1 00:00:15,380 --> 00:00:20,020 So the previous talk, right. Told you maybe. 2 00:00:20,030 --> 00:00:23,810 Okay, if we have these large scale quantum simulators. 3 00:00:24,980 --> 00:00:30,380 What kind of questions can we answer as condensed matter theorists by using them? 4 00:00:31,600 --> 00:00:36,490 But of course, we also know that we shouldn't only be asking what can quantum computing do for us, 5 00:00:37,090 --> 00:00:40,960 but we should also be asking what can we do for quantum computing? Right. 6 00:00:41,200 --> 00:00:44,910 And. Well, we can do a whole lot, it turns out. 7 00:00:45,210 --> 00:00:50,700 And this would be the topic of my talk, which is the miracle that is quantum error correction. 8 00:00:50,700 --> 00:00:53,969 And I hope I will convince you that it is not surprising that it works. But it does. 9 00:00:53,970 --> 00:01:00,120 It does. It does work. And this is really the reason why we can can have any hope of building these machines. 10 00:01:02,330 --> 00:01:06,530 Okay, so you should know your enemy, but you should also know what you are protecting. 11 00:01:06,560 --> 00:01:10,250 So let's talk about what is actually classical and quantum information. 12 00:01:11,380 --> 00:01:13,840 So classical information is maybe a bit dull. 13 00:01:13,840 --> 00:01:22,629 It's zeros and ones that list the unit that we usually use is the bit which is just as you're in one and your computers, right. 14 00:01:22,630 --> 00:01:27,790 Your phones in your pockets, they all internally calculate with strings of zeros and ones. 15 00:01:29,850 --> 00:01:35,310 And zeros and ones. Okay. They either zero one yes or no. If you have been turning on, maybe your cat is dead or alive. 16 00:01:36,220 --> 00:01:43,270 And then we're already seeing where I'm getting here with quantum information, because quantum information is much more subtle. 17 00:01:44,690 --> 00:01:49,249 So quantum information, like the minimum information to describe it, 18 00:01:49,250 --> 00:01:54,020 needed to describe a quantum system of two levels, like an atom of two levels or something similar. 19 00:01:55,470 --> 00:02:01,980 Is a cubits and a cubit cannot only be in zero and one, it can be in a superposition we say. 20 00:02:02,760 --> 00:02:08,490 Right. And as we write it mathematically and this kind of well, it's a vector, technically speaking, 21 00:02:08,510 --> 00:02:14,340 we can write it as alpha zero plus better one where alpha and beta are complex numbers. 22 00:02:14,880 --> 00:02:18,600 Right. The magnitude has to their magnitude squared. It has to sum to one. 23 00:02:19,140 --> 00:02:24,120 And I can like visually represent this kind of state and the so-called block sphere. 24 00:02:24,180 --> 00:02:27,570 You might remember seeing this picture in in your quantum mechanics class. 25 00:02:28,740 --> 00:02:29,040 Right. 26 00:02:29,460 --> 00:02:39,510 That means that I can associate basically every con, every combination of efforts and betters that describe a valid quantum state with some angles. 27 00:02:40,560 --> 00:02:48,630 Like this. And then I can identify each point on the surface of the three dimensional sphere with one quantum state of a of a single cubits. 28 00:02:49,620 --> 00:02:52,900 Right. So this is much more complicated than just zero and one, right? 29 00:02:52,960 --> 00:02:57,630 Actually, there's a continuity of states. Right. Any point on on this fear. 30 00:02:58,350 --> 00:03:03,930 And it gets even weirder when you think about how we as classical beings interact with this quantum information. 31 00:03:04,890 --> 00:03:10,500 Because now. Right. If you want to look at something, right, look at the cubit, we have to measure it. 32 00:03:11,190 --> 00:03:18,929 And it turns out if you measure this thing, then it is indeed with certainty afterwards, it is a certainty. 33 00:03:18,930 --> 00:03:22,860 Zero or one. Right. So we get a probabilistic outcome, right? 34 00:03:23,280 --> 00:03:28,290 We get zero one. Yes, I know that. Live with probability alpha squared and beta squared. 35 00:03:28,530 --> 00:03:33,250 So it's a populist outcome. But after the measurement, the state of the cubit is fixed. 36 00:03:33,810 --> 00:03:39,660 Right. So once you open the box letting us cat is dead or alive, it's only both as long as you do not look. 37 00:03:42,970 --> 00:03:46,720 And this witness, in fact, can be utilised. 38 00:03:46,730 --> 00:03:50,480 But this is, of course, why I'm telling you about this today. 39 00:03:51,130 --> 00:03:57,100 Right. And actually the way it can be utilised. Right. Okay. And I see already told you a great bunch about these things, but I want to say okay, 40 00:03:57,250 --> 00:04:02,469 actually the first very first people to think about or want some of the very first people to think about the problem, 41 00:04:02,470 --> 00:04:08,650 how to utilise this witness of quantum mechanics actually where an Oxford blockchain user published the dojos algorithm, 42 00:04:08,650 --> 00:04:14,590 which is kind of a toy problem in some sense, but it was one of the first problems that actually did establish that quantum. 43 00:04:14,590 --> 00:04:21,610 We just can solve problems that classical computers cannot solve. Then of course, sure, they would need to show up on time. 44 00:04:21,610 --> 00:04:27,560 And it came along in 1994 and he published his prime factoring algorithm, which is maybe the most famous application, 45 00:04:27,580 --> 00:04:32,080 quote unquote, that we have today, at least from for exact quantum computers. 46 00:04:32,680 --> 00:04:37,720 But I also want to say, if I was under a cat, for example, here in Oxford, in the math department, 47 00:04:38,320 --> 00:04:45,610 he was the first person to think about using quantum computers, not for breaking encryption, but for actually encrypting things. 48 00:04:45,760 --> 00:04:52,370 Very savvy. And so this is quantum property. And each of these topics, honestly, would be a 45 minute lecture on their own. 49 00:04:53,210 --> 00:04:56,780 So I would not talk about that much more. You have to ask me afterwards. 50 00:04:57,890 --> 00:05:05,390 But what I will instead. Trying to tell you is how can we implement or how can we protect this information? 51 00:05:05,390 --> 00:05:08,750 Because if we want to use it, we have to be able to store it in some way. 52 00:05:11,690 --> 00:05:17,360 Okay, so maybe before I go too deep into this, how would you present Quantum de quantum? 53 00:05:17,360 --> 00:05:24,140 Who does look like? So what is the quantum information? Maybe the most similar version of quantum computers to to the machines. 54 00:05:24,170 --> 00:05:28,040 We could call it classical computers. Is the superconducting Hubert machines. 55 00:05:28,040 --> 00:05:35,590 Right. This chip flies out from the Google lab and it looks kind of like you would expect from a silicon processor, 56 00:05:36,440 --> 00:05:40,540 although it's made actually from superconducting aluminium instead. 57 00:05:42,290 --> 00:05:47,930 I mean, this is a very rich platform, so it's always been oxide. This is these kind of devices are built by the lab of Peter Leek. 58 00:05:48,320 --> 00:05:52,490 This picture is from Google Quantum and there's a major industry players now Quanta, 59 00:05:52,610 --> 00:05:57,349 Google Quantum, IBM quantum and it's involved being from a start up from France. 60 00:05:57,350 --> 00:06:01,010 So these things also are now get into the Start-Up space. 61 00:06:01,130 --> 00:06:05,000 But of course, the pioneering work, for example, a lot of pioneering was done in academia, 62 00:06:05,720 --> 00:06:12,920 mostly in the Yale Quantum Institute and also at Zurich at the Quantum Devices lab there. 63 00:06:14,140 --> 00:06:19,090 So that another major problem is trapped. IONS Unless he already mentioned them, 64 00:06:19,240 --> 00:06:27,310 kind of this is actually a picture of a trapped ion taken just in the basement of the Beecroft Building and the level of David Lucas. 65 00:06:27,820 --> 00:06:30,400 It's very beautifully actually captured with with the iPhone, I believe. 66 00:06:31,060 --> 00:06:35,650 And if you listen very closely to actually see the atom, this is this is kind of fascinating. 67 00:06:36,160 --> 00:06:39,640 And this is this is an ionised atom. So you trap it with a magnetic strip. 68 00:06:39,820 --> 00:06:47,200 This is why we call them them trapped ions. And actually, this was historically the first platform where quantum computing was performed. 69 00:06:47,710 --> 00:06:55,000 Right. David Wineland, who was at Nist's when he performed the work, received a Nobel Prize for that in 2012. 70 00:06:56,170 --> 00:07:02,110 Today, of course, there's there's again, major industry players continuing with close relations between University of Oxford, 71 00:07:02,890 --> 00:07:10,330 especially the computer science department here and then Ion. Q I think being actually the Quantum Start-Up was the biggest market capitalisation. 72 00:07:10,330 --> 00:07:15,520 I think so it was, I think, the first quantum unicorn. I'm not sure they're still that valued, but there were at some point. 73 00:07:17,500 --> 00:07:21,649 So. Uh, uh, another platform. 74 00:07:21,650 --> 00:07:24,910 And see, I already mentioned that our reconfigure atom arrays. 75 00:07:24,920 --> 00:07:30,890 So right here the atoms are neutral and you track them with a laser beam with something called an optical tweezer. 76 00:07:31,580 --> 00:07:37,250 And actually these I mean, this is maybe a more boring simulation than the than the Mario that was shown before. 77 00:07:38,030 --> 00:07:40,520 But if you want to do computation, this is more like what you will do, 78 00:07:40,730 --> 00:07:44,240 because you see that there is like these arrays and they get brought close together. 79 00:07:44,240 --> 00:07:47,780 Basically OnDemand when you want to perform at a gate, when you want to want to qubits, 80 00:07:47,780 --> 00:07:51,620 to interact, you bring them close together physically and otherwise you have them far apart. 81 00:07:52,250 --> 00:07:57,960 And this is kind of fascinating, has created a lot of trade recently because of this great flexibility. 82 00:07:57,980 --> 00:08:00,530 So this is a really promising platform that's emerging right now. 83 00:08:01,250 --> 00:08:05,059 And this also means that this I mean, at the moment, I would say there's one major player, 84 00:08:05,060 --> 00:08:11,600 which is the laboratory of looking at it, and they have also funded a company which is a computing. 85 00:08:12,290 --> 00:08:14,390 There's a bunch of other platforms. 86 00:08:15,260 --> 00:08:23,569 All of them have advantages, disadvantages, uh, I won't talk about these too much less, but tiny quantum windows as quantum unless they're spin cube. 87 00:08:23,570 --> 00:08:26,299 It's actually quantum motion and major play. And thank you. 88 00:08:26,300 --> 00:08:33,110 It's also again being founded actually at Oxford in the Material Science Department by Simon Benjamin. 89 00:08:33,720 --> 00:08:41,060 And so there's a lot of platforms that, that all have promises and none of them works just yet. 90 00:08:41,420 --> 00:08:48,710 And this is because all of them share one simple problem, and that is noise by quantum states are inherently fragile. 91 00:08:49,720 --> 00:08:53,960 And this is really also, of course, the reason why we don't see quantum mechanics around us, Right? 92 00:08:54,040 --> 00:08:59,140 Because quantum mechanics is fragile, quantum information decoherence, and it becomes classical. 93 00:08:59,800 --> 00:09:05,920 So in the end, if you screw things up and you do it without being very careful. 94 00:09:06,190 --> 00:09:11,400 Right. That is to say it becomes just classical information. And well, 95 00:09:11,430 --> 00:09:16,379 hopefully we convince you that there is a way to do it to prevent this classification of of 96 00:09:16,380 --> 00:09:20,730 the quantum information and maybe to really drive the point home of how important this is. 97 00:09:21,210 --> 00:09:23,670 Let me tell you the thing that is not often appreciated, 98 00:09:23,670 --> 00:09:29,580 which is that quantum computing is by far not the first alternative idea to implement computers beyond classical ones. 99 00:09:31,030 --> 00:09:32,530 Beyond the ones we built today. 100 00:09:33,100 --> 00:09:41,620 For example, my my fellow German and I should hug in 1979, published a beautiful paper on the power of random access machines. 101 00:09:42,310 --> 00:09:43,960 And he basically asked the question. 102 00:09:45,280 --> 00:09:51,130 Imagine you had something called a random access machine, which is basically a computer that operates on floating point arithmetic. 103 00:09:52,190 --> 00:09:55,459 The idea is that you have a memory that instead of bits stores, 104 00:09:55,460 --> 00:10:02,180 floating point numbers of arbitrary precision is not something we can do if we are on a computer, if we have a finite memory. 105 00:10:02,840 --> 00:10:04,340 Right. This is why it's different. 106 00:10:04,910 --> 00:10:11,239 And imagine also imagine you had this floating point point that you can store and you can also perform arbitrary arithmetic on them again, 107 00:10:11,240 --> 00:10:13,310 with arbitrary precision. Right. 108 00:10:13,610 --> 00:10:19,810 His idea actually, I think, was basically to build an analogue computer based on some I don't know, some way, some coherence. 109 00:10:22,040 --> 00:10:25,580 I think some oscillations in a in a tube or something like this. 110 00:10:25,730 --> 00:10:31,470 Right. Right, which then would correspond to a continuous number. 111 00:10:32,780 --> 00:10:36,490 But then he proved, okay, if you do this right, actually, 112 00:10:36,530 --> 00:10:42,990 then these random access machines will be vastly more powerful even than the quantum computers we build we try to build today, Right? 113 00:10:43,040 --> 00:10:45,290 So for random access machines is NPR. 114 00:10:46,010 --> 00:10:51,830 If that tells you anything, what it really means is that you can basically really solve optimisation problems you can solve. 115 00:10:51,860 --> 00:10:55,550 Travelling salesman All of this in polynomial time. That's the idea. 116 00:10:56,420 --> 00:10:59,840 So it would be extremely powerful, but they have never been built. And why is that? 117 00:10:59,870 --> 00:11:04,880 Because of noise. We have just not figured out a way to do floating point arithmetic well enough. 118 00:11:05,960 --> 00:11:13,460 But there's no known wire that this sends no known way to fall tolerant li implement these machines. 119 00:11:14,120 --> 00:11:19,050 And surprising fact about quantum computers is that they can be implemented for currently. 120 00:11:21,750 --> 00:11:24,780 Okay, So now before I talk about Quentin, who does, let me tell you. 121 00:11:24,780 --> 00:11:29,610 Okay, what does work anyway? So we want to protect classical information, maybe first. 122 00:11:29,640 --> 00:11:33,320 Let's start with that. Let's start with sending a bit. Right. 123 00:11:33,390 --> 00:11:38,480 So imagine Alice comes into a bar and she wants to order coffee at Bob and 124 00:11:38,480 --> 00:11:43,580 assemble by the names we usually give to these two people in computer science. 125 00:11:44,330 --> 00:11:46,909 And she said, okay, I want coffee. And then there's some noise. 126 00:11:46,910 --> 00:11:51,660 And actually business turns on his tea and she's good at getting a tea and this happens is probably deep. 127 00:11:52,600 --> 00:11:58,450 We can formalise this by saying, okay, you're sending a zero and it's probability P, it flips in this photo, one minus B it doesn't. 128 00:11:58,510 --> 00:11:59,980 Right? So this is the final version of that. 129 00:12:00,250 --> 00:12:05,620 And there's one simple form of error correction that we all do intuitively in a noisy bar we repeat ourselves. 130 00:12:07,120 --> 00:12:10,410 And it could be decisive because the repetition could. Right. 131 00:12:10,620 --> 00:12:13,710 So instead of saying I want coffee, I say, I want coffee, I want coffee, I want coffee. 132 00:12:14,340 --> 00:12:19,620 And they'd say, okay, now it that three times. And Bob was on a sensor, one's iced coffee, coffee. 133 00:12:19,860 --> 00:12:23,670 And he goes like, I guess she meant coffee. Right. 134 00:12:24,180 --> 00:12:28,680 Again, the former version of this, you implement a repetition code code to find something called a logical bit. 135 00:12:29,610 --> 00:12:32,830 And the logic in zero is just and times zero in this time. 136 00:12:32,880 --> 00:12:36,330 And here's and it's three times one. 137 00:12:37,020 --> 00:12:41,790 And if noise flips half of your bits right, then you can do something called majority. 138 00:12:41,790 --> 00:12:45,210 But the thing that is you asked, are there more zeros and ones in the message I received? 139 00:12:45,480 --> 00:12:51,600 You just assume it was the origin of man has had all ones or one zeros, right? 140 00:12:53,080 --> 00:12:56,230 And well, the pain probability then is lower because of this. 141 00:12:56,290 --> 00:13:00,550 Right. If this can still go wrong, if Bob actually misunderstands you twice. 142 00:13:01,330 --> 00:13:06,670 And of course this happens with probability P squares and if P is small, P squared is smaller than P. 143 00:13:07,800 --> 00:13:13,080 So that's that's the idea, right? Erick Erickson improves the fidelity of classic information. 144 00:13:13,080 --> 00:13:16,530 Even if you send it through a noisy channel, that would be maybe the form and statement here. 145 00:13:18,680 --> 00:13:25,460 Okay. But quantum error correction. So so really, people in the early nineties thought it was a futile effort. 146 00:13:26,820 --> 00:13:31,530 And there's a bunch of reasonable objections to to quantum error correction being possible. 147 00:13:32,100 --> 00:13:35,280 And maybe the most damning one is we cannot copy quantum information. 148 00:13:36,150 --> 00:13:40,680 And that seems to be a major objection to what I just told you that. Okay, how do you protect information? 149 00:13:40,680 --> 00:13:45,229 You copy it and send it multiple times. But for quantum information, actually, we cannot do that. 150 00:13:45,230 --> 00:13:50,750 And the proof is very simple. So let's attempt a proof. Okay, it goes the following. 151 00:13:51,170 --> 00:13:54,890 We have a proof by contradiction. So imagine you could do it. 152 00:13:56,050 --> 00:14:00,400 I imagine there was an operation. You. This is some unitary metric that Exxon tool. 153 00:14:00,710 --> 00:14:04,300 Cubits. Right. We want to copy an arbitrary state. 154 00:14:04,300 --> 00:14:09,130 Sire into this state e e can be whatever zero. 155 00:14:10,330 --> 00:14:13,930 And then after you apply, the operation has to be beside sire. And this has to work. 156 00:14:13,930 --> 00:14:17,259 Not for arbitrary states. Right? Because all copy machine should copy. 157 00:14:17,260 --> 00:14:21,499 Whatever. Right. Then we can write the following, right? 158 00:14:21,500 --> 00:14:24,050 You can rearrange this. So I'm working in bracket notation here. 159 00:14:24,060 --> 00:14:30,890 I hope you do somewhat remember, but what it means really is you consider the overlap of two states fly and say, 160 00:14:31,580 --> 00:14:35,239 and we assume that both of them can be copied by the machine. Right? 161 00:14:35,240 --> 00:14:41,270 And then this is just the overlap. So this is but it will be zero if they are very different and it will be one if they are the same. 162 00:14:41,900 --> 00:14:46,550 And then what I can do is I can insert this you dagger you here, which is just the identity. 163 00:14:46,910 --> 00:14:50,750 This just to say okay quantum and in quantum operations are unitary. 164 00:14:51,020 --> 00:14:54,319 This is mathematically the precise statement. But then I can also say, okay, 165 00:14:54,320 --> 00:14:58,879 now I can imagine dissecting to the left and dissecting to the right and what I would get upside 166 00:14:58,880 --> 00:15:04,100 PSI and Phi Phi and is of course this again the overlapped but squared because now I have a twice. 167 00:15:07,070 --> 00:15:12,260 And okay, so we have proof that the overlap between the two states is equal to the square of its overlap. 168 00:15:12,260 --> 00:15:16,140 But there's not a bunch of numbers. There's not many numbers where equal to their square, right? 169 00:15:16,900 --> 00:15:26,460 There's two of them, actually. This is zero and one. So either the states identical or they are orthogonal, but they are certainly not arbitrary. 170 00:15:27,190 --> 00:15:30,479 But and that is a simple proof that a sensible quantum operation, 171 00:15:30,480 --> 00:15:34,530 a unitary time evolution, cannot copy quantum information, at least not arbitrary one. 172 00:15:36,370 --> 00:15:40,210 Okay, so that is bad, but it gets even worse. It's a square. 173 00:15:41,860 --> 00:15:45,759 So measurements are destructive. I told you right after you looked at at the cat. 174 00:15:45,760 --> 00:15:49,120 Right. It's not that or alive anymore. Right. It's. 175 00:15:49,360 --> 00:15:52,910 I mean, it's either the two, but it's not both. Sorry. That way. Right. 176 00:15:52,930 --> 00:15:57,340 That means. Okay, we measured this thing, we measure that family. So we are asking, is it 0a1? 177 00:15:58,000 --> 00:16:01,360 Right? And then it will be zero or one with a respective probabilities. 178 00:16:02,460 --> 00:16:06,430 And that's. Well, so we how do we do majority voting then, Right. 179 00:16:06,450 --> 00:16:11,610 I told you. Right. You look at what you've got and you look what you've got more. But now you kind of look at your state because that will destroy it. 180 00:16:12,300 --> 00:16:19,660 That, again, seems very bad. But it gets even worse because all of the errors are continuous. 181 00:16:19,660 --> 00:16:25,660 Right? So I told you, the state of Cuba, just like can be identified with this point on the three dimensional sphere. 182 00:16:27,080 --> 00:16:31,250 But of course, errors are arbitrary. Small rotations then, Right. Any sensible operations. 183 00:16:31,250 --> 00:16:33,950 Right. This is why it's unitary. It's a rotation on the sphere. 184 00:16:35,290 --> 00:16:40,629 And of course, now this can be very small and this starts to look a lot like these floating points problem, right. 185 00:16:40,630 --> 00:16:43,900 Where we cannot even do floating point arithmetic. Correct. 186 00:16:44,440 --> 00:16:49,330 How can we do that correctly? But it seems like very, very, very bad. 187 00:16:49,930 --> 00:16:53,140 But again, surprisingly, no. 188 00:16:53,500 --> 00:16:57,640 I hope you don't get the surprise of people that really happened in 1994. 189 00:16:58,060 --> 00:17:01,330 Peter Shore and actually independently. Andrew Stephen of Oxford University. 190 00:17:02,710 --> 00:17:05,470 So it's that kind of error correction is indeed possible. 191 00:17:06,500 --> 00:17:13,069 But what I want you to appreciate is that we really need all of the witnesses and the beauty, I think, of quantum mechanics to make it work. 192 00:17:13,070 --> 00:17:17,240 So we need everything. We need entanglement. We need measurements. Superpositions. 193 00:17:18,050 --> 00:17:25,580 So, yeah. So buckle up. And I hope that I will try to be able to will try to explain to you how it works. 194 00:17:26,030 --> 00:17:30,920 But I will lose the shortcut because of the shortcut we have to for the steam code. 195 00:17:30,980 --> 00:17:37,340 The single works just as well. Right. And I would love to explain this to you because we are in Oxford, but unfortunately we need for that. 196 00:17:37,460 --> 00:17:43,190 You need a bit more information theory, background. So I think this is a bit more intuitive, I'm afraid. 197 00:17:44,350 --> 00:17:48,690 But okay, so let's do it. And let's start with the quantum repetition code. 198 00:17:49,180 --> 00:17:52,200 So what's the quantum version of our repeated coffee order? 199 00:17:53,830 --> 00:17:59,560 And the first observation is that cloning is not necessary because we can do entanglement instead. 200 00:18:00,260 --> 00:18:04,730 Okay. So unless you're. Fortunately already did some circuits. 201 00:18:04,940 --> 00:18:12,410 So again, just to repeat. Right. So each horizontal line here in this diagram by this this diagram presents an operation on a quantum computer. 202 00:18:12,860 --> 00:18:16,630 Each horizontal line is a qubit. Here we have in the nation state. 203 00:18:16,870 --> 00:18:20,719 And this is a gate. This is the gate is called the Senate. It's a control. 204 00:18:20,720 --> 00:18:24,490 It's not This gate actually makes perfect sense. Classically, right. 205 00:18:24,520 --> 00:18:28,080 It just says you apply a not to this cubit. 206 00:18:28,090 --> 00:18:31,990 If this cubit is one and you don't do it if it's zero. 207 00:18:33,360 --> 00:18:37,740 Right. So that means, okay, if this is zero, if the input would be zero zero, Right. 208 00:18:37,860 --> 00:18:43,170 Output would be again, zero zero. If the input would be one zero, the output would be one one. 209 00:18:43,200 --> 00:18:47,610 Because the first bit because it indicates where they should flip. So now this is a quantum computer. 210 00:18:48,180 --> 00:18:51,360 So it gets a bit more complicated than that because I can get as an input. 211 00:18:51,990 --> 00:18:57,780 Right. I can put a superposition of things and how does the gate X then we get this by linearity. 212 00:18:58,110 --> 00:19:05,160 So we can think here about what happens to each turn. And I'll just point out it's not okay. 213 00:19:05,560 --> 00:19:09,440 Right. We can think about what happens to each time separately and we can ask if it's zero. 214 00:19:09,440 --> 00:19:15,440 Right? I already told you zero zero becomes zero zero and this is the output first output term, and then here there is a plus better one. 215 00:19:15,920 --> 00:19:19,700 And okay, if the first Q it is one, the second gets flipped, so the output will be one one. 216 00:19:20,240 --> 00:19:27,060 And this is the second term in the output. This is somewhat clear. 217 00:19:27,240 --> 00:19:32,640 But the important thing is not that the output indeed looks a bit like a repeated information. 218 00:19:32,850 --> 00:19:36,270 Right. Because I've I've not copy it, but I've entangled the two qubit. 219 00:19:36,600 --> 00:19:41,910 Right. And entanglement to means just that. Okay. On the left hand side, you see, I can write the state of the two cubed separately. 220 00:19:42,030 --> 00:19:45,120 On the right hand side, I have to write this joint state. Right. This is what I. 221 00:19:45,120 --> 00:19:51,770 What I mean by putting it here. So we produce redundancy by entanglement instead of by cloning. 222 00:19:52,780 --> 00:19:56,860 And then the quantum version of our redundancy of us just be makes two with this twice, right? 223 00:19:56,890 --> 00:20:01,450 So we copy and we entangle upside with the second and the third qubit. 224 00:20:01,450 --> 00:20:07,120 And this now will be our logic, a quantum state of the quantum repetition got. 225 00:20:08,420 --> 00:20:13,000 This is not yet. We are not dead yet, but we are getting there. So what about measurements? 226 00:20:13,540 --> 00:20:18,670 So I told you, measurements are another problem. Right. So you see already going one by one through our objections. 227 00:20:19,840 --> 00:20:25,120 What about measurements? Okay, we have this logical stage IFR, 000 plus better one, one one. 228 00:20:25,120 --> 00:20:29,950 Right. This is a superposition of a superposition and an entangled state. 229 00:20:30,610 --> 00:20:35,500 Right. And imagine you measure Now, is that one. So you ask for the first qubit is a zero or one. 230 00:20:35,980 --> 00:20:40,040 Right? So formally, this means we measure the operators and one. Right. 231 00:20:40,190 --> 00:20:44,270 And then you. And then the thing is, of course, this has not a well-defined answer within the stage. 232 00:20:44,690 --> 00:20:48,770 Right. And what it means is, right, that the answer is different for the for the two terms. 233 00:20:48,800 --> 00:20:51,230 Right. And the first term would be zero. And the second one be one. 234 00:20:51,770 --> 00:20:58,040 So if you meant if we ask this question and get an answer, the state afterwards will be either 000 or 111. 235 00:20:58,370 --> 00:21:01,670 That's the destructive nature of a measurement. 236 00:21:03,600 --> 00:21:07,290 But instead what we can ask are the two other first two bids equal? 237 00:21:07,920 --> 00:21:11,220 I would ask a slightly different question and then actually it's okay, it turns out, 238 00:21:11,730 --> 00:21:16,050 because if you just ask formal you again you measure operate and it is that one's a two. 239 00:21:16,470 --> 00:21:22,110 But the question you are asking does is are the first two bids equal or equivalently are the last two bids equal? 240 00:21:22,170 --> 00:21:26,370 And that actually has a well defined answer within the state because it's the same in both the right. 241 00:21:26,790 --> 00:21:31,440 The first is 000. So all are the same. So the first two and the last two are the same. 242 00:21:31,860 --> 00:21:35,490 And the second term, again, they have a different value but that they are the same. 243 00:21:36,240 --> 00:21:43,890 So if you ask that question, we get a clear answer. And what this means physically is that if we measure this, if we really, 244 00:21:43,980 --> 00:21:49,440 really ask this question to nature and we get an answer, we can get this answer without destroying the state. 245 00:21:50,490 --> 00:21:53,490 Right, because we have asked the right question. 246 00:21:54,090 --> 00:21:59,180 And this is maybe the first witness of quantum mechanics that comes in here. And this is not important classically. 247 00:21:59,190 --> 00:22:05,160 We could also do this classically right in the sense of the question. But in classical four, classically we don't have to do it. 248 00:22:05,940 --> 00:22:11,530 Quantum mechanics forces us to do that and nothing else. Okay. 249 00:22:12,610 --> 00:22:17,020 Second part of the invitation. Quote, This is just a reminder of what we are doing so far. 250 00:22:17,830 --> 00:22:21,040 Now, let's consider that is actually protect us from an error. 251 00:22:22,130 --> 00:22:25,740 And let's consider one very special error. The big flip. 252 00:22:25,890 --> 00:22:29,190 Is this just the thing that flips 0 to 1 and 1 to 0? Hence the name. 253 00:22:29,400 --> 00:22:33,660 It's also politics symmetric. If you remember quantum mechanics one. 254 00:22:34,830 --> 00:22:40,830 So what happens to the encoded state? Let's do a table and ask how do these measurements outcomes change? 255 00:22:41,310 --> 00:22:45,330 Imagine we apply a bit, flip to the first cubit. What would be the state after? 256 00:22:45,330 --> 00:22:48,780 Once again, it follows from linearity might be 100011. 257 00:22:49,320 --> 00:22:53,770 The first bit is now flipped in both. And of course I can again ask, are the first two bits equal? 258 00:22:53,790 --> 00:22:58,669 And now the answer is no. But for the second one, the answer is still yes. 259 00:22:58,670 --> 00:23:02,389 The last two bits are still equal. Equivalently. Now. Right. I can ask the AK. 260 00:23:02,390 --> 00:23:06,800 What if I flipped the second cubits? Huh? I already see a pattern here. 261 00:23:06,830 --> 00:23:10,460 Right. So now the first two are not equal, and also the last two are not equal. 262 00:23:11,000 --> 00:23:14,240 So I get a different answer, actually, if I measure these things. 263 00:23:14,990 --> 00:23:21,710 And lastly. Okay, what is the last thing that happened? The third bit could flip and now the first four equal, but the last two are not. 264 00:23:22,490 --> 00:23:30,470 And of course, this is a very important observation here, is that all these different possible errors have a different outcome for these measurements. 265 00:23:31,580 --> 00:23:37,489 Right. So we can correct actually a single bit flip here in this coat by doing these two measurements. 266 00:23:37,490 --> 00:23:42,340 And then depending on the outcome. Right. If it was both equal, we would see nothing happened. 267 00:23:42,350 --> 00:23:46,100 We are good. If one of you get one of these outcomes right, 268 00:23:46,100 --> 00:23:49,880 then you know you have to do something and you even know what to do because you just have to reverse that error. 269 00:23:50,060 --> 00:23:54,530 So you apply another bit. Flip to cancer. The first one that happened accidentally. 270 00:23:56,300 --> 00:24:01,610 All right. No, this is not quite yet enough. 271 00:24:02,360 --> 00:24:05,840 So you have to bear with me for another second. So let's do another repetition. 272 00:24:06,030 --> 00:24:09,349 So what do I mean by this? Classically, there's only one. Right? 273 00:24:09,350 --> 00:24:12,649 And I want to stress the fact that so far, all the circuits that I've drawn to you. 274 00:24:12,650 --> 00:24:16,309 Right. But flip C, they all make perfectly sense on a classical computer. 275 00:24:16,310 --> 00:24:21,020 Actually, the only way quantum mechanics could came in was through the input state. 276 00:24:21,260 --> 00:24:23,630 Right. Because we put an entangled state in the beginning. 277 00:24:25,090 --> 00:24:31,090 And now it turns out and well, it's not surprising maybe that if you want to protect quantum information, we have to actually do something quantum. 278 00:24:32,660 --> 00:24:38,569 And for that, let's start maybe with the circuit of the bit flip code that we had before right side the dashed line here, 279 00:24:38,570 --> 00:24:44,180 the state of the circuit will be this logical state, 4000 plus better 111. 280 00:24:45,710 --> 00:24:52,520 And then let's add a layer of so-called hot air maggots. So what I had I might get you probably maybe you have never heard of them. 281 00:24:52,520 --> 00:24:57,290 And this is because they have no correspondence on a classical computer because they do superposition. 282 00:24:57,560 --> 00:25:02,330 So they take the zero stage to a superposition of zero and one with relative phase plus 283 00:25:02,960 --> 00:25:06,500 and the one state to another superposition with a different relative phase minus. 284 00:25:06,650 --> 00:25:12,320 So on the block sphere, this is a rotation by 90 degrees around the new Y axis. 285 00:25:13,070 --> 00:25:18,350 And this is of course, why it has not an equivalent operation on a classical computer, because the classical media only lives on the poles. 286 00:25:18,350 --> 00:25:21,710 Right. And now suddenly we are here on the equator of the blogosphere. 287 00:25:24,700 --> 00:25:28,509 Okay, so what does I do? So then I'll state afterwards this. 288 00:25:28,510 --> 00:25:31,710 I have a plus. Plus, plus, plus better. Minus, minus, minus. 289 00:25:31,760 --> 00:25:35,950 Right. This is just from the definition. Basically, this follows. But what does it mean? 290 00:25:36,760 --> 00:25:41,559 What it means is. Right. Maybe if its simplest on a set, maybe on the blogosphere picture. 291 00:25:41,560 --> 00:25:43,630 Right. Because you can think of it as a rotation, really. 292 00:25:43,930 --> 00:25:49,090 And if you think about now, this is little labels, X and Z on the block to your right one at the end. 293 00:25:49,120 --> 00:25:52,599 Unfortunately, my point is not. I did this one. 294 00:25:52,600 --> 00:25:56,950 And now. Okay. Anyway, I hope this is the next level and the rotation. 295 00:25:56,950 --> 00:26:00,460 Will Smith will swap the roles of the two. Maybe the intuitive answer. 296 00:26:01,920 --> 00:26:08,010 But what I can tell you certainly is that the harassment does change the role of X and Zs. 297 00:26:08,640 --> 00:26:16,340 So it makes arrows into checks and checks into errors. That means, okay, instead of these Z operators, we will measure a different kind of operators. 298 00:26:16,730 --> 00:26:20,000 If you tell us, you know, and they can actually do it right. 299 00:26:20,240 --> 00:26:26,420 And now suddenly, instead of a bit flip, I can correct the so-called face flip, which is a zero. 300 00:26:26,600 --> 00:26:31,790 Because I told you. Right. There's different areas and a bit flip is the only thing that can happen classically quantum mechanically, 301 00:26:31,790 --> 00:26:39,830 a lot more things can happen right again on the block sphere a bit Flip is a rotation of 180 degrees around the y axis. 302 00:26:40,340 --> 00:26:43,880 The face flip would be a 180 degree rotation around a different axis. 303 00:26:44,120 --> 00:26:48,860 Again, no correspondence classically. But again, it's a different era and we can do it. 304 00:26:50,570 --> 00:26:53,450 Okay, so now we have a different code. The corrects a different kind of error. 305 00:26:54,930 --> 00:26:59,549 And our OC watch audit now is here to find something called the shortcode. 306 00:26:59,550 --> 00:27:05,370 And he said, Why don't we do both? And this works the following if you will concatenate the two. 307 00:27:05,390 --> 00:27:08,540 So we start with the face of code. Right? This is just the circuit I had before. 308 00:27:09,970 --> 00:27:13,600 But drawn very largely and wide to have drawn this very large. 309 00:27:13,610 --> 00:27:19,229 Right. So this court, we know, corrects this face flip error. So now we want to do also a bit flip. 310 00:27:19,230 --> 00:27:24,209 And what we simply do is we encode each of the constituent cubits of the face flip coat in a bit. 311 00:27:24,210 --> 00:27:30,620 Flip coat. So concatenation just means that we have these kind of. 312 00:27:31,040 --> 00:27:35,499 Well. This hierarchical structure, right? 313 00:27:35,500 --> 00:27:38,770 We have three copies of a bit flip code and then a single Facebook code outside. 314 00:27:40,050 --> 00:27:45,180 So this is not a simple position. The output stage will be a simple position of like nine cubits. 315 00:27:45,360 --> 00:27:50,310 So it's a tangled state of nine cubits. And now it looks like the face of code. 316 00:27:50,320 --> 00:27:58,500 But basically each constituent here, the minus bar, the bar indicates that these minus states are now themselves states on three cubits. 317 00:28:00,900 --> 00:28:07,050 But okay. And it turns out this code can correct a single bit flip and a face flip. 318 00:28:07,260 --> 00:28:10,600 So now to air us. So we have done a bit better. Right. 319 00:28:10,760 --> 00:28:15,180 And look, again, if you want to tell this to a colleague in the lab, they have to measure a bunch of stuff, right? 320 00:28:15,200 --> 00:28:21,499 They have to measure all of the they have to always ask the key is not the first but equal, the second equal and the fourth and the fifth and so on. 321 00:28:21,500 --> 00:28:22,790 Right. So this is the inner checks. 322 00:28:23,060 --> 00:28:28,910 And then you also have to measure these big things, which are the checks of the outer coat, as if we have to ask, are the first two pluses. 323 00:28:29,210 --> 00:28:35,210 Right. But it's now you're asking ideological pluses. So actually all you have to measure is like this product of six operators. 324 00:28:35,840 --> 00:28:39,440 It's more complicated for the arbitrary. They will not like that. 325 00:28:39,440 --> 00:28:44,080 But okay, we are serious for now. We can do that. Okay. 326 00:28:45,420 --> 00:28:49,860 So now the Mirror. The real America that I promised you is that this is enough. 327 00:28:51,780 --> 00:28:57,810 Right. Correcting bits and face time is enough. We can now correct an arbitrary single cubit error actually. 328 00:28:58,380 --> 00:29:01,530 That is the really fascinating thing I think about it. 329 00:29:02,600 --> 00:29:07,370 And the essential insight for that is the following. So there is a math version and the informal version. 330 00:29:07,910 --> 00:29:12,410 The math version is that the party apparatus that has exist and the product, 331 00:29:12,410 --> 00:29:18,140 which is the party of biometrics and the identity, they form a basis of two by two matrices. 332 00:29:19,300 --> 00:29:21,280 And what does it mean physically and formally? 333 00:29:21,280 --> 00:29:26,960 That means that any operation, anything that happens on a single qubit is actually a superposition of nothing happening. 334 00:29:26,980 --> 00:29:31,090 The identity of a bit flip, a face flip, and both together. 335 00:29:32,140 --> 00:29:39,340 So I can view anything that happens on a single cubit as a combination of these two choices and both of them together and nothing. 336 00:29:40,640 --> 00:29:43,770 Right. So now apply this to a to a state. 337 00:29:43,790 --> 00:29:47,119 What does it mean? It really means, okay, anything happens to the shore code state, right? 338 00:29:47,120 --> 00:29:50,599 Then it's either not the so-called state or it's a bit flip. 339 00:29:50,600 --> 00:29:53,960 Apply to it. It's a face flip, fly to it or it's both apply to it. 340 00:29:54,680 --> 00:29:58,770 They told you if you can correct a bit and a face flip. But what does it mean? 341 00:29:58,780 --> 00:30:03,850 We might be correct these things by measuring operators and collecting the outcomes of the measurements. 342 00:30:04,120 --> 00:30:11,050 And we can correct these things because all the possibilities, all these three different possibilities correspond to different measurement outcomes. 343 00:30:11,710 --> 00:30:16,480 And what did I tell you about measurements? Right. Once you look, the cat is dead or alive. 344 00:30:17,560 --> 00:30:22,030 So this is a very process of measuring these operators as this criticises the error. 345 00:30:22,690 --> 00:30:28,839 So we have this gigantic continuity continuum of possible errors, but by measuring, we collapse. 346 00:30:28,840 --> 00:30:36,079 This set onto a different set of outcomes. It will be after the measurements, but the cat will be dead and alive and well. 347 00:30:36,080 --> 00:30:43,080 Here. Our arrows will either be nothing happened. A bid for a patent face for the patent or both together happened. 348 00:30:44,130 --> 00:30:47,370 Right. And this is really, really the miracle of quantum error correction, that kind of. 349 00:30:48,690 --> 00:30:52,360 Quantum mechanics is a combination of the continuous and the discrete right, 350 00:30:52,380 --> 00:30:55,800 and this is why it works here, but didn't work in the floating point case. 351 00:30:56,950 --> 00:31:00,030 Okay, maybe a preliminary summary, because in the second part of the talk, 352 00:31:00,360 --> 00:31:03,420 what I want to do is I want to also give you a flavour of what people are asking today. 353 00:31:03,570 --> 00:31:07,110 These are questions that people have answered in the nineties, which I think is very fascinating. 354 00:31:07,610 --> 00:31:12,440 Look, let's summarise, right? So we have done the chalkboard and the Steam coach has the same property. 355 00:31:12,460 --> 00:31:17,490 I should think it is a better quote to some extent, but again, it's somewhat difficult to understand, I believe. 356 00:31:18,420 --> 00:31:21,710 The shortcut can correct an arbitrary single cubit error. Right. 357 00:31:21,790 --> 00:31:26,739 And there are versions of it now we can basically. But we we were in the chocolate we repeated three times. 358 00:31:26,740 --> 00:31:31,310 Right. We can repeat it five times, seven times. And we will actually be able to correct one more. 359 00:31:32,080 --> 00:31:37,120 So this basically already gives a blueprint for how to do fault tolerant quantum computation, 360 00:31:37,510 --> 00:31:41,020 because now we have to make all our constituents these big superpositions of stuff. 361 00:31:42,570 --> 00:31:49,040 Right. And again, I want to remember that this is how, you know, trivia because falter and random access machines, for example, not exist. 362 00:31:49,820 --> 00:31:54,110 And quantum error correction is really possible because quantum mechanics is not just wave mechanics. 363 00:31:54,590 --> 00:32:03,770 You might have heard that statement is not true. Kind of mechanics is really this dance of continuous superpositions of things entangled states, 364 00:32:04,160 --> 00:32:07,190 but then discrete that is technically projective measurements. 365 00:32:08,240 --> 00:32:11,890 And it's really both that we need to to do it at. 366 00:32:13,710 --> 00:32:19,770 All right. Very good. Should I take a question at this point or. 367 00:32:20,650 --> 00:32:28,830 To. Maybe I can ask whether there's any questions already about the first part, because then afterwards we can go to, uh, to the current research. 368 00:32:30,320 --> 00:32:34,650 How. And also your. 369 00:32:36,870 --> 00:32:41,520 Fantastic question. Not very efficient. It's possible. 370 00:32:41,970 --> 00:32:46,350 Okay, so it's two statements. There's a statement about numbers in their statement about scaling. 371 00:32:46,950 --> 00:32:50,880 The lecture being very important. So the overheads is. 372 00:32:52,570 --> 00:32:56,590 It's big, but it doesn't grow. If you build a larger and larger computer. 373 00:32:57,550 --> 00:33:01,680 But that's the key insight, I think. Right. So it's kind of bounded. 374 00:33:01,980 --> 00:33:06,150 It's a big number. As said, if you want to do short algorithm, also you need a thousand qubits. 375 00:33:06,150 --> 00:33:09,420 And that means a million logical cubits, something like that. 376 00:33:09,430 --> 00:33:10,860 Right. So this is like the ballpark. 377 00:33:11,220 --> 00:33:18,420 Every cubit will have a thousand, every logical cubit actually qubit and you can go to will have a thousand of these atoms. 378 00:33:19,400 --> 00:33:23,840 Right. But if you build a bigger quantum computer, you will not have to use more than that. 379 00:33:24,260 --> 00:33:28,880 And if you want a million quantum logical qubits, you still need thousands more. 380 00:33:30,910 --> 00:33:37,030 Yeah, not yet, but hopefully one day I will show you what is the best state. 381 00:33:37,230 --> 00:33:41,260 But the best we can do. Then you can judge. Yes, please. 382 00:33:41,870 --> 00:33:45,880 Ridiculous question. Appropriate. So is the. 383 00:33:47,090 --> 00:33:52,100 And the measurement. You learn something noises and entropy is not is the lack of knowledge. 384 00:33:53,800 --> 00:33:59,020 Right? No, no. I mean, literally. Right. So we learned something about the system by measuring. 385 00:33:59,890 --> 00:34:05,630 That's where the entropy goes. And you learned a bit. That's how you I mean, it's called entropy evacuation, I think, technically. 386 00:34:06,590 --> 00:34:13,100 So literally. Is that clear? What I mean? S.O.B. is basically an entropy is a bunch of boxes, right? 387 00:34:13,100 --> 00:34:19,520 You say in the magnet, right? You say, Ah, we have a mechanisation and there's a lot of micro states, but we don't know the micro state. 388 00:34:19,520 --> 00:34:22,880 So the so we only know the mechanisation. That is the sum of all of these things. 389 00:34:23,780 --> 00:34:28,010 And that sense entropy is a lack of knowledge because we don't know what is what every atom does. 390 00:34:29,360 --> 00:34:35,900 Right. And also here, right. We basically make noise at uncertainty, but we learn something by measuring. 391 00:34:42,750 --> 00:34:47,170 Look. It is a fun series to kill. 392 00:34:47,710 --> 00:34:52,730 Yes, it is a monstrous memory if you go through refresh cycles. 393 00:34:53,110 --> 00:34:56,560 You can stabilise like a healthy kid. 394 00:34:57,190 --> 00:34:57,669 Yeah, exactly. 395 00:34:57,670 --> 00:35:05,290 So the idea is basically that you intersperse your computation with measurement of these called syndrome operators write these are the check, these, 396 00:35:05,620 --> 00:35:13,720 these operators that I've written like that ones that to you if repeatedly asked these questions and between the asking you can actually do stuff. 397 00:35:15,530 --> 00:35:20,439 Right. But you have to do this actually quite frequently. There is a space and a time. 398 00:35:20,440 --> 00:35:25,210 All that it's called, right? All right. 399 00:35:26,170 --> 00:35:29,290 So then let's look at what people are up to right now. 400 00:35:30,330 --> 00:35:32,820 Okay. So the first observation that I want to say to you is have. 401 00:35:33,060 --> 00:35:37,350 I mean, I try to do stuff mostly with the colleagues in the lab because I already told you. 402 00:35:37,350 --> 00:35:44,340 Right. The so-called intuitive, but not very practical, because for son to take the one size larger, right where we repeated five stuff five times. 403 00:35:45,600 --> 00:35:50,260 Then does that Shakespeare be okay? Right. Other industry. You always only have to ask the neighbours. 404 00:35:50,910 --> 00:35:54,090 But I already hinted at this fact that the other check of the outer code, 405 00:35:54,120 --> 00:36:00,420 they grow because you have to ask things about logical states and these operators make it very hard to measure. 406 00:36:01,820 --> 00:36:05,090 So so it's not a scalable thing if you want to do it in the lab. 407 00:36:05,810 --> 00:36:13,370 Instead, you want to do something different, which is a bit hard to understand. But let me attempt to give you an idea of what people are doing. 408 00:36:13,640 --> 00:36:18,020 So what you want to do instead of the chalkboard is something called a low density parity checkout. 409 00:36:18,740 --> 00:36:24,380 So because these are called parity checks and low density implies that they have support only on few qubits. 410 00:36:24,620 --> 00:36:28,520 So you're not asking a question. Are is the product of these five qubits? 411 00:36:28,820 --> 00:36:33,080 Of these million qubits the same? But you only have to ask it about like a finite set. 412 00:36:34,280 --> 00:36:35,060 And how do you do this? 413 00:36:35,570 --> 00:36:43,010 The most well-studied example is probably something called the surface code, where you arrange qubit on such a checkerboard pattern here. 414 00:36:43,550 --> 00:36:47,750 And okay, this has some size L So this works for a for any, any size of rich. 415 00:36:47,750 --> 00:36:49,610 This is a grade of size five five sample. 416 00:36:50,150 --> 00:36:55,790 And then okay, you put the cubits on the, on the, on the, on the vertices here, right on these little suckers. 417 00:36:56,600 --> 00:37:00,140 And then for each face you have an associated check. 418 00:37:01,030 --> 00:37:04,810 Right. And you have to see that the faces are. Well, they are. 419 00:37:04,990 --> 00:37:07,990 It's a checkerboard, right? So they are in this case, they are blue and red. 420 00:37:09,460 --> 00:37:16,900 And the blue checks correspond to measuring basically a Z check on the on the on the corners of the face. 421 00:37:17,410 --> 00:37:23,470 Whereas the red things correspond to measuring an X operator on the corners of the face. 422 00:37:24,100 --> 00:37:30,399 And now this. Okay, you well, hopefully you see that at least in the set up you will at most measure for buddy operator. 423 00:37:30,400 --> 00:37:37,990 So that's much better for all experimental colleagues. And also it will protect for an arbitrary sizeable protects l have rounded down. 424 00:37:37,990 --> 00:37:45,820 Right. So this is the floor function, this angle of brackets it will protect against any error acting on at most l have cubits. 425 00:37:45,880 --> 00:37:49,420 So you build this off size five. You can, for example correct two errors. 426 00:37:50,720 --> 00:37:54,960 All right. So. Yes. 427 00:37:55,440 --> 00:38:03,380 So this looks much more tractable and deep. The suspended. So this is a fantastic experiment done very recently, actually. 428 00:38:03,500 --> 00:38:13,640 At least now we are jumping ahead a long time from 1995, which was 22,022, where the age group this is the group of hundreds by the RAF, 429 00:38:14,150 --> 00:38:18,380 the quantum devices left that I mentioned before, they have actually built a three by three version of this. 430 00:38:18,860 --> 00:38:24,620 This is a sketch. This is the actually a picture of the circuit, right? Because this academic group, we actually get a picture in a publication. 431 00:38:24,620 --> 00:38:29,720 That's fantastic. So I can show you this. And you see how these little stars are basically the qubits. 432 00:38:31,430 --> 00:38:36,100 There's a few more. I would probably not have the time to fully explain why, 433 00:38:36,250 --> 00:38:41,890 but let's let's just superposed this thing with the was a sketch that I do this, so I would rotate about 45 degrees. 434 00:38:42,520 --> 00:38:45,819 And so this is really a chip. These are aluminium wires. 435 00:38:45,820 --> 00:38:49,180 Basically they get superconducting, this is how you perform the computation. 436 00:38:49,630 --> 00:38:54,610 And then all these coloured lines are microwave electronics that you use to control this chip. 437 00:38:56,370 --> 00:39:00,180 Okay, Google has much more money than academia, so they have even the two of these things. 438 00:39:01,250 --> 00:39:05,330 And they have built a size three one and a size five one. 439 00:39:06,360 --> 00:39:10,950 And this is a plot, so we don't have a picture because they're not academia, unfortunately. 440 00:39:11,850 --> 00:39:15,050 So they don't give you pictures anymore that chips of the newer models. 441 00:39:15,480 --> 00:39:19,510 So we don't know how this device looks in practice, but they have on their on their block, 442 00:39:19,560 --> 00:39:26,100 they actually did publish this this nice plot where they compare the performance of the small and the bigger court. 443 00:39:26,550 --> 00:39:29,610 Right now, told you all. Maybe now going back to the coffee example. 444 00:39:29,610 --> 00:39:35,340 Right you want if you repeat more, you want to be better at correcting and then you see, okay, 445 00:39:35,580 --> 00:39:40,409 this is so they built this chip and they have actually built a bunch of chips and each chip has a distance. 446 00:39:40,410 --> 00:39:46,320 Three or so a size three could enter a size five, right? Because you can just ignore part of the five code and you have a31. 447 00:39:47,010 --> 00:39:51,600 Right. And each of these dots, this thing in this plot is a single device. 448 00:39:51,930 --> 00:39:58,530 And on the x axis we have the fidelity of the distance three record. And on the y axis you have the fidelity of the distance five. 449 00:39:58,530 --> 00:40:03,990 So distance the same as the size. And so this is a technical term and you see, okay, you want to be above the direction of that. 450 00:40:04,020 --> 00:40:11,400 If quantum error correction is supposed to work and you see their very best device is a bit better if you scale it up. 451 00:40:12,230 --> 00:40:14,930 So they are really just scratching the surface of quantum error correction being 452 00:40:15,260 --> 00:40:20,330 being possible and why it actually can can get worse if you make it bigger. 453 00:40:20,420 --> 00:40:26,410 Basically, if your device is too bad, you are making it was actually if you if you scale it up because errors accumulate even more. 454 00:40:26,420 --> 00:40:31,979 And do you have to fight that? And now the question is maybe connecting. 455 00:40:31,980 --> 00:40:37,879 So I promise in the abstract, I believe, to connect quantum out correction to what my original speciality is. 456 00:40:37,880 --> 00:40:43,070 That is. His metaphysics, and that is known as metaphysics. 457 00:40:43,070 --> 00:40:48,380 Come in if you want to model the statistics of these devices, because now you can ask how many errors are actually too much. 458 00:40:48,590 --> 00:40:54,400 So how good as do your constituents have to be? Two for error correction to work. 459 00:40:56,430 --> 00:41:01,680 And formally the statement is that there's a threshold, so you have some form of the logical parameter describing, 460 00:41:02,040 --> 00:41:06,570 you know, noise that is pro and it has to be smaller than some critical value. 461 00:41:07,110 --> 00:41:08,429 And actually then the statement is, okay, 462 00:41:08,430 --> 00:41:14,370 if you make your call bigger and bigger and you are below the threshold, so your your computer is good enough, 463 00:41:14,580 --> 00:41:20,010 then encoding the help and the failure rate of the of the code will go to zero as you make the code larger and larger. 464 00:41:20,010 --> 00:41:25,260 Right. So as you scale up your your surface code to you on the left, right, your cat gets sharper and sharper. 465 00:41:26,180 --> 00:41:30,330 Right. And this is really a quantum state that you are stabilising, right? So the cat is both that in the life you. 466 00:41:31,570 --> 00:41:35,920 But if you are not good enough, then actually you make it worse by encoding. 467 00:41:37,020 --> 00:41:40,139 Right. And this. Some of you have background comments made up of these. 468 00:41:40,140 --> 00:41:44,130 This might remind you of the behaviour at a phase transition. And this is because this is a phase transition. 469 00:41:45,210 --> 00:41:48,270 And in fact, okay, this is a very, very complicated process to model. 470 00:41:48,630 --> 00:41:53,940 But remarkably, you can map it Exactly. And certain assumptions on a condensed matter system and a very old one. 471 00:41:54,810 --> 00:42:01,040 So the certainty that certain assumptions of the noise, this process for the surface court, for this thing that has only very recently been built. 472 00:42:01,340 --> 00:42:07,800 That's exactly on the awareness and model in physics as you can describe this process and looking at something called the random bond ising model. 473 00:42:08,520 --> 00:42:10,709 And this is a remarkably because a random analysing model is a very, 474 00:42:10,710 --> 00:42:15,550 very old model of condensed matter physics and has been devised by by Philip Byrne and is an of Princeton, 475 00:42:15,570 --> 00:42:23,760 one of the giants of the field, to describe the interactions of magnetic impurities in magnetic iron in metal alloys. 476 00:42:24,810 --> 00:42:27,900 Right. And this model remarkably describes this phase transition as well. 477 00:42:28,500 --> 00:42:33,900 The form the statement is, okay, you draw a phase diagram of this model that describes like magnetic impurities. 478 00:42:34,440 --> 00:42:39,510 And this magnetic impurities can become basically they are magnetic and is a known magnetic phase. 479 00:42:39,930 --> 00:42:42,210 There's a pheromone phase and there's a known pheromone phase. 480 00:42:42,570 --> 00:42:48,450 And then it turns out if you compute the phase some of this model, right, it has to exist as a disorder strength. 481 00:42:48,450 --> 00:42:51,570 So you have like some dirt in the model and then it's temperature. 482 00:42:51,690 --> 00:42:56,819 And basically the statement is that the largest amount of noise that your court can tolerate 483 00:42:56,820 --> 00:43:03,660 is actually exactly equal to the largest extent of this or that phase that I indicated in RET, 484 00:43:03,780 --> 00:43:07,140 you projected onto the x axis, which is it disorder in this in this model? 485 00:43:09,520 --> 00:43:13,680 This is a remarkable connection where you can learn something new about this quantum system, 486 00:43:14,040 --> 00:43:17,940 something that exactly to a very old model, the system used in condensed matter physics. 487 00:43:20,640 --> 00:43:26,250 Maybe now, in my last slides, I want to tell you a bit about what I do. 488 00:43:27,230 --> 00:43:31,190 So. Um. 489 00:43:32,220 --> 00:43:39,160 So. What is the active research question in kinematic correction is the. 490 00:43:40,700 --> 00:43:48,090 Is developing better codes. And one example of that that I want to tell you about is hyperbolic surface quads and what is the recipe. 491 00:43:48,120 --> 00:43:51,360 So they are complicated beast. They look mesmerising as I like them. 492 00:43:51,390 --> 00:43:55,890 Like these patterns as well. But really, what you should remember is this is basically your circuit layout. 493 00:43:56,580 --> 00:44:03,180 So you have to put a cube. It's not now on a square grid, but you have to put them on each edge of this this picture. 494 00:44:04,080 --> 00:44:07,440 And then the statement is, okay, of course this is harder to build. You also have to define some checks. 495 00:44:07,440 --> 00:44:09,540 I don't want to go into too much detail in this, but again, 496 00:44:09,540 --> 00:44:15,719 they are local and there's some description based on the graph and they are harder to build because they are not naturally embedded. 497 00:44:15,720 --> 00:44:18,900 Right. It gets very crowded to the edge if you go to the edges of this plot. 498 00:44:19,110 --> 00:44:22,620 But if it is built, it has a much reduced overhead. 499 00:44:22,620 --> 00:44:25,020 So this connects maybe to one of the previous questions that we had. 500 00:44:25,410 --> 00:44:32,440 The topic, quote has a lot of all that a single qubits will be encoded into this grid, right? 501 00:44:32,520 --> 00:44:37,709 If you have a ten by ten grid, right. 100 atoms, for example, it will be one logical qubit. 502 00:44:37,710 --> 00:44:42,930 Effectively. The problem is this if you do it by this, not on this gregori, but on this geometry, 503 00:44:43,260 --> 00:44:48,180 what you will get is if you make it bigger and bigger, you will not only protect information better. 504 00:44:49,230 --> 00:44:55,890 But you will also get more cubits and call it. So the information gets better and you encode more. 505 00:44:56,610 --> 00:45:00,600 But this one is quite maybe some of, you know, the code series called a finite rate cut. 506 00:45:02,170 --> 00:45:05,559 And of course, in principle, this can be built. So this is a beautiful picture of an artificial, 507 00:45:05,560 --> 00:45:14,140 hyperbolic lettuce also made from superconducting resonators in the laboratory of Andrew Hauck in Princeton from this publication. 508 00:45:14,620 --> 00:45:19,509 So in principle you can realise such a geometry even in the lab, and there may be one reside that I went that I did, 509 00:45:19,510 --> 00:45:22,270 for example, is now you can also claim the again for these quote model, 510 00:45:22,930 --> 00:45:26,379 the error correction process can be mapped to some model of condensed matter physics 511 00:45:26,380 --> 00:45:32,170 and indeed you can in maps on a very complicated model that lives in in curved space. 512 00:45:32,560 --> 00:45:37,810 So on a on a negatively curved space, these models that maybe in the past had been only studied, 513 00:45:37,810 --> 00:45:44,080 but mostly studied actually by and by high energy physicists because the universe kind of is a negatively curved. 514 00:45:44,080 --> 00:45:51,159 Well it's supposed to be maybe negative because surface and and turns out okay we could map it. 515 00:45:51,160 --> 00:45:58,450 Exactly. And and this mapping kind of then allows us to calculate for these kind of codes the optimal information, 516 00:45:58,450 --> 00:46:02,470 theoretically optimal performance that you can get at a certain size and a certain strength of noise. 517 00:46:04,150 --> 00:46:09,910 And remarkably, also, this actually yielded some new insights into the statistical mechanics of these of these models. 518 00:46:10,540 --> 00:46:13,180 So you can not only learn basically about new systems. 519 00:46:14,280 --> 00:46:20,430 By an old theory, but you can actually also learn from new systems about an old theory that that is very, very nice, I thought. 520 00:46:21,530 --> 00:46:26,939 Okay. And was that I reached my summary. So hopefully I'll convince the quantum of corrections in you. 521 00:46:26,940 --> 00:46:35,310 Possible, which is surprising. And Alice introduced the Threshold Theorem, which says, okay, QC, the constituents have to be good enough. 522 00:46:36,620 --> 00:46:42,900 The experimental state of the field is. We are just scratching the surface, maybe off the threshold. 523 00:46:44,270 --> 00:46:51,170 And I've also told you about some current research that that I am doing just to connect with the team. 524 00:46:51,870 --> 00:46:53,480 Oh, thank you so much. I want.