistic and Alexa oversee you have to rerun it several times. 322 00:56:57.684 --> 00:57:00.144 And what you do is that you get a. 323 00:57:01.260 --> 00:57:05.394 Probably, and probably an answer. 324 00:57:11.335 --> 00:57:11.784 Now. 325 00:57:15.235 --> 00:57:18.804 So, some, some things that go into this. 326 00:57:19.974 --> 00:57:29.784 Are greatest common Divisor and so you so I mentioned this silly, simple way. 327 00:57:29.784 --> 00:57:44.184 You test if a number's Prime or not, you pick a test advisor and see if the test device divides into N if it does and it's definitely conscious that if it does not, you still don't know. And you pick another test Divisor. 328 00:57:44.695 --> 00:57:56.425 Obviously, that's not a practical method. You'd have to, you know, something like square root to then test devisors. So however, it shows you the flavor of what's happening and. 329 00:57:59.934 --> 00:58:12.505 Relating to this, so one of the things coming in is greatest common Divisor. So, let me just demo that for you. In case people. Are we on that time? For the. 330 00:58:16.019 --> 00:58:19.344 Okay, let's see. 331 00:58:23.394 --> 00:58:31.375 So, tell me, so, let me pop up the chat window. 332 00:58:32.244 --> 00:58:46.105 I can find it here. Cool. So so. 333 00:58:52.764 --> 00:59:03.655 So question, you see. 334 00:59:07.260 --> 00:59:10.735 I spend a few minutes saying what it is or not. 335 00:59:15.025 --> 00:59:18.264 So while you're answering them, give examples. 336 00:59:20.184 --> 00:59:27.954 You know, give some examples. So, say, PCD, for example, let's say six, any equals to. 337 00:59:29.514 --> 00:59:34.974 A of eight, nine cause one. 338 00:59:36.869 --> 00:59:42.954 That and the twenty plus ten and so on. So it's used. 339 00:59:44.574 --> 00:59:57.295 So, greatest com advisor, it's the largest number. Well, we're all talking natural numbers here. These are integers from one zero off, depending on you to find them and it's the largest energy that divides both input. 340 01:00:10.855 --> 01:00:18.385 So, let's say GCD, so I'm gonna say twenty, twenty four was born. 341 01:00:24.539 --> 01:00:30.625 I don't don't take a problem. I said, okay, so this goes into the. 342 01:00:31.710 --> 01:00:33.204 i'll address in here 343 01:00:36.804 --> 01:00:36.985 Oh, 344 01:00:36.985 --> 01:00:37.284 okay, 345 01:00:39.954 --> 01:00:48.655 so what we're doing is so you pick a random a, 346 01:00:48.744 --> 01:00:52.525 and if it divides and we're done, 347 01:00:53.574 --> 01:00:54.445 otherwise. 348 01:00:57.264 --> 01:00:59.605 We're using some manager exponentiates here. 349 01:01:00.594 --> 01:01:08.125 And what we're doing is we're taking powers of. 350 01:01:10.074 --> 01:01:12.414 We're taking powers of a mod and. 351 01:01:13.434 --> 01:01:25.704 And seeing do they divide into so, let's suppose, say any costs a one on one. 352 01:01:27.389 --> 01:01:41.635 Let's say a close up to a smaller one here say N equals, say nineteen and equals three or something. 353 01:01:42.474 --> 01:01:54.534 So g. C. D three and nineteen equals one. Okay. And now what we do is we look at. 354 01:01:59.514 --> 01:02:05.304 Or make it smaller seven. Okay. Oh, so we look at three. 355 01:02:06.750 --> 01:02:18.684 We look at three square equals nine. That becomes two seven. We looked at three. Q. equals twenty seven. That becomes six months seven. 356 01:02:19.710 --> 01:02:27.054 What we're doing here, launch seven so, you know, the four equals eighty one equals. 357 01:02:32.820 --> 01:02:36.204 For March, seven, it's great as a fifth. 358 01:02:37.855 --> 01:02:41.275 Equals twelve week course, I. 359 01:02:43.260 --> 01:02:51.204 It is a six since fifteen which happens we won seventy equals three and so. 360 01:02:52.019 --> 01:02:52.409 Three, 361 01:02:52.405 --> 01:02:53.545 the one equal suited, 362 01:02:53.545 --> 01:02:54.144 the seven, 363 01:02:55.284 --> 01:02:56.005 seven, 364 01:02:56.579 --> 01:02:57.210 period, 365 01:02:59.094 --> 01:03:06.295 six and okay, 366 01:03:06.295 --> 01:03:07.315 so what we're doing. 367 01:03:11.099 --> 01:03:17.994 So so, so we can use these and factors of. 368 01:03:23.880 --> 01:03:26.934 So, I got the period being six, three. 369 01:03:28.050 --> 01:03:33.744 To six, eighteen or twelve for twelve. 370 01:03:37.494 --> 01:03:38.755 So now we can say. 371 01:03:40.019 --> 01:03:54.355 We can work our way through here. So the period finding subroutines is these slow part, but I talked about that, you know, two days ago how to find a period of a function. So we're gonna use that here. In any case. 372 01:03:59.250 --> 01:04:06.355 We can look at our was six so well, it's five. Here are over. Two is three. 373 01:04:08.605 --> 01:04:19.554 Doesn't work, I'm a times are over to a was three nine that doesn't work. 374 01:04:19.554 --> 01:04:28.914 So the factories event are, but in case crying here. Gcd. 375 01:04:30.869 --> 01:04:33.594 Free time, three, plus or minus one over to. 376 01:04:34.914 --> 01:04:44.635 I don't know worry about that Thursday any case. So we're using some some modern algebra here. So, Matt, and we go through and. 377 01:04:47.244 --> 01:04:56.094 So, the hard part is the period finding with this and this is probably as we keep having to run this many times, perhaps go back to step one. 378 01:04:56.760 --> 01:05:09.204 And so we have to find a period, but we saw a little about finding periods and things in it for you transform. Okay. 379 01:05:13.764 --> 01:05:23.514 Are just going through here somewhat. I'll just wait by hand a little if you look at that hit more detail Thursday in any case. 380 01:05:27.809 --> 01:05:31.405 So, this is the period finding thing. Like I said, we saw a week ago, so. 381 01:05:33.750 --> 01:05:48.655 And you can read this, this is reading assignment for Thursday you can look at that. So talk about that. Okay. We can also look at the textbook textbook if you wish, but any case. 382 01:05:50.130 --> 01:06:02.905 So this, so what I showed you is several different inputs so you've got the textbook, which is good but should you prefer different entries? Different ways describing things is the which can be nice. 383 01:06:02.905 --> 01:06:09.025 Sometimes is Wikipedia, which is nice sometimes. And there's kiss. Good, which is website. 384 01:06:10.344 --> 01:06:20.574 Now, what we'll be doing, so we're migrating towards IBM and we'll see some of these things. So what I have here at, we've got the videos here, feel free to look at that. 385 01:06:20.905 --> 01:06:35.514 And if I've totally confused you with grover's and so on, you can watch some of these videos, it's this we're not a web based class. I'd show some of these videos now, but it's so it's silly for me to running videos through webx for you. 386 01:06:35.905 --> 01:06:50.215 You can look at them on your own and then I might ask questions about that in any case. So, now, Thursday, so I have a draft here what I'll be talking about Thursday and you can look at. You can go ahead of me if you want. 387 01:06:50.789 --> 01:06:53.485 So, I've got some videos here of. 388 01:06:56.454 --> 01:07:07.914 I've got some videos from IBM. Some of them are on YouTube, some of their own website, and you can look at some of them getting into specifically IBM stuff. 389 01:07:26.364 --> 01:07:28.014 you can also go to the 390 01:07:31.764 --> 01:07:41.934 Well, this this is the get help site for casket so you can have fun anticipating me if you wish and downloading it trying to install it will walk through some of that on Thursday. 391 01:07:43.315 --> 01:07:51.925 And that's basically, so, what I've done is, I've got a little light on some of the hard details. 392 01:07:51.954 --> 01:08:02.394 What I'm trying to do is give you a sense of where these algorithms fit in the universe is somebody wishes. We can go through some of the details. 393 01:08:02.425 --> 01:08:11.695 But but what I'd like to do is give you the higher level thing and let you look at the details yourself. 394 01:08:11.695 --> 01:08:24.085 I mean, I can dig down into the details if you wish, but other than that, yeah, I think the broad perspective is more important. Someone. So. 395 01:08:27.234 --> 01:08:33.625 So, I've tried right overview. 396 01:08:42.510 --> 01:08:52.435 If you wish and then Thursday. 397 01:08:53.725 --> 01:08:58.975 You know, getting into IBM, just scheduling someone. 398 01:09:00.869 --> 01:09:02.425 Maybe some more algorithm it's. 399 01:09:12.835 --> 01:09:13.465 Okay, 400 01:09:21.685 --> 01:09:23.154 any questions and things, 401 01:09:23.154 --> 01:09:29.484 and I'll probably fix the thing so my photograph is not occupying half the screen all the time next time. 402 01:09:30.119 --> 01:09:35.635 So and this is what I have two computers in front of me, 403 01:09:36.055 --> 01:09:36.385 you know, 404 01:09:36.385 --> 01:09:37.375 my new computer, 405 01:09:37.375 --> 01:09:38.215 something hangs, 406 01:09:38.215 --> 01:09:39.715 and rather than restarting it, 407 01:09:40.135 --> 01:09:42.895 which to add just move over to the second computer, 408 01:09:43.560 --> 01:09:44.814 second computers actually, 409 01:09:44.814 --> 01:09:46.255 about sixty years old but hey, 410 01:09:46.255 --> 01:09:47.635 to think bad and still works. 411 01:09:48.449 --> 01:09:51.175 So and computers are crazy. 412 01:09:55.979 --> 01:10:01.795 So, and also just to remind you. 413 01:10:04.914 --> 01:10:08.755 On watching videos. 414 01:10:09.895 --> 01:10:20.154 Oh, okay, good. 415 01:10:23.100 --> 01:10:23.699 Okay. 416 01:10:46.765 --> 01:10:55.284 Professor, yes, I just have a suggestion for the popping the chat window box. I think you're able to do that. 417 01:10:56.305 --> 01:11:09.534 It's like one of the, and might be part of yours, you know, the main screen where I'm not floating now I'm running it inside of Firefox. 418 01:11:10.260 --> 01:11:21.864 So, okay, no, I'll tell you what I should do, and you can badly criticize me for not getting this better set up. What I need is a separate computer, just showing the chat window. 419 01:11:23.095 --> 01:11:28.284 Gotcha, and that's that's what I should have done. So, you know, I actually have lots of computers. 420 01:11:31.074 --> 01:11:36.354 But, yeah, no, sounds good. All right. Thank you. Let's try next time you keep the suggestions coming, so. 421 01:11:40.194 --> 01:11:41.034 Anything else. 422 01:11:53.609 --> 01:11:56.515 Well, if not okay. 423 01:12:01.319 --> 01:12:01.859 Okay.