WEBVTT 1 00:01:46.290 --> 00:01:53.185 Let's see what happens? Good afternoon. We. 2 00:02:02.935 --> 00:02:16.104 Okay, good afternoon. The usual question is and you hear me. Okay. Wow, we'll see how long that last and. 3 00:02:17.490 --> 00:02:31.705 So, as a backup in case, you need to communicate with me, I'm going to give you my phone number and you can text me or something. So, in any case, this is approximately class five. 4 00:02:37.194 --> 00:02:40.104 Okay, so what you can try to do. 5 00:02:43.914 --> 00:02:44.664 Let's see. 6 00:02:58.104 --> 00:03:04.705 The second I'm getting, so it could try something like this. 7 00:03:11.544 --> 00:03:12.444 Okay, 8 00:03:13.465 --> 00:03:25.194 so you can text as if you have to get hold of me because something else is not nothing else is working, 9 00:03:25.555 --> 00:03:28.164 but I encourage you to chat and to unmute. 10 00:03:28.194 --> 00:03:29.905 And so also. 11 00:03:31.620 --> 00:03:44.485 Trying to go on mute and talk. If that works. The problem is, I cannot tell if my audio is coming through, as you figured out by now. 12 00:03:45.629 --> 00:03:54.534 Okay, so I can see what the video you're seeing, but I haven't figured out a way to tell what audio you're hearing. 13 00:03:54.835 --> 00:04:04.435 I can't just put a headset on, let's say, because there would be a significant delay and that would drive me crazy. Okay. 14 00:04:05.965 --> 00:04:17.935 So, what's happening today is I'm going to continuing onto the textbook quantum computing for computer scientists and I think I might get to chapters four and five. 15 00:04:17.935 --> 00:04:24.264 Actually, if that sounds like a lot, it's because we've seen a lot of the ideas in it before. 16 00:04:24.954 --> 00:04:33.175 And the reason I was spending time on them, it presents these ideas in. 17 00:04:34.319 --> 00:04:48.714 You might say a more detailed and more fundamentally and more and more detailed way. So you see if I can here, put this on so you can. 18 00:04:49.644 --> 00:04:59.394 Okay. And what I so, I'll be going through this and I, what I've got here is I typed as before. 19 00:04:59.574 --> 00:05:13.855 I've typed in some things so that you don't have to completely rely onto my handwriting and I'll put some notes on over here. Let me just bring up a copy of the book. But what I'll be doing is walking you through the book to. 20 00:05:15.120 --> 00:05:17.214 To a large extent and. 21 00:05:22.495 --> 00:05:23.185 Quantum. 22 00:05:26.365 --> 00:05:33.894 Let's see, okay. Walking into the park. Someone. 23 00:05:40.074 --> 00:05:44.694 Okay, so we've seen this before. 24 00:05:44.875 --> 00:05:45.144 I mean, 25 00:05:45.144 --> 00:05:51.295 the book goes through some history from some history of quantum computing, 26 00:05:51.295 --> 00:05:56.185 which you can read some of the experiment of validation about why we believe this, 27 00:05:56.334 --> 00:06:01.045 this thing and let me put up also my notes over here. 28 00:06:01.045 --> 00:06:14.665 Some of this. Okay. So so it motivates with a couple of motivate a couple of examples. The first one will be a particle on a line. 29 00:06:15.240 --> 00:06:18.894 It shows you this the particle could be in different positions on the line. 30 00:06:19.769 --> 00:06:21.024 X zero X, 31 00:06:21.024 --> 00:06:21.324 zero, 32 00:06:21.324 --> 00:06:22.735 two X minus one, 33 00:06:23.185 --> 00:06:30.444 and the way this bracket notation the way we represent it is something like, 34 00:06:30.745 --> 00:06:32.425 if something like that over here, 35 00:06:32.425 --> 00:06:35.875 this means that the particles in position setback side. 36 00:06:36.295 --> 00:06:36.535 So, 37 00:06:36.535 --> 00:06:43.404 we represent the state by a complex sector and and, 38 00:06:43.435 --> 00:06:49.404 and elements Senate and which just says that initially the, 39 00:06:50.035 --> 00:06:50.665 that one of them is, 40 00:06:50.665 --> 00:06:51.985 one is the rest is zero. 41 00:06:52.345 --> 00:06:56.214 So if I put something over here, let's say. 42 00:06:59.185 --> 00:07:09.535 I mean, let's suppose I didn't know it, so suppose that equals for or something and if the, you know, the particle. So if we're in, say state. 43 00:07:10.529 --> 00:07:21.355 Two, then this is gonna be the vector. It's not in state zero. It's out in state one. Yes, it's in stage two and it's not in stage three. For example, that's how we would represent that. 44 00:07:24.564 --> 00:07:38.274 And we would also represent is say next two, and that goes over that this means this is where I presented to ask that arrow thing on the end. Okay. 45 00:07:38.935 --> 00:07:43.944 So that is, that's my almost city is classical, but the. 46 00:07:46.105 --> 00:07:55.225 In quantum mechanics, we allow any there, not just all zeros, except for one run with quantum mechanics we allow anything. 47 00:07:55.225 --> 00:08:08.004 So it could be, for example, say, you know, three, four, plus minus to see the whole story. 48 00:08:10.345 --> 00:08:11.035 Okay. 49 00:08:11.970 --> 00:08:19.524 Thank you. Okay, so with quantum mechanics, it's this is a four state system classically it's zero. 50 00:08:19.524 --> 00:08:31.915 So for one one thing, where the particle is quantum mechanically, it's any for complex numbers make a state and that's that's what it's showing. That's what it's showing here. 51 00:08:32.095 --> 00:08:46.914 And, of course, what we had before is that we normalize the so we have probabilities. So, and not just what the thing, the second here there. Okay. 52 00:08:48.325 --> 00:08:53.455 And we get the probabilty formula here, which you've seen before. Okay. So it's just a review. 53 00:08:55.585 --> 00:09:00.565 And so I had a case here, the probability that it's in state to would be. 54 00:09:05.965 --> 00:09:18.595 That state to hear, is there a one two three would be minus two square divided by very squared plus four plus where it costs and so on. Okay. 55 00:09:25.980 --> 00:09:37.524 Can rep, I'm, I'm making a points over here. Okay. The point is all vectors and CNE are legal States right here and we have a general quantum state. Okay. 56 00:09:38.304 --> 00:09:52.044 And something like this over here called superposition. It's in to the four states with the different probabilities. Okay. You know, normalize it, so we multiply everything there by the same number. 57 00:09:52.225 --> 00:09:59.184 It doesn't change the state because you normalize it. Okay. Going through there. 58 00:10:02.904 --> 00:10:09.595 The next mode, so that was the motivating example from article on points on the line. The next motivating. 59 00:10:11.995 --> 00:10:23.634 Example brings in something interesting. Most quantum particles have a new property called spin. Some of them. I've only spent zero, but. 60 00:10:25.014 --> 00:10:29.154 A lot of them have is spin, it's actually it's of one half, but that doesn't matter. 61 00:10:29.365 --> 00:10:31.559 It's some typically, 62 00:10:31.554 --> 00:10:31.764 like, 63 00:10:31.764 --> 00:10:33.985 it's upwards down the thing is, 64 00:10:33.985 --> 00:10:38.365 you measured the spin with respect to some access you get to pick the access well, 65 00:10:38.365 --> 00:10:51.205 depending on your equipment and measure equipment allows and with respect to that access it will appear to be up or down or clockwise and they represented as something like this, 66 00:10:51.504 --> 00:10:56.429 this stage is the superposition of an upstate and down state was here amplitude, 67 00:10:56.455 --> 00:10:57.654 zero and see run. 68 00:10:57.654 --> 00:11:09.325 And again, the probabilities side to square, not the square the length of that. So, actually, I did this, I, I actually was wrong here. 69 00:11:09.325 --> 00:11:23.004 What I should have said was minus to complement times to divided by. Well, three doesn't matter. Plus four. Plus I compliment four. Plus I, and so on. 70 00:11:25.284 --> 00:11:34.434 Context complex conjugate times that oh, okay. And I'm Heather wise. I don't know. It was negative numbers up there perhaps. 71 00:11:34.465 --> 00:11:41.034 So if I take the I take the magnitude modifying complex conjugate times the number. 72 00:11:42.149 --> 00:11:53.034 Sick always comes out non negative. Okay. So nothing. So, this talks about spin there because it comes up it talks about this geometry. 73 00:11:53.065 --> 00:12:05.335 I actually don't think that this geometry so interesting they mentioned, but this is an important concept here, the transition aptitudes and so. 74 00:12:07.195 --> 00:12:18.085 I tried to write it down a little better over here so I mentioned spin the second motivating example. So okay. 75 00:12:20.335 --> 00:12:23.965 So, it's happening here with transitions when you measure it. 76 00:12:25.044 --> 00:12:39.745 The measurement causes the state to transition actually, to one thing turns out one of the Aiken vectors of the, the measurement. 77 00:12:39.745 --> 00:12:53.934 Obviously, the measurement operator information operator. It operates on the state, and it causes the state to transition here to something. There's a little actually prime up the site prime you can't really see the prime. Okay. 78 00:12:54.654 --> 00:13:06.294 And so if there's several possible states, it might, it might transition to then the probabilities depend on things called transition aptitudes. 79 00:13:06.865 --> 00:13:11.184 So, the transition aptitude is the probability that. 80 00:13:12.054 --> 00:13:24.715 This quantum state will transition into whichever of the possible quantum states that will result from the from the measurement operator. And the way this works is it's an inter product. 81 00:13:26.184 --> 00:13:38.664 And so what we see. So, over here in the book is that we have the initial state of the system, and then it will be in a final stake. 82 00:13:38.934 --> 00:13:41.424 So the, all the way to change and. 83 00:13:43.225 --> 00:13:46.134 And then, of course, if we take for these. 84 00:13:47.394 --> 00:13:58.764 You take zero complement conjugate from zero, and you get probability get the probability of be observed in any of these. But the thing is, the measurement changes the state of the system. 85 00:13:59.399 --> 00:14:08.215 It's some matrix modification by her mission matrix. So so what I put here, it causes it to transition to another state. 86 00:14:08.485 --> 00:14:18.684 And the probability of it going to any particular state is in our product, or a DoD product of the state, and the possible and state. 87 00:14:20.394 --> 00:14:25.225 And the possible end state, if it's done right and probabilities or sum up to one. 88 00:14:27.235 --> 00:14:36.205 So, let me do this, all of these three sections here together the book sort of goes back and forth in a sense. 89 00:14:37.529 --> 00:14:41.154 So, I tried to make it Oopsy let me try to make a little clearer here. 90 00:14:45.085 --> 00:14:59.304 That what we're doing is that we have observable observable. Like, if we have the particle could be in places on the line, the observer, what we're measuring is its position and. 91 00:15:00.240 --> 00:15:03.865 So, the observe is an abstract content. We're measuring position. 92 00:15:03.865 --> 00:15:07.134 Let's say when we measure position, 93 00:15:07.375 --> 00:15:09.774 we're going to see one end possible values, 94 00:15:09.774 --> 00:15:10.044 zero, 95 00:15:10.044 --> 00:15:10.164 one, 96 00:15:10.164 --> 00:15:10.315 two, 97 00:15:10.315 --> 00:15:10.465 three, 98 00:15:10.465 --> 00:15:16.884 four or five up to a minus one and the possible values we might see are Aiken, 99 00:15:16.884 --> 00:15:19.105 values of the matrix. 100 00:15:19.794 --> 00:15:26.335 And the positions are Aiken vectors while the I position, we made into a vector. Okay. 101 00:15:27.149 --> 00:15:31.495 And the result is, it changes things so. 102 00:15:33.355 --> 00:15:36.715 And what they're showing here, I'm jumping back and forth. 103 00:15:39.534 --> 00:15:48.835 Okay, so here, it's in some state and again that these are the weights again, this is a particle on the line. These are weights for the various things and. 104 00:15:51.024 --> 00:15:58.254 And then you would end up with these, and the probability of going from here to here is the inter product of these two factors. 105 00:16:00.565 --> 00:16:13.075 And the inner product of two vectors, when their complex is, you take the complex conjugate to the first one, and you multiply by the second one. Oops, I will left. Okay. Okay, cool. 106 00:16:14.845 --> 00:16:25.945 And they use this notation down here line, four, thirty two to say we're going from stateside means we're transitioning to stateside Prime. 107 00:16:26.274 --> 00:16:31.465 And the probability of that is given by the inter product upside side Prime. 108 00:16:33.384 --> 00:16:39.325 Okay, and you could have a case of the product gives you a zero or a two stage perpendicular to each other. 109 00:16:40.409 --> 00:16:43.945 Yeah, okay. That's what they're talking about here. 110 00:16:46.735 --> 00:16:50.455 And so you're doing a measurement and there's. 111 00:16:51.330 --> 00:16:52.134 Say here, 112 00:16:52.914 --> 00:16:54.264 you're initially in stateside, 113 00:16:54.264 --> 00:16:55.014 you're doing a measurement, 114 00:16:55.014 --> 00:17:03.090 the possible states you might end up in are zero to be in minus one and the probability of each one of these and different states, 115 00:17:03.085 --> 00:17:06.809 after the measurement are these transition amplitude, 116 00:17:06.835 --> 00:17:08.634 transition probabilities here. 117 00:17:08.845 --> 00:17:15.805 All these little in our product things. Okay. And. 118 00:17:19.799 --> 00:17:22.734 I give an example, let me work through. 119 00:17:26.335 --> 00:17:30.805 I'll walk you through this example. 120 00:17:32.789 --> 00:17:34.494 Four one six up here. 121 00:17:43.765 --> 00:17:46.585 Okay, so it is. 122 00:17:48.055 --> 00:18:00.085 Excuse me so I'm working on. 123 00:18:05.664 --> 00:18:11.934 Example, four one, six page one one three. 124 00:18:12.505 --> 00:18:24.384 Okay so we're transitioning from one stay here one eye and we're going to this state, 125 00:18:24.990 --> 00:18:33.115 which is high minus one and what's the probability? 126 00:18:33.115 --> 00:18:33.414 So. 127 00:18:41.009 --> 00:18:54.835 Of that transition. So this here is the state on before the measurement and this is one of the possible states active dimension. 128 00:18:54.835 --> 00:19:00.505 Let me write this down to the left hand. Side is the state of some system. 129 00:19:03.029 --> 00:19:09.894 Before the measurement, the right hand side. 130 00:19:10.680 --> 00:19:17.214 Hey, guys, one possible state. 131 00:19:19.375 --> 00:19:27.565 After, and we assume things things have been normalize. Okay. So. 132 00:19:30.299 --> 00:19:43.674 So, we will do is that it doesn't actually matter which one which one we put on the left and which one we put on the. Right but we would say, maybe I put the initial one on the left. 133 00:19:43.825 --> 00:19:45.355 That doesn't matter which one you do. 134 00:19:46.345 --> 00:19:49.194 We'd get the complex conjugate so we say one over Square, 135 00:19:49.255 --> 00:19:49.674 two, 136 00:19:50.365 --> 00:20:02.875 minus square root or to that'd be the college you get here and dot square would have to thing minus one and that's going to be on one half. 137 00:20:03.839 --> 00:20:05.214 One times I. 138 00:20:15.234 --> 00:20:16.134 Let's see. 139 00:20:18.329 --> 00:20:19.585 Plus, I. 140 00:20:25.680 --> 00:20:27.295 See, I did. Just right. 141 00:20:31.559 --> 00:20:37.285 If I got my minus signs correct? And so this will give a probabilty a thought. 142 00:20:38.244 --> 00:20:42.984 And we'll give a probability of one. So let's see what the book says. 143 00:20:45.805 --> 00:20:50.275 In your product mind I got a minus because I did it the other way. Okay. 144 00:20:52.015 --> 00:21:02.845 So so that's what's happening here and and again at the aptitude of the transition. 145 00:21:06.115 --> 00:21:17.664 Okay, so that's the basic thing there now so that was a section on, on measurement. The section on let me go back here. I need a bigger, probably gonna transition. Okay. 146 00:21:17.664 --> 00:21:27.684 Now, so observable is just something like position that we're measuring and observable is the name of the item. 147 00:21:27.684 --> 00:21:40.434 We're measuring mass position momentum, whatever, and we do it again, but with a modifying. And is it or we should operator I have that written down here. 148 00:21:42.355 --> 00:21:44.755 And the biggest thing here. 149 00:21:45.865 --> 00:21:55.464 Is that if we do, we might do several success of observations of the system and each observation affect the system. 150 00:21:55.825 --> 00:22:08.184 So, if we do two observations and sequence, then the order that the order that we do the two observations effects, what we see in the second observation. 151 00:22:09.414 --> 00:22:11.634 And a nice example. 152 00:22:14.005 --> 00:22:24.535 This is this way they show up down here, I won't write it down because books needed the type setters needed in my hand writing is this thing you saw in the physics. 153 00:22:24.535 --> 00:22:35.335 Course, I think where you take a couple polarizing filters in the order, you put two polarizing filters in affects what you see how much gets to the result. So. 154 00:22:36.535 --> 00:22:51.174 Any case this thing I mentioned here okay now this is a whole section here on Heisenberg uncertainty, and so on chime ignoring you don't have to getting too far away from the course. 155 00:22:51.174 --> 00:22:53.305 So I mentioned here that. 156 00:22:56.605 --> 00:22:58.315 skip those pages there 157 00:23:51.505 --> 00:24:05.605 I like to look at it is measuring, is that you're in tagged in the system, the outside world so the system you're isolated, you're federally isolated system plus the rest of the universe will be a unitary offer. 158 00:24:05.605 --> 00:24:09.505 Its transition would be unitary operator. It's only looking at this. 159 00:24:10.440 --> 00:24:25.194 System and I, you know, on its own the measurement operator causes it to collapse into one of the possible conductors of the operator. So I like to look at it is that the measurement is the system to interact with the rest of the world. 160 00:24:25.559 --> 00:24:28.914 I put that blurb in here to see what else the book has. 161 00:24:31.704 --> 00:24:34.404 I basically gave you the information content there. 162 00:24:37.045 --> 00:24:48.085 Yeah, and here they mentioned the polarizing thing. I hear you take it in physics. 163 00:24:48.085 --> 00:24:57.714 One you see this thing and if you could correct me, you have three polarizing sheets and, like, go through them. 164 00:24:58.615 --> 00:25:07.134 And two things happen when the, like, goes through one of these polarizing sheets here, polarizing filters, number one, is it. 165 00:25:08.154 --> 00:25:11.785 The light it comes to a smaller intensity, depending on. 166 00:25:13.015 --> 00:25:15.115 While the traditional light was not polarized, 167 00:25:15.355 --> 00:25:18.535 then the amount of tickets to is one half initial, 168 00:25:18.535 --> 00:25:18.775 like, 169 00:25:18.805 --> 00:25:29.095 is polarized the amount that gets through depends on the angle between the initial polarization and the polarization of the sheet and the about product in there. 170 00:25:29.634 --> 00:25:31.825 And so that's one thing is that. 171 00:25:32.934 --> 00:25:36.265 The amount of like to get through is reduced. The second thing is I liked it. 172 00:25:37.434 --> 00:25:41.244 Exits from the sheet is now polarized in the direction of the sheet. 173 00:25:42.295 --> 00:25:53.755 And so, if we have this here, that was say, horizontally polarize, she liked it gets through is now it's less intense and it's all horizontally polarized. 174 00:25:54.924 --> 00:26:09.835 Then they say it's the second sheet is tilted a forty five degree angle again, less light will get through, depending on the top products, or the angles. But the light that gets through this middle sheet is now polarized at forty five degrees. 175 00:26:11.154 --> 00:26:24.144 And if the filter is polarized vertically, then the White tickets through the third filter, it's penalized vertically. So it affects the system is effective. 176 00:26:24.894 --> 00:26:27.384 And the crazy thing here is that. 177 00:26:28.710 --> 00:26:41.994 I don't know, give or take an eighth the right gets through all three filters, but if you remove the middle filter now, like, it's through so in searching the filter calls in the middle filter, cause it's more like to get to. 178 00:26:41.994 --> 00:26:56.815 Because if you remove the middle filter, the like, that exits, the first sheet filter is horizontally polarized it hits the last filter, which is a vertically polarized filter and that absorbs all the horizontally polarized light. 179 00:26:56.815 --> 00:27:02.065 So that's what they're talking about here with measurements. And so on, okay. 180 00:27:04.404 --> 00:27:06.625 All of the measuring section. 181 00:27:11.724 --> 00:27:24.414 Next section dynamic, so measurement is applying a information operator and it collapses the system into one of the I can factors of the operator. Okay. The next thing dynamics. 182 00:27:24.414 --> 00:27:34.045 So dynamics is talking about operating on your quantum system with irreversible operators in their unitary matrices. That are in vertical. 183 00:27:35.670 --> 00:27:39.714 Information matrices may not be in vertable. Okay. 184 00:27:41.484 --> 00:27:47.095 And so I just have a little summary here and again, 185 00:27:47.515 --> 00:27:49.019 so these, 186 00:27:49.555 --> 00:27:55.434 this evolution thing there and and so so, 187 00:27:55.434 --> 00:28:02.065 notation here and so each change state change is matrix multiply. 188 00:28:02.095 --> 00:28:13.434 We've seen this enough times. The ultimate evolution thing is showing, or is the equation that talks about for continuous systems we're talking about discreet system, you know, time, zero time, one time to. 189 00:28:13.765 --> 00:28:17.904 But this would be the physics thing, which we're not gonna talk about anymore. 190 00:28:18.775 --> 00:28:20.724 And so, 191 00:28:20.724 --> 00:28:22.884 this describes how the, 192 00:28:22.884 --> 00:28:24.865 all of the complexity is embedded, 193 00:28:25.134 --> 00:28:26.184 and that H, 194 00:28:26.184 --> 00:28:27.355 operator here, 195 00:28:27.714 --> 00:28:28.585 which that H, 196 00:28:28.585 --> 00:28:36.625 operator captures the physics of the system and it turns out you can solve this for a hydrogen atom, 197 00:28:36.984 --> 00:28:37.855 but not for anything. 198 00:28:37.855 --> 00:28:43.615 That's more complicated. So not exactly. So, okay. 199 00:28:46.404 --> 00:28:52.494 Okay, so now, so that was that anemic section. This is a complete summary here. 200 00:28:53.279 --> 00:29:08.095 You might apply to state by unitary matrix and matrix hesitant and inverse. You could go backwards. And this all comes down to equation. If we won't talk about anymore. Yeah. Okay. Assembling. Okay. 201 00:29:08.095 --> 00:29:16.644 So we've seen this before in great detail. I just wanted to hit you again. With this, because the book has at the section four point five. 202 00:29:18.535 --> 00:29:20.515 Well, if you wanna go, like, crazy numbering here. 203 00:29:22.644 --> 00:29:34.855 My contact management system numbers I told it to number every section. So this is my section two point five, but I type in the book section number just to help you find things and put the page number and Dell find things. Okay. 204 00:29:35.549 --> 00:29:45.595 So this is where we combine smaller quantum systems to make bigger quantum system, this is where attentive product comes in, and we get things that are entangled. So I showed you last time. 205 00:29:46.200 --> 00:29:54.295 I just hit it again, because for a few minutes, it's important. So, you've got two separate systems. Here's the system. 206 00:29:54.295 --> 00:30:06.954 Particle X could be and different places and difference and different states classically or two to the different states and quantum mechanically. Here's a different, unrelated system. 207 00:30:06.954 --> 00:30:10.644 Its particle Y, and M, different places and. 208 00:30:12.414 --> 00:30:23.964 Again, so classically as different states quantum mechanics would have to to the different states. Now, look you want to do is now talk about these two systems as a unit as one combined system. 209 00:30:24.509 --> 00:30:27.744 So that's why you bring in the tens of product and. 210 00:30:29.039 --> 00:30:38.994 So, the access that were to, to the end possible X stage to the impossible, why States? And to the M, plus and combo States because every possible. 211 00:30:40.319 --> 00:30:45.144 Super Super post date, and X can combine with every possible Super Bowl state for why. 212 00:30:46.345 --> 00:30:48.565 Okay, so that's what they're talking about. 213 00:30:50.519 --> 00:31:02.545 Here so that's your tenser product and they're writing it down here as again every possible X date zero, two X, minus one, combined with every possible Y state. 214 00:31:02.545 --> 00:31:17.365 So the number of states multiplies, these are the separate basis States, but then it is and and time some of these, but then they are any set a weights here. So that's how you get the power thing. 215 00:31:18.204 --> 00:31:22.194 Okay and you can write into something like this. 216 00:31:24.174 --> 00:31:36.204 There are end times and coefficients here the sees seize time. These are your basis factors, every combo over your size or basis factors for the X system. Your Y, supervise. 217 00:31:36.539 --> 00:31:50.214 There are basically just subjective basis factors for the Y, system, and the basic sector for the combo. Are here every combination of spaces Dr with the wide basis factor? That's what the Tensor product does. Okay. 218 00:31:50.214 --> 00:31:55.884 So, this end times and combos of the basis factors. So there's any times and coefficients. 219 00:31:57.924 --> 00:32:04.224 And so so the things that grows exponentially. 220 00:32:05.184 --> 00:32:16.914 And the amplitude squared will give us a probability of it being in one of these. And again, the amplitude Square, just Congress conjugate times the thing. Okay. And. 221 00:32:18.119 --> 00:32:29.214 Give it specific example here is two policy places wrecks and two places for why? And we've got four combos for the basic basis things here. Okay. 222 00:32:34.525 --> 00:32:42.505 And for something like this, where there are four again to two possibilities for two possibilities for why. 223 00:32:44.190 --> 00:32:58.224 But the extra can be super, super, both wise. You get something like this. So you for waits are I one minus? I, to and minus one minus side and the probability of being and say. 224 00:33:02.634 --> 00:33:10.765 One one, we have two probability of over here on the right then it's the magnitude to minus one minus high. 225 00:33:12.569 --> 00:33:25.615 Divided by that and so on. No, no, no this is not minus one minus. I squared because the vertical bars here means find the link. Okay and we get something like that. 226 00:33:30.474 --> 00:33:31.134 Okay. 227 00:33:33.474 --> 00:33:45.894 And now, one thing again, just seeing tanglement again is sad, I will go back up to this thing right here. Then. 228 00:33:48.595 --> 00:34:00.835 Well, could these for weights up here if I bring him up here, could they be created by doing attends or property of the two separate systems if they could untangle well, what again? 229 00:34:00.835 --> 00:34:11.724 So, what and tanglement means that if you observe the first article, it now constraints what you are allowed to see when you observe the second particle. 230 00:34:13.255 --> 00:34:20.994 I have a nice example here on the left put in then we take spend now. 231 00:34:22.434 --> 00:34:36.204 You can sometimes in nuclear physics, create a pair, a particles, like a pair of electrons or electron positron from nothing I'd say and or from some neutral article with no spin. 232 00:34:36.235 --> 00:34:50.965 I mean, some particles of no spend. So, let's suppose you create a pair of articles and the, it's a type of article that has a spin. Now, the initial neutral thing you started with had no spin no. Spin is conserve. 233 00:34:50.965 --> 00:35:04.644 It's a fundamental concern, property and atomic physics. So, if you create two particles, they're going to have opposite spins because they spins have to add up to zero. So, now those two particles are entangled. 234 00:35:05.545 --> 00:35:19.525 So, if you observe the spin of one particle, that will then mean that the spin of the other particle has to be the opposite if you're observing it with respect to the same access. So that's what's. 235 00:35:21.985 --> 00:35:33.505 The total spend is conserve and again, this would apply if you took the other particle move to the thousand miles away, if you could keep it isolated for the rest of the universe. 236 00:35:36.264 --> 00:35:42.355 As you move to a thousand miles away, and you observed it, it's been would be the opposite. So that's in tanglement. So. 237 00:35:43.500 --> 00:35:50.514 Okay, couldn't walk to the example, maybe that somebody wants, but okay. 238 00:35:53.695 --> 00:36:02.934 So, that's what I hit this enough times that may be you sort of have the theme to the feeling of it. Oh, okay. And that's what they're talking about there. 239 00:36:03.929 --> 00:36:12.925 Again, and again, and again, now, if we had something like down here, this would be a case where. 240 00:36:13.824 --> 00:36:25.795 It's well, could somebody tell me if somebody would like to talk exercise four or five three here that I've got up? Is that a separable system? 241 00:36:29.250 --> 00:36:34.554 What do you think if anyone would like to unmute themselves and chime in. 242 00:36:37.679 --> 00:36:38.605 Any idea. 243 00:37:01.014 --> 00:37:06.565 Anyone yes, I don't think there will be they will be separate. 244 00:37:08.034 --> 00:37:15.054 Why not? You don't okay. Cargo anyone disagree. Do you agree with Ricardo? Or do you disagree with them? 245 00:37:24.625 --> 00:37:25.675 At say. 246 00:37:28.590 --> 00:37:40.405 I got one, two, three, four or five six. I've got seven other people in class. Anyone else. 247 00:37:43.795 --> 00:37:46.554 I think you could factor out the one. 248 00:37:48.594 --> 00:37:52.585 Totally hearing you, John. Did you factor out? Why one. 249 00:37:53.934 --> 00:38:00.894 Well, that's so you disagree you could factor out why one? Let's see. 250 00:38:02.639 --> 00:38:09.235 Amanda, what do you think anyone chase Cohen. 251 00:38:12.690 --> 00:38:24.715 Potter gunner, MCO and anyone else. I saw a lot more people not to send the whole pile. Someone who's been quiet. Do you think you can factor it out? 252 00:38:24.715 --> 00:38:33.744 We've got one person saying this cannot be factored in the second person saying it can be factored, but someone else, like to give an opinion and go for two to three. 253 00:38:36.025 --> 00:38:50.005 I would assume you can't because they're entangled, right? Oh, that's a question. Not a statement. The previous paragraph up. He was talking about the previous set of numbers for five three. 254 00:38:50.005 --> 00:38:51.474 Now change the weights here. 255 00:38:55.195 --> 00:38:57.985 So example, 256 00:38:57.985 --> 00:38:59.184 four or five two, 257 00:38:59.875 --> 00:39:01.704 and if I scroll back up, 258 00:39:03.355 --> 00:39:05.514 had different way, 259 00:39:05.635 --> 00:39:07.284 because four or five to if you look at that, 260 00:39:07.525 --> 00:39:09.445 you see it that says zero Y, 261 00:39:09.445 --> 00:39:10.974 must be zero one. 262 00:39:10.974 --> 00:39:13.974 Why must be one? And here. 263 00:39:20.695 --> 00:39:21.355 What do you think. 264 00:39:24.025 --> 00:39:35.335 Any ideas in that case, maybe we could fact that I'm honestly not sure. Yeah. Okay. I'm hearing to, for two to three that we can, I would agree with the majority opinion here, because right. 265 00:39:35.364 --> 00:39:47.815 Why has to be what, why, the second part of goes in state? Y, one, the first could be one equals equal probability. So reserving. 266 00:39:50.155 --> 00:39:59.065 Either one doesn't affect the other one. Yeah. It's a little tricky because why is only one state there? Why? So one? So I would say they are separable. Yeah. 267 00:40:01.199 --> 00:40:04.375 Yeah, okay. I don't have a I don't have an answer book at the moment so. 268 00:40:06.059 --> 00:40:18.085 My opinion, okay. Any case. So now they talk about spins and so I gave the intellectual content of spin and that you have the summary here? 269 00:40:19.500 --> 00:40:22.224 Well, we've seen this twice before maybe. So. 270 00:40:23.429 --> 00:40:23.789 Yeah, 271 00:40:23.784 --> 00:40:25.224 okay now okay now, 272 00:40:26.545 --> 00:40:29.485 if you remember what's different classical to quantum, 273 00:40:29.815 --> 00:40:38.695 because it can you go you can do tens of products a classical stage what you cannot do with a classical system and tangled them. 274 00:40:40.260 --> 00:40:52.855 So that's the difference. I think the math would be the same, except there's no classical operators that entangle the classical stage. So, okay, that was chapter four. 275 00:40:54.150 --> 00:40:57.985 By what did I type about chapter five. 276 00:41:00.144 --> 00:41:07.135 Okay, so we've seen again chapter five before and let me sprinkler touch. 277 00:41:11.695 --> 00:41:12.295 Okay. 278 00:41:15.864 --> 00:41:25.974 Cool. Okay so we seen and are good. Yeah, I could fix my typing. So we've seen this before chapter. 279 00:41:28.650 --> 00:41:32.184 There's the stuff here, but it's good to see. Is it maybe more. 280 00:41:33.360 --> 00:41:46.465 Systematically, I just mentioned a spelling thing about too, but people have strong opinions about how it should be spelled to travelers half the people, put you in strongly and have the people strongly feel you should not put a you. And so, okay. 281 00:41:47.155 --> 00:41:50.335 That's the cool thing about this chapter. 282 00:41:51.144 --> 00:42:01.375 It uses the quantum mechanical location on classical Gates and why this is nice is if you see and stuff, 283 00:42:01.375 --> 00:42:04.554 you already understand if you so, 284 00:42:04.554 --> 00:42:08.965 by whatever color or whatever I put as a prerequisite. 285 00:42:10.554 --> 00:42:16.164 You understand the classical Gates, and you see them written with the quantum notation then this. 286 00:42:17.605 --> 00:42:26.184 May help you understand the quantum location better and key classical you got a bit zero or run. 287 00:42:27.534 --> 00:42:42.144 And but now we're using the quantum notation for the classical bit, and getting classical bits, you know, the gates and stuff like switches and stuff like that. So, the classical bit, it could be a zero or a one. 288 00:42:43.074 --> 00:42:57.235 And think about equation five one. Okay, so the bracket location. This is the stage zero and state run. So, what this means is a hundred percent chance of being in stage zero. I know. 289 00:42:57.565 --> 00:43:03.204 Zero percent chance of being in state one here. So it's so touch confusing. 290 00:43:03.204 --> 00:43:17.335 But the build phase are the basis States there are what possibly run and then inside the matrix is the weight of it being in the possible basis state. 291 00:43:17.335 --> 00:43:27.054 So, this is probably wanna being in stage zero. The bottom one probably zero being in state zero. Okay. So, it's using this quantum notation for the classical States. 292 00:43:30.025 --> 00:43:31.074 Oh, okay. 293 00:43:34.945 --> 00:43:41.695 The difference with the quantum states, classically the way towards zero and one quantumly the weights or complex numbers that. 294 00:43:43.409 --> 00:43:53.034 Is probability some to one and again, the vertical bar here is meaning finding like, the length of a thing. Okay. The probability. 295 00:43:57.324 --> 00:44:06.355 What's happening here is that or observing the this cute but. 296 00:44:08.724 --> 00:44:22.585 Something that has not been emphasized much before is when you will deserve wrecked, you might possibly see depends on what the observation operator and the measurement operator is. 297 00:44:23.605 --> 00:44:37.769 So this particular measurement operator that they haven't written down as possible observations, one, zero or is there a one? A different measurement operator would a different possible things. 298 00:44:37.795 --> 00:44:41.485 You could see this becomes important later. 299 00:44:41.815 --> 00:44:56.755 It hasn't really been emphasize up to now that, you know, all these are not necessarily the possible things that you see the possible things that you see, because of this specific measurement matrix that you're using. They're add to the icon values. 300 00:44:57.715 --> 00:45:00.025 The measurement matrix. 301 00:45:03.144 --> 00:45:17.125 Okay, any case so the squiggly line means you see one or the other, and when you observe, and again with probability of going to the one or the other are, is that okay you've seen that before. 302 00:45:20.695 --> 00:45:35.454 And again, things are not necessarily some normalize them. They normalize them where they don't depending, which is easier. It doesn't affect anything normalizing. Okay. 303 00:45:40.434 --> 00:45:43.704 Oops, okay cool. A little thing down here. 304 00:45:46.164 --> 00:45:56.664 Is that we haven't talked about building a quantum computer yet so the thing it was worked out, it's interesting. Looking at the progress of science and engineering. 305 00:45:58.764 --> 00:46:10.914 Sometimes the experimentation that you call it, the hacking in the sense of white hat hacking comes first, and after the hacking, the thier additions come in to explain it. And sometimes it's the irritation. 306 00:46:10.914 --> 00:46:11.994 Theoretically, 307 00:46:12.054 --> 00:46:16.704 do something predict something and then later on people try to build it, 308 00:46:19.735 --> 00:46:27.445 give you an example of a lot of the advances in physics where the experimental was observed something, 309 00:46:27.445 --> 00:46:29.335 the observed orbits of the planets. 310 00:46:29.844 --> 00:46:35.425 And then they had to try to put some theory into that. 311 00:46:35.425 --> 00:46:48.534 So the experimentation came first electro magnetism, observing the effect of a moving magnet on a wire or occur and so on moving maggot would induce occurred and the wire occurred. 312 00:46:48.534 --> 00:46:57.114 And the wire wouldn't cause a magnetic field this was observed, and then they had to come up with things ultimately, and to explain. 313 00:46:57.114 --> 00:47:09.625 So hacking came before theory and other times it comes before hacking, or becomes before implementation let's say and so in the. 314 00:47:10.110 --> 00:47:19.554 thirty's mathematicians, computer scientists look at the theory of universal computers and universe. 315 00:47:19.554 --> 00:47:30.565 So turning machine, for example, I mean, turning machine was not something to build it was it I theoretical construct. They'd said you could have one. 316 00:47:31.914 --> 00:47:37.885 Computer with machine, that would compute any computable form. She didn't have to have a separate machine for every function. You wanted to compute. 317 00:47:39.414 --> 00:47:43.014 And so that was worked out theoretically, before computers were built. 318 00:47:45.204 --> 00:47:53.454 With quantum computing. Same thing the theory was worked out a few decades ago. Richard. Fine, man, some of you have heard. 319 00:47:53.454 --> 00:48:01.135 I was one of the people involved, and it's being worked out thirty four years ago, give or take before any machines were built. 320 00:48:01.135 --> 00:48:14.454 So, where people said is elected, they look at quantum physics, and they said within tangle States and so on and they said this is interesting if we could ever actually build it. 321 00:48:14.485 --> 00:48:16.585 We might have a powerful computer. 322 00:48:18.090 --> 00:48:30.565 Okay, so the question is good. If we could build it this would be nice. How can we build it? We got to somehow it work with Quanta and so these are possible ideas. 323 00:48:31.014 --> 00:48:44.695 How you would do it, and someone can work with electrons, photons, various other particles and then people sort of fanned out and worked on all these different ideas competing with Joe to see. Are you get an IBM some machine with? 324 00:48:44.695 --> 00:48:50.454 You can read up ahead if you want uses one technology, there are competitors that use other technologies. 325 00:48:53.094 --> 00:49:04.344 Okay, so the question is, how do you have and they talk about it more chapter eleven and you're free to read ahead. Okay. Any case so we combine you've seen this so much of it maybe. 326 00:49:05.005 --> 00:49:19.675 Okay, so classically again, one of them is one, the rest are zero. These are the tens of product of all the various to make a bite. Let's say, classically quantumly. They're all. 327 00:49:20.760 --> 00:49:31.585 You're all. Wait. So okay, you've seen that again and again and again. Okay. Class come back? No. Okay. No section type two. 328 00:49:31.795 --> 00:49:46.585 They're talking about classical Gates now while you're talking about, it isn't going to represent them as unitary matrices. So, and I'm not gonna hand write the stuff here and I'll walk you through it, because writing down and matrices takes too long and you can't read my handwriting. Anyway. 329 00:49:47.125 --> 00:49:58.105 So so so if I page back, Here's an update, turns a one into zero and vice versa. So this could be done as a matrix here. 330 00:49:58.105 --> 00:50:02.215 So so so this is our state of the system. 331 00:50:02.215 --> 00:50:13.675 This means that stage zero, because the, and it's hundred percent chance when you say zero we apply the not matrix and we get this converts it other Gates. 332 00:50:13.675 --> 00:50:26.545 And gate would be this matrix, it's a four by it's a two by four and not was two by two because he had his two inputs. So we write it like this. And again, it's classically so, only one of the. 333 00:50:29.215 --> 00:50:38.574 So only one of the in this vertical vector column back to here, only one of the entries is one and the rest are zero. And this means that both inputs are one and we. 334 00:50:39.539 --> 00:50:40.914 Okay, so it's. 335 00:50:44.880 --> 00:50:59.815 So, it's using the quantum rotation we've seen in the classical case, and we have this notation down here, for example, over here as an example, I get the bracket notation on the left is left of the vertical bars a matrix, right? The vertical bar. 336 00:51:00.804 --> 00:51:15.684 Is the state of the system could be written as a column vector? Etc is here for them to record. And this is the output and here is two inputs in one output. All this doesn't. It's not invulnerable, but was not a square matrix. So, you can figure that out. 337 00:51:16.195 --> 00:51:16.974 Okay. 338 00:51:21.355 --> 00:51:33.864 Of course, you could put in some quantum mechanics or something like that. You gotta get numbers out, but they're meaningless. Okay. Or Gates a different matrix navigate is a different matrix still. 339 00:51:40.974 --> 00:51:50.664 No. And so on. And you could, if you have to make receipts one after the other two operations and a circuit, like a, then you just multiply the matrices and tricks. 340 00:51:50.664 --> 00:52:04.914 Okay, hopefully that notation there, the little slash with the number of guess how many bits table? How many it's there this is all totally trivial. Yeah, they don't the order matters. 341 00:52:06.954 --> 00:52:21.835 Don't commute in general do do do do oh, okay. This is go back. Now. This is cool up here. Okay so we've got three separate things. Three separate operators. 342 00:52:23.579 --> 00:52:36.445 So, we got a knock on one variable we've got the end, which combines to and makes one and so up until here going into the, or we've got two separate systems. 343 00:52:36.445 --> 00:52:45.204 Now, what we do is we do a tenth of product of these two separate systems to make one system with two variables, and then apply this matrix to it. 344 00:52:45.744 --> 00:52:52.135 So, this example, five to two shows combining matrices and also. 345 00:52:54.025 --> 00:52:55.675 Doing cancer products up here, 346 00:52:57.355 --> 00:52:59.755 so they're not in the end don't affect each other, 347 00:52:59.784 --> 00:53:01.855 but we want to consider them as one system, 348 00:53:01.855 --> 00:53:03.594 because they're feeding in to the, 349 00:53:03.594 --> 00:53:08.275 or and okay so not in the end, 350 00:53:08.695 --> 00:53:11.094 they're not supplying to one variable. 351 00:53:11.094 --> 00:53:20.965 The end is applying to different variables so we get this matrix here. That's the combo. That's the Tensor product of them. 352 00:53:22.585 --> 00:53:27.235 And and again, 353 00:53:27.235 --> 00:53:30.804 the size is because you've got three inputs into outputs here, 354 00:53:31.315 --> 00:53:39.985 and then you combine it with the or you multiplied by the or and you're gonna get this edit shape is because you got three inputs in one output. 355 00:53:39.985 --> 00:53:48.445 So, if you've got three inputs, you're gonna have two to the three columns, one output. You've got two to the one rows. 356 00:53:49.409 --> 00:53:55.855 Okay, this is a cool example where it shows for the classical case, tens of products and composing matrices. 357 00:54:00.264 --> 00:54:15.144 So, to understand the quantum notation, and you can also improve the Morgan's laws by just Matrix, spotless matrix modification and so on, by the way, if you were going to do this very much, you probably want to find a nice matrix package. 358 00:54:15.175 --> 00:54:19.284 I would recommend something like this map Labs possibly easiest way to do it. So. 359 00:54:21.864 --> 00:54:22.675 Okay, 360 00:54:26.244 --> 00:54:33.534 here's another example two separate not now we want you to the tens of product of these two variables, 361 00:54:33.534 --> 00:54:38.545 because we're gonna bring them together into the and so the tens of product of these to not get that. 362 00:54:39.594 --> 00:54:53.454 And then we apply, we're gonna apply it and we're going to apply and not and this should get the same effect as the or and again. So. 363 00:54:54.715 --> 00:55:06.324 Our system stayed as a column factor, so it's gonna be on the. Right? So, these agencies are being applied with the right? Most one first second. Third. Okay so this is the right most matrix serious attentive product is the two not. 364 00:55:06.594 --> 00:55:13.945 The second matrix is the and the third matrix is the, not if we multiply the three major SES, we should get a matrix. That's the or. 365 00:55:14.880 --> 00:55:23.364 Okay, so I suppose that you had an exam and you're asked approved Mark and this law. This would be a valid proof if you could ever convince the greater. 366 00:55:24.420 --> 00:55:38.755 Perfectly valid proof in fact, but okay til we got that I gave you a next exercise here on the homework. Another homework. I forgot to mention as an exercise to do this on the one bit at or just to have fun. 367 00:55:39.295 --> 00:55:42.025 Okay, that was classical gate. 368 00:55:45.235 --> 00:55:56.275 Okay, so the classical gates are mostly not reversible. We're not square. They got fewer inputs and outputs actually. Oh, Here's a cool idea. Here. This land hours principle. 369 00:55:59.695 --> 00:56:07.914 This principle is that what creates entropy and dissipates energy is a racing stuff, 370 00:56:08.605 --> 00:56:10.105 and you can read it, 371 00:56:10.344 --> 00:56:13.074 but basically writing information, 372 00:56:13.074 --> 00:56:13.974 thousand times, 373 00:56:16.675 --> 00:56:19.045 use energy to eracing information. 374 00:56:19.974 --> 00:56:31.644 Because I can't bring two pages up at the same time here. If something is reversible that doesn't create enter Peter and then theory doesn't use energy. 375 00:56:32.489 --> 00:56:46.764 So the theory here is that if you did reversible computations, even classically, then the computer would use less energy, which means you get miniaturized more and make them faster. So any case you can go to that here. But Tom. 376 00:56:48.534 --> 00:57:01.554 Okay. So that so there's a nice example. Here, so if you write on a blackboard, that's reversible. You can erase the blackboard, but if you erase the blackboard, that's not reversible because you don't know what you were raised and anticipating for the quantum computing. 377 00:57:01.554 --> 00:57:12.684 We work with a reversible operators. So Here's a concept here. Not gate by itself erosive all the immediacy and the control knock gate here. 378 00:57:14.219 --> 00:57:21.445 It's also reversible, but the broader point of this gate here is controlled not gate. 379 00:57:22.795 --> 00:57:23.125 Well, 380 00:57:23.125 --> 00:57:23.605 controlled, 381 00:57:23.605 --> 00:57:24.474 not being, 382 00:57:24.505 --> 00:57:38.125 is that why is the control and access the control and effects is zero then then I disclose to to the right if is one then why gets inverted? 383 00:57:38.400 --> 00:57:52.795 So the not here is only active effects is true. One is the way writing it here. It's explored why? And it's reversible, but this is a general concept. We'll see. Later you've got some gate, which could have one, two, three as many inputs as you want. 384 00:57:53.034 --> 00:57:55.315 And again is controlled by another input. 385 00:57:56.244 --> 00:58:08.844 And so if the gauges and inputs, we take an plus one input, the controls, whether the gate is active or not, it's a controlling input as false. The gate does nothing. It just identity is a controlling input is true. 386 00:58:08.844 --> 00:58:13.494 Then the gate operates, and this is an idea. Excuse me? That we see. 387 00:58:15.264 --> 00:58:21.894 Okay, this is a controlled not as a matrix edits. 388 00:58:21.894 --> 00:58:34.824 Reversible matrix can be inverted and this is reversible and actually to control it's own. Inverse. Actually important. 389 00:58:36.054 --> 00:58:48.505 You apply to control? Not send it that's out to the identity. So, he took the matrix up here, five, fifty, nine and you square did you get the identity Matrix? Okay. 390 00:58:51.355 --> 00:58:53.335 You do, is that's what it talks about there. 391 00:58:55.045 --> 00:58:55.255 No, 392 00:58:55.255 --> 00:59:07.074 it starts you mentioned a couple of specific gates that I've mentioned before the toppling gate has two controlling inputs and if they're both true then Z, 393 00:59:07.795 --> 00:59:08.244 or is it, 394 00:59:08.244 --> 00:59:12.114 depending where you're from gets inversion and that's how it would be written. 395 00:59:13.585 --> 00:59:22.945 And, okay, and again, it's if you square multiplied by itself, it comes back to the identity. 396 00:59:23.880 --> 00:59:32.394 So, it's reversible, it's its own reverse and you gotta matrix here for the eight by eight because you got three inputs in three outputs. So. 397 00:59:35.514 --> 00:59:48.625 And this example, five, three run what it's doing is, is it taking three toppling Gates and making a three controlling betting. 398 00:59:48.625 --> 01:00:03.565 So hopefully gate had two controllers. If I put three of them like this. What I get is a, not with three controllers and W gets inverted. If and only X Y, and Z are all true. 399 01:00:26.304 --> 01:00:26.485 Oh, 400 01:00:26.485 --> 01:00:28.045 this universal concepts of the, 401 01:00:29.335 --> 01:00:30.655 from the toffee gate, 402 01:00:30.744 --> 01:00:40.554 you can make any other gate just like again and your computer engineering fundamental course with a not big gate you can, 403 01:00:40.554 --> 01:00:40.885 from an, 404 01:00:41.724 --> 01:00:44.664 you can make every other gate or from Amanda gate, 405 01:00:44.664 --> 01:00:45.804 you can make every other gate, 406 01:00:46.105 --> 01:00:47.215 so their universal. 407 01:00:49.195 --> 01:00:51.445 The top gate universal. Okay. 408 01:00:54.655 --> 01:01:04.855 And they show here so it's the one I taught you have we people talk about the top to get because having two inputs gives you some freedom. Excuse me. 409 01:01:05.905 --> 01:01:20.005 So, here, if you want to do, and you make the third input zero and post the inputs X, and Y are true, then zero gets reversed and to X and Y. 410 01:01:21.420 --> 01:01:29.184 The one, but it's one only X and Y are to so this is using a softly to implement and but not gay. 411 01:01:29.184 --> 01:01:41.635 Tom is just that if you make X and Y run, then Z always gets inverted. So it's a not great. Okay. So here's a cool thing here. This is. 412 01:01:43.110 --> 01:01:47.394 The span out, this is a difference between. 413 01:01:50.065 --> 01:01:51.744 Classical and. 414 01:01:54.204 --> 01:02:03.864 And to him, this is a pat out this circuit here or five sixty six well duplicate it's second input. 415 01:02:04.980 --> 01:02:11.125 But I told you for quantum quantum computation is no cloning allowed. 416 01:02:12.000 --> 01:02:15.594 So, in computation, you cannot duplicate a Cupid. 417 01:02:16.469 --> 01:02:31.135 So, if you're anticipating me, this is not gonna work, right? For in the crowd him case we'll see. Oh, it fails. But in the classical case, what this does is, this would duplicate why? 418 01:02:31.824 --> 01:02:46.014 So we make X always one. Why is whatever you want Z is always one and so what will happen is that it will complement. This is the zero G is always zero. I'm sorry it will complement zero. 419 01:02:46.014 --> 01:02:53.425 If is one. So, why is one and the output here becomes one also. 420 01:02:53.425 --> 01:03:04.704 So what happens is the input ad to fix things and then why could be there on the output why got now copied handout to two separate outputs. 421 01:03:05.789 --> 01:03:15.565 So so this is using the classical portfolio gate to do the classical the gate to do cloning. 422 01:03:16.764 --> 01:03:28.525 Okay and it's gonna have a matrix and so on. But that's sort of cool is another game here. The Fred can gate. 423 01:03:33.204 --> 01:03:35.545 This does a controlled swap. 424 01:03:36.715 --> 01:03:50.364 Little X is there means swap effects zero wide passes through and see passes through effects is wrong. Y, and Z interchange. Excuse me too much talking. 425 01:03:53.724 --> 01:03:55.824 And what we get is here. 426 01:03:57.085 --> 01:04:09.744 Okay estimate to extend it to universal from so the Fred can engage the controlled swap gate and we can use it to get and Gates and stuff like that. 427 01:04:10.405 --> 01:04:17.844 Okay, that was classical. Now, using the quad notation on classical computer engineering. 428 01:04:19.195 --> 01:04:27.114 Totally simple for everyone. Okay. Now, quantumly you've got Darius operators. 429 01:04:29.275 --> 01:04:37.525 These are called poly, spend matrices for purpose of this. Course I don't care why, but these are matrices and. 430 01:04:40.375 --> 01:04:43.914 They're operating on one Cupid and flip. 431 01:04:47.454 --> 01:04:51.655 And so X is the not made the classical, not. 432 01:04:52.554 --> 01:05:05.905 And Y, and Z rotate into a complex space. I don't ask for some regional find the block sphere that interesting to myself. But, let me scroll up to do my comments here. 433 01:05:06.114 --> 01:05:12.534 But you may if you find it interesting, the wind is either rotating the cube on the blocks sphere. 434 01:05:16.139 --> 01:05:25.465 Can do lots of other matrices and so on rotate by an arbitrary angle. That doesn't have to be I Pi of report data. 435 01:05:27.780 --> 01:05:40.974 Okay, various things you can do we've seen this before the not Matrix has a square root. This is the square root of a not. You apply it twice you get a not okay. Sort of cool. We've seen that before. 436 01:05:41.905 --> 01:05:53.605 Okay, Bob. These are unitary matrices now measurement. It's not reversible. This is getting into the block sphere. I'm gonna page over it. 437 01:06:02.034 --> 01:06:02.844 Okay. 438 01:06:06.715 --> 01:06:14.425 Yeah, that's the thing can't use the box here for more than one cube. Okay. 439 01:06:15.539 --> 01:06:26.454 So this is the fiber run here. All probably one on one is what I showed you before mentioned, you can take any unitary matrix and control it with one more input. 440 01:06:27.565 --> 01:06:36.625 And then, but so, X controls, whether or not the carry matrix is active effects is zero. It's a straight pass through effects is one and it's active and. 441 01:06:38.400 --> 01:06:52.945 To show what you're doing here this and this is the operation you want to control and this is how they notate it a little Super scripts. See to the left of the, you of the matrix. 442 01:06:52.945 --> 01:06:59.184 So yeah. Okay, mathematicians like putting Super sub scripts on all sides of the main. 443 01:07:00.295 --> 01:07:02.934 Variable so okay. 444 01:07:09.835 --> 01:07:10.735 And. 445 01:07:12.324 --> 01:07:16.195 Header Mark will come back to this is controlling. 446 01:07:17.250 --> 01:07:25.704 Let's skip this, so these, I would get to more general rotation. Okay. 447 01:07:29.454 --> 01:07:32.425 Okay, what is happening here. 448 01:07:33.534 --> 01:07:47.635 I showed you that you could use a to clone a classical bit, but I told you and quantum beds. Cubics cannot be cloned. This is a fear. 449 01:07:48.894 --> 01:08:00.204 So, the question is what would happen if I apply that classical cloning circuit to acute it's a circuit. You know, you could apply what happens. 450 01:08:03.894 --> 01:08:04.764 And. 451 01:08:05.844 --> 01:08:08.125 Well, I'll give you the executive summary. It doesn't clone it. 452 01:08:09.684 --> 01:08:17.664 It works it through here and again so the executive summary is that if the inputs or zero and run. 453 01:08:18.654 --> 01:08:30.385 It is the input that you want to clone is zero and one it gets clone, but it's if it's a cubic with complex non, zero one rates that doesn't get cloned. 454 01:08:31.645 --> 01:08:39.444 And this works through the mass of that, so oh, yeah. 455 01:08:39.989 --> 01:08:46.734 And giving you, the executive summary might work out Thursday might not but if anyone's interested okay. 456 01:08:49.585 --> 01:09:04.524 So what you can do is, you can swap to bets can gate, but you cannot clone a bet. So, Star Trek, got it correct you could try to support someone, but you couldn't clone clerk. 457 01:09:04.944 --> 01:09:15.984 So, that's what I mentioned here. There's that gate, which will clone the classical bit, but it does not clone a Cupid. 458 01:09:16.920 --> 01:09:24.984 It'd be a nice homework edge question to play with maybe next time. Okay. Now they're talking about in detail here. 459 01:09:27.234 --> 01:09:37.284 So this is the cloning and if you apply quantum superposition things, so it doesn't pan out. 460 01:09:37.404 --> 01:09:49.770 That's the clone on a classical fit, but doesn't clone a Cupid so much of that showing. Okay. 461 01:09:53.604 --> 01:10:05.664 So that was, and one day I did chapters four and five, and again, there's a homework over here. That is to remind you what was happening. 462 01:10:06.414 --> 01:10:19.704 And so chapter four, we're just talking more about quantum states and to motivating examples of particle on the line and superposition. Going from classical quantum we've seen in several times. 463 01:10:20.154 --> 01:10:32.395 Spin is a big topic in quantum mechanics not so big in this course, but I mentioned it and again, so a measurement operator. There's lots of measurement. Make you sort of measure operator does it's a matrix. 464 01:10:32.395 --> 01:10:46.435 It's not an vertable and it's a matrix it's got eigen factors and I can values and it takes a quantum state and transitions that to one of the measurement operators eigen factors. 465 01:10:47.395 --> 01:10:49.494 And what's the value of, 466 01:10:49.765 --> 01:10:51.204 like the value so, 467 01:10:51.534 --> 01:11:02.664 and the Pro and so which eigen factor does this initial state get transition to is the doc product is the inner product of the initial state and the possible output States. 468 01:11:04.765 --> 01:11:15.265 And if it, perhaps the input was perpendicular to one of the output States, then zero observable is something you're observing the name or something or reserving, like position. 469 01:11:15.385 --> 01:11:19.885 So, and possible values their values. 470 01:11:20.274 --> 01:11:22.524 And what it ends up in is one of the, 471 01:11:22.645 --> 01:11:31.439 what's the state ends up in is one of the factors that's measuring with it dynamics is the state changes by applying these reversible convertible, 472 01:11:31.435 --> 01:11:43.074 unitary matrices and you assemble small quantum systems in the big quantum system was the Tensor product and but after you apply Matrix, 473 01:11:43.074 --> 01:11:43.645 it's possible. 474 01:11:43.645 --> 01:11:53.454 You cannot pick them apart again. If they're entangled another Matrix would make them separable. Okay. And then this architecture, where we had to make is to do operators. 475 01:11:53.880 --> 01:12:04.914 We saw it with the classical case, and then we took the quantum notation on the classical computer engineering, and then quantum thing. 476 01:12:06.085 --> 01:12:18.595 So he said there's a number of separate, a couple of Gates ones that are really common are not a portfolio red can. I mentioned how to mark we'll come back to that next chapter thing. 477 01:12:19.524 --> 01:12:31.135 It's what entangled stuff, and you can make any gate, a control gate and you can clone there's a gate that will clone a classical bit. But if you apply that gate to acute, but it doesn't clone it. It messes it up. 478 01:12:31.529 --> 01:12:34.645 No, so you can transport and swap stuff. You can clone stuff. 479 01:12:35.454 --> 01:12:35.635 Oh, 480 01:12:35.635 --> 01:12:39.085 this transporting by the way is the basis for, 481 01:12:39.600 --> 01:12:40.135 you know, 482 01:12:40.225 --> 01:12:45.564 on top of all communications because if you send me a cue bit, 483 01:12:45.564 --> 01:12:46.314 let's say, 484 01:12:46.704 --> 01:12:47.215 then, 485 01:12:49.284 --> 01:12:51.654 and I observed at a random angle. 486 01:12:51.685 --> 01:13:01.284 Well, if even the middle tries to ease drop, she's gonna change, it's gonna get project down one of the factors of her observation of our measurement matrix. 487 01:13:01.765 --> 01:13:10.409 And then, if I then observe it at some random angle, I can tell that it was projected down to some other thing that will affect what I see statistically. 488 01:13:10.795 --> 01:13:23.274 And if she, if my source sends me a number of Q bits, I can statistically determine whether or not, they were observed or not. This can also be used actually to establish the secret key. 489 01:13:25.104 --> 01:13:34.975 If somebody sends me some cute bits and I pick some of them. And send them back and do somebody changing like this a cue bits at the end of it? 490 01:13:35.005 --> 01:13:45.414 We have decided, on the secret key we can use the future communication so I'm not gonna exponentially speed up in the future. If I do two chapters today. 491 01:13:45.414 --> 01:13:53.875 Does not mean I do four chapters on Thursday, so you're safe, but any case feel free to read ahead. 492 01:13:55.225 --> 01:14:03.175 Okay, so what's happening chapter six I put a little bit on blurb that I typed in a while ago. We're gonna see algorithms to do things like. 493 01:14:04.824 --> 01:14:08.274 Determine which inputs and inputs which one and one of them is true. 494 01:14:08.274 --> 01:14:23.095 Which one is it factor and so now these algorithms are much more complicated than the classical algorithm and so so chapter six six starts 495 01:14:23.095 --> 01:14:26.545 getting a little more complicated now, 496 01:14:27.085 --> 01:14:29.725 given rolled into a false sense of security thinking, 497 01:14:29.725 --> 01:14:30.654 the thing is easy. 498 01:14:30.654 --> 01:14:32.154 So this gets more complicated. 499 01:14:33.024 --> 01:14:33.715 Okay, 500 01:14:33.984 --> 01:14:34.555 and then, 501 01:14:34.555 --> 01:14:34.944 of course, 502 01:14:34.944 --> 01:14:36.295 then with leading so, 503 01:14:36.689 --> 01:14:38.244 so the first part of the courses, 504 01:14:38.244 --> 01:14:38.604 like this, 505 01:14:38.604 --> 01:14:40.045 last two weeks, 506 01:14:40.045 --> 01:14:40.675 some general, 507 01:14:41.784 --> 01:14:48.295 general quantum stop the second part of the course a quantum computing algorithm to the third part of the course will be the Q machine and so, 508 01:14:48.295 --> 01:14:48.475 on, 509 01:14:49.104 --> 01:14:49.585 okay, 510 01:14:50.454 --> 01:14:54.835 excuse me open if anyone wants to talk to me, 511 01:14:54.835 --> 01:14:55.975 I'm available. 512 01:14:55.975 --> 01:15:03.715 If you don't then have a good weekend talk to you Thursday. So. 513 01:15:09.505 --> 01:15:14.965 Question okay, so I can. 514 01:15:17.250 --> 01:15:30.385 You know, see, okay and right. 515 01:15:32.784 --> 01:15:37.524 And again, if you're wondering why I'm on two different, why I'm signed in twice. 516 01:15:37.555 --> 01:15:40.465 It's because I have the, 517 01:15:42.295 --> 01:15:48.114 I've got one computer here is showing the hover cam and the text book, 518 01:15:48.564 --> 01:15:59.154 and my blog and the second computer is showing me the chat window and giving the approximation of what you see. 519 01:15:59.670 --> 01:16:03.925 So too much technology in front of me. 520 01:16:06.354 --> 01:16:20.694 You know, what it does is the reliability, which in the course of everything working as a product of their liabilities of all the separate machines. So, you know, the more machines I got, the lower, the probability the system actually works but, hey, that's computers. 521 01:16:21.085 --> 01:16:30.774 There's no redundancy here. Really? Redundancy would be a go to another computer. Okay. But no, and so thank you. You're welcome. 522 01:16:30.805 --> 01:16:36.444 And if there's put this up and it says no questions and. 523 01:16:39.324 --> 01:16:42.265 We'll stay around a minute or two in case. People have questions. 524 01:17:46.375 --> 01:17:49.885 Okay, good. He was down to two percent.