So the previous talk, right. Told you maybe. Okay, if we have these large scale quantum simulators. What kind of questions can we answer as condensed matter theorists by using them? But of course, we also know that we shouldn't only be asking what can quantum computing do for us, but we should also be asking what can we do for quantum computing? Right. And. Well, we can do a whole lot, it turns out. And this would be the topic of my talk, which is the miracle that is quantum error correction.
And I hope I will convince you that it is not surprising that it works. But it does. It does. It does work. And this is really the reason why we can can have any hope of building these machines. Okay, so you should know your enemy, but you should also know what you are protecting. So let's talk about what is actually classical and quantum information. So classical information is maybe a bit dull.
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. Your phones in your pockets, they all internally calculate with strings of zeros and ones. 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. And then we're already seeing where I'm getting here with quantum information, because quantum information is much more subtle.
So quantum information, like the minimum information to describe it, needed to describe a quantum system of two levels, like an atom of two levels or something similar. Is a cubits and a cubit cannot only be in zero and one, it can be in a superposition we say. Right. And as we write it mathematically and this kind of well, it's a vector, technically speaking, we can write it as alpha zero plus better one where alpha and beta are complex numbers.
Right. The magnitude has to their magnitude squared. It has to sum to one. And I can like visually represent this kind of state and the so-called block sphere. You might remember seeing this picture in in your quantum mechanics class. Right. That means that I can associate basically every con, every combination of efforts and betters that describe a valid quantum state with some angles.
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. Right. So this is much more complicated than just zero and one, right? Actually, there's a continuity of states. Right. Any point on on this fear. And it gets even weirder when you think about how we as classical beings interact with this quantum information.
Because now. Right. If you want to look at something, right, look at the cubit, we have to measure it. And it turns out if you measure this thing, then it is indeed with certainty afterwards, it is a certainty. Zero or one. Right. So we get a probabilistic outcome, right? We get zero one. Yes, I know that. Live with probability alpha squared and beta squared. So it's a populist outcome. But after the measurement, the state of the cubit is fixed.
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. And this witness, in fact, can be utilised. But this is, of course, why I'm telling you about this today.
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, actually the first very first people to think about or want some of the very first people to think about the problem, how to utilise this witness of quantum mechanics actually where an Oxford blockchain user published the dojos algorithm,
which is kind of a toy problem in some sense, but it was one of the first problems that actually did establish that quantum. We just can solve problems that classical computers cannot solve. Then of course, sure, they would need to show up on time. And it came along in 1994 and he published his prime factoring algorithm, which is maybe the most famous application, quote unquote, that we have today, at least from for exact quantum computers.
But I also want to say, if I was under a cat, for example, here in Oxford, in the math department, he was the first person to think about using quantum computers, not for breaking encryption, but for actually encrypting things. Very savvy. And so this is quantum property. And each of these topics, honestly, would be a 45 minute lecture on their own. So I would not talk about that much more. You have to ask me afterwards.
But what I will instead. Trying to tell you is how can we implement or how can we protect this information? Because if we want to use it, we have to be able to store it in some way. Okay, so maybe before I go too deep into this, how would you present Quantum de quantum? Who does look like? So what is the quantum information? Maybe the most similar version of quantum computers to to the machines. We could call it classical computers. Is the superconducting Hubert machines.
Right. This chip flies out from the Google lab and it looks kind of like you would expect from a silicon processor, although it's made actually from superconducting aluminium instead. 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. This picture is from Google Quantum and there's a major industry players now Quanta, Google Quantum, IBM quantum and it's involved being from a start up from France.
So these things also are now get into the Start-Up space. But of course, the pioneering work, for example, a lot of pioneering was done in academia, mostly in the Yale Quantum Institute and also at Zurich at the Quantum Devices lab there. So that another major problem is trapped. IONS Unless he already mentioned them, 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.
It's very beautifully actually captured with with the iPhone, I believe. And if you listen very closely to actually see the atom, this is this is kind of fascinating. And this is this is an ionised atom. So you trap it with a magnetic strip. This is why we call them them trapped ions. And actually, this was historically the first platform where quantum computing was performed. Right. David Wineland, who was at Nist's when he performed the work, received a Nobel Prize for that in 2012.
Today, of course, there's there's again, major industry players continuing with close relations between University of Oxford, especially the computer science department here and then Ion. Q I think being actually the Quantum Start-Up was the biggest market capitalisation. 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. So. Uh, uh, another platform. And see, I already mentioned that our reconfigure atom arrays.
So right here the atoms are neutral and you track them with a laser beam with something called an optical tweezer. And actually these I mean, this is maybe a more boring simulation than the than the Mario that was shown before. But if you want to do computation, this is more like what you will do, because you see that there is like these arrays and they get brought close together.
Basically OnDemand when you want to perform at a gate, when you want to want to qubits, to interact, you bring them close together physically and otherwise you have them far apart. And this is kind of fascinating, has created a lot of trade recently because of this great flexibility. So this is a really promising platform that's emerging right now.
And this also means that this I mean, at the moment, I would say there's one major player, which is the laboratory of looking at it, and they have also funded a company which is a computing. There's a bunch of other platforms. 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. It's actually quantum motion and major play. And thank you.
It's also again being founded actually at Oxford in the Material Science Department by Simon Benjamin. And so there's a lot of platforms that, that all have promises and none of them works just yet. And this is because all of them share one simple problem, and that is noise by quantum states are inherently fragile. And this is really also, of course, the reason why we don't see quantum mechanics around us, Right?
Because quantum mechanics is fragile, quantum information decoherence, and it becomes classical. So in the end, if you screw things up and you do it without being very careful. Right. That is to say it becomes just classical information. And well, hopefully we convince you that there is a way to do it to prevent this classification of of the quantum information and maybe to really drive the point home of how important this is.
Let me tell you the thing that is not often appreciated, which is that quantum computing is by far not the first alternative idea to implement computers beyond classical ones. Beyond the ones we built today. For example, my my fellow German and I should hug in 1979, published a beautiful paper on the power of random access machines. And he basically asked the question.
Imagine you had something called a random access machine, which is basically a computer that operates on floating point arithmetic. The idea is that you have a memory that instead of bits stores, floating point numbers of arbitrary precision is not something we can do if we are on a computer, if we have a finite memory. Right. This is why it's different. And imagine also imagine you had this floating point point that you can store and you can also perform arbitrary arithmetic on them again,
with arbitrary precision. Right. His idea actually, I think, was basically to build an analogue computer based on some I don't know, some way, some coherence. I think some oscillations in a in a tube or something like this. Right. Right, which then would correspond to a continuous number. But then he proved, okay, if you do this right, actually, then these random access machines will be vastly more powerful even than the quantum computers we build we try to build today, Right?
So for random access machines is NPR. If that tells you anything, what it really means is that you can basically really solve optimisation problems you can solve. Travelling salesman All of this in polynomial time. That's the idea. So it would be extremely powerful, but they have never been built. And why is that? Because of noise. We have just not figured out a way to do floating point arithmetic well enough.
But there's no known wire that this sends no known way to fall tolerant li implement these machines. And surprising fact about quantum computers is that they can be implemented for currently. Okay, So now before I talk about Quentin, who does, let me tell you. Okay, what does work anyway? So we want to protect classical information, maybe first. Let's start with that. Let's start with sending a bit. Right.
So imagine Alice comes into a bar and she wants to order coffee at Bob and assemble by the names we usually give to these two people in computer science. And she said, okay, I want coffee. And then there's some noise. And actually business turns on his tea and she's good at getting a tea and this happens is probably deep. 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.
Right? So this is the final version of that. And there's one simple form of error correction that we all do intuitively in a noisy bar we repeat ourselves. And it could be decisive because the repetition could. Right. So instead of saying I want coffee, I say, I want coffee, I want coffee, I want coffee. And they'd say, okay, now it that three times. And Bob was on a sensor, one's iced coffee, coffee. And he goes like, I guess she meant coffee. Right.
Again, the former version of this, you implement a repetition code code to find something called a logical bit. And the logic in zero is just and times zero in this time. And here's and it's three times one. And if noise flips half of your bits right, then you can do something called majority. But the thing that is you asked, are there more zeros and ones in the message I received? You just assume it was the origin of man has had all ones or one zeros, right?
And well, the pain probability then is lower because of this. Right. If this can still go wrong, if Bob actually misunderstands you twice. And of course this happens with probability P squares and if P is small, P squared is smaller than P. So that's that's the idea, right? Erick Erickson improves the fidelity of classic information. Even if you send it through a noisy channel, that would be maybe the form and statement here.
Okay. But quantum error correction. So so really, people in the early nineties thought it was a futile effort. And there's a bunch of reasonable objections to to quantum error correction being possible. And maybe the most damning one is we cannot copy quantum information. And that seems to be a major objection to what I just told you that. Okay, how do you protect information? You copy it and send it multiple times. But for quantum information, actually, we cannot do that.
And the proof is very simple. So let's attempt a proof. Okay, it goes the following. We have a proof by contradiction. So imagine you could do it. I imagine there was an operation. You. This is some unitary metric that Exxon tool. Cubits. Right. We want to copy an arbitrary state. Sire into this state e e can be whatever zero. And then after you apply, the operation has to be beside sire. And this has to work. Not for arbitrary states. Right? Because all copy machine should copy.
Whatever. Right. Then we can write the following, right? You can rearrange this. So I'm working in bracket notation here. I hope you do somewhat remember, but what it means really is you consider the overlap of two states fly and say, and we assume that both of them can be copied by the machine. Right? 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.
And then what I can do is I can insert this you dagger you here, which is just the identity. This just to say okay quantum and in quantum operations are unitary. This is mathematically the precise statement. But then I can also say, okay, now I can imagine dissecting to the left and dissecting to the right and what I would get upside PSI and Phi Phi and is of course this again the overlapped but squared because now I have a twice.
And okay, so we have proof that the overlap between the two states is equal to the square of its overlap. But there's not a bunch of numbers. There's not many numbers where equal to their square, right? 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.
But and that is a simple proof that a sensible quantum operation, a unitary time evolution, cannot copy quantum information, at least not arbitrary one. Okay, so that is bad, but it gets even worse. It's a square. So measurements are destructive. I told you right after you looked at at the cat. Right. It's not that or alive anymore. Right. It's. I mean, it's either the two, but it's not both. Sorry. That way. Right.
That means. Okay, we measured this thing, we measure that family. So we are asking, is it 0a1? Right? And then it will be zero or one with a respective probabilities. And that's. Well, so we how do we do majority voting then, Right. 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. That, again, seems very bad. But it gets even worse because all of the errors are continuous.
Right? So I told you, the state of Cuba, just like can be identified with this point on the three dimensional sphere. But of course, errors are arbitrary. Small rotations then, Right. Any sensible operations. Right. This is why it's unitary. It's a rotation on the sphere. And of course, now this can be very small and this starts to look a lot like these floating points problem, right. Where we cannot even do floating point arithmetic. Correct.
How can we do that correctly? But it seems like very, very, very bad. But again, surprisingly, no. I hope you don't get the surprise of people that really happened in 1994. Peter Shore and actually independently. Andrew Stephen of Oxford University. So it's that kind of error correction is indeed possible. 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.
So we need everything. We need entanglement. We need measurements. Superpositions. 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. But I will lose the shortcut because of the shortcut we have to for the steam code. 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.
You need a bit more information theory, background. So I think this is a bit more intuitive, I'm afraid. But okay, so let's do it. And let's start with the quantum repetition code. So what's the quantum version of our repeated coffee order? And the first observation is that cloning is not necessary because we can do entanglement instead. Okay. So unless you're. Fortunately already did some circuits.
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. Each horizontal line is a qubit. Here we have in the nation state. And this is a gate. This is the gate is called the Senate. It's a control. It's not This gate actually makes perfect sense. Classically, right. It just says you apply a not to this cubit. If this cubit is one and you don't do it if it's zero.
Right. So that means, okay, if this is zero, if the input would be zero zero, Right. Output would be again, zero zero. If the input would be one zero, the output would be one one. Because the first bit because it indicates where they should flip. So now this is a quantum computer. So it gets a bit more complicated than that because I can get as an input. Right. I can put a superposition of things and how does the gate X then we get this by linearity.
So we can think here about what happens to each turn. And I'll just point out it's not okay. Right. We can think about what happens to each time separately and we can ask if it's zero. 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. And okay, if the first Q it is one, the second gets flipped, so the output will be one one. And this is the second term in the output. This is somewhat clear.
But the important thing is not that the output indeed looks a bit like a repeated information. Right. Because I've I've not copy it, but I've entangled the two qubit. 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. On the right hand side, I have to write this joint state. Right. This is what I. What I mean by putting it here. So we produce redundancy by entanglement instead of by cloning.
And then the quantum version of our redundancy of us just be makes two with this twice, right? So we copy and we entangle upside with the second and the third qubit. And this now will be our logic, a quantum state of the quantum repetition got. This is not yet. We are not dead yet, but we are getting there. So what about measurements? So I told you, measurements are another problem. Right. So you see already going one by one through our objections.
What about measurements? Okay, we have this logical stage IFR, 000 plus better one, one one. Right. This is a superposition of a superposition and an entangled state. Right. And imagine you measure Now, is that one. So you ask for the first qubit is a zero or one. Right? So formally, this means we measure the operators and one. Right. And then you. And then the thing is, of course, this has not a well-defined answer within the stage.
Right. And what it means is, right, that the answer is different for the for the two terms. Right. And the first term would be zero. And the second one be one. So if you meant if we ask this question and get an answer, the state afterwards will be either 000 or 111. That's the destructive nature of a measurement. But instead what we can ask are the two other first two bids equal?
I would ask a slightly different question and then actually it's okay, it turns out, because if you just ask formal you again you measure operate and it is that one's a two. But the question you are asking does is are the first two bids equal or equivalently are the last two bids equal? And that actually has a well defined answer within the state because it's the same in both the right. The first is 000. So all are the same. So the first two and the last two are the same.
And the second term, again, they have a different value but that they are the same. 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, really ask this question to nature and we get an answer, we can get this answer without destroying the state. Right, because we have asked the right question. And this is maybe the first witness of quantum mechanics that comes in here. And this is not important classically.
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. Quantum mechanics forces us to do that and nothing else. Okay. Second part of the invitation. Quote, This is just a reminder of what we are doing so far. Now, let's consider that is actually protect us from an error. And let's consider one very special error. The big flip. Is this just the thing that flips 0 to 1 and 1 to 0? Hence the name.
It's also politics symmetric. If you remember quantum mechanics one. So what happens to the encoded state? Let's do a table and ask how do these measurements outcomes change? Imagine we apply a bit, flip to the first cubit. What would be the state after? Once again, it follows from linearity might be 100011. The first bit is now flipped in both. And of course I can again ask, are the first two bits equal? And now the answer is no. But for the second one, the answer is still yes.
The last two bits are still equal. Equivalently. Now. Right. I can ask the AK. What if I flipped the second cubits? Huh? I already see a pattern here. Right. So now the first two are not equal, and also the last two are not equal. So I get a different answer, actually, if I measure these things. 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.
And of course, this is a very important observation here, is that all these different possible errors have a different outcome for these measurements. Right. So we can correct actually a single bit flip here in this coat by doing these two measurements. And then depending on the outcome. Right. If it was both equal, we would see nothing happened.
We are good. If one of you get one of these outcomes right, then you know you have to do something and you even know what to do because you just have to reverse that error. So you apply another bit. Flip to cancer. The first one that happened accidentally. All right. No, this is not quite yet enough. So you have to bear with me for another second. So let's do another repetition. So what do I mean by this? Classically, there's only one. Right?
And I want to stress the fact that so far, all the circuits that I've drawn to you. Right. But flip C, they all make perfectly sense on a classical computer. Actually, the only way quantum mechanics could came in was through the input state. Right. Because we put an entangled state in the beginning. 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.
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, the state of the circuit will be this logical state, 4000 plus better 111. 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. And this is because they have no correspondence on a classical computer because they do superposition.
So they take the zero stage to a superposition of zero and one with relative phase plus and the one state to another superposition with a different relative phase minus. So on the block sphere, this is a rotation by 90 degrees around the new Y axis. 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. Right. And now suddenly we are here on the equator of the blogosphere.
Okay, so what does I do? So then I'll state afterwards this. I have a plus. Plus, plus, plus better. Minus, minus, minus. Right. This is just from the definition. Basically, this follows. But what does it mean? What it means is. Right. Maybe if its simplest on a set, maybe on the blogosphere picture. Right. Because you can think of it as a rotation, really. And if you think about now, this is little labels, X and Z on the block to your right one at the end.
Unfortunately, my point is not. I did this one. And now. Okay. Anyway, I hope this is the next level and the rotation. Will Smith will swap the roles of the two. Maybe the intuitive answer. But what I can tell you certainly is that the harassment does change the role of X and Zs. 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. If you tell us, you know, and they can actually do it right.
And now suddenly, instead of a bit flip, I can correct the so-called face flip, which is a zero. Because I told you. Right. There's different areas and a bit flip is the only thing that can happen classically quantum mechanically, 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. The face flip would be a 180 degree rotation around a different axis.
Again, no correspondence classically. But again, it's a different era and we can do it. Okay, so now we have a different code. The corrects a different kind of error. And our OC watch audit now is here to find something called the shortcode. And he said, Why don't we do both? And this works the following if you will concatenate the two. So we start with the face of code. Right? This is just the circuit I had before. But drawn very largely and wide to have drawn this very large.
Right. So this court, we know, corrects this face flip error. So now we want to do also a bit flip. And what we simply do is we encode each of the constituent cubits of the face flip coat in a bit. Flip coat. So concatenation just means that we have these kind of. Well. This hierarchical structure, right? We have three copies of a bit flip code and then a single Facebook code outside. So this is not a simple position. The output stage will be a simple position of like nine cubits.
So it's a tangled state of nine cubits. And now it looks like the face of code. But basically each constituent here, the minus bar, the bar indicates that these minus states are now themselves states on three cubits. But okay. And it turns out this code can correct a single bit flip and a face flip. So now to air us. So we have done a bit better. Right. And look, again, if you want to tell this to a colleague in the lab, they have to measure a bunch of stuff, right?
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. Right. So this is the inner checks. 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. Right. But it's now you're asking ideological pluses. So actually all you have to measure is like this product of six operators.
It's more complicated for the arbitrary. They will not like that. But okay, we are serious for now. We can do that. Okay. So now the Mirror. The real America that I promised you is that this is enough. Right. Correcting bits and face time is enough. We can now correct an arbitrary single cubit error actually. That is the really fascinating thing I think about it. And the essential insight for that is the following. So there is a math version and the informal version.
The math version is that the party apparatus that has exist and the product, which is the party of biometrics and the identity, they form a basis of two by two matrices. And what does it mean physically and formally? That means that any operation, anything that happens on a single qubit is actually a superposition of nothing happening. The identity of a bit flip, a face flip, and both together.
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. Right. So now apply this to a to a state. What does it mean? It really means, okay, anything happens to the shore code state, right? Then it's either not the so-called state or it's a bit flip. Apply to it. It's a face flip, fly to it or it's both apply to it. They told you if you can correct a bit and a face flip. But what does it mean?
We might be correct these things by measuring operators and collecting the outcomes of the measurements. And we can correct these things because all the possibilities, all these three different possibilities correspond to different measurement outcomes. And what did I tell you about measurements? Right. Once you look, the cat is dead or alive. So this is a very process of measuring these operators as this criticises the error.
So we have this gigantic continuity continuum of possible errors, but by measuring, we collapse. This set onto a different set of outcomes. It will be after the measurements, but the cat will be dead and alive and well. Here. Our arrows will either be nothing happened. A bid for a patent face for the patent or both together happened. Right. And this is really, really the miracle of quantum error correction, that kind of.
Quantum mechanics is a combination of the continuous and the discrete right, and this is why it works here, but didn't work in the floating point case. Okay, maybe a preliminary summary, because in the second part of the talk, what I want to do is I want to also give you a flavour of what people are asking today. These are questions that people have answered in the nineties, which I think is very fascinating.
Look, let's summarise, right? So we have done the chalkboard and the Steam coach has the same property. I should think it is a better quote to some extent, but again, it's somewhat difficult to understand, I believe. The shortcut can correct an arbitrary single cubit error. Right. And there are versions of it now we can basically. But we we were in the chocolate we repeated three times. Right. We can repeat it five times, seven times. And we will actually be able to correct one more.
So this basically already gives a blueprint for how to do fault tolerant quantum computation, because now we have to make all our constituents these big superpositions of stuff. Right. And again, I want to remember that this is how, you know, trivia because falter and random access machines, for example, not exist. And quantum error correction is really possible because quantum mechanics is not just wave mechanics.
You might have heard that statement is not true. Kind of mechanics is really this dance of continuous superpositions of things entangled states, but then discrete that is technically projective measurements. And it's really both that we need to to do it at. All right. Very good. Should I take a question at this point or. 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. How. And also your.
Fantastic question. Not very efficient. It's possible. Okay, so it's two statements. There's a statement about numbers in their statement about scaling. The lecture being very important. So the overheads is. It's big, but it doesn't grow. If you build a larger and larger computer. But that's the key insight, I think. Right. So it's kind of bounded. It's a big number. As said, if you want to do short algorithm, also you need a thousand qubits.
And that means a million logical cubits, something like that. Right. So this is like the ballpark. Every cubit will have a thousand, every logical cubit actually qubit and you can go to will have a thousand of these atoms. Right. But if you build a bigger quantum computer, you will not have to use more than that. And if you want a million quantum logical qubits, you still need thousands more. Yeah, not yet, but hopefully one day I will show you what is the best state.
But the best we can do. Then you can judge. Yes, please. Ridiculous question. Appropriate. So is the. And the measurement. You learn something noises and entropy is not is the lack of knowledge. Right? No, no. I mean, literally. Right. So we learned something about the system by measuring. 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.
So literally. Is that clear? What I mean? S.O.B. is basically an entropy is a bunch of boxes, right? 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. So the so we only know the mechanisation. That is the sum of all of these things. And that sense entropy is a lack of knowledge because we don't know what is what every atom does.
Right. And also here, right. We basically make noise at uncertainty, but we learn something by measuring. Look. It is a fun series to kill. Yes, it is a monstrous memory if you go through refresh cycles. You can stabilise like a healthy kid. Yeah, exactly.
So the idea is basically that you intersperse your computation with measurement of these called syndrome operators write these are the check, these, 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. Right. But you have to do this actually quite frequently. There is a space and a time. All that it's called, right? All right. So then let's look at what people are up to right now.
Okay. So the first observation that I want to say to you is have. I mean, I try to do stuff mostly with the colleagues in the lab because I already told you. 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. Then does that Shakespeare be okay? Right. Other industry. You always only have to ask the neighbours.
But I already hinted at this fact that the other check of the outer code, they grow because you have to ask things about logical states and these operators make it very hard to measure. So so it's not a scalable thing if you want to do it in the lab. 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. So what you want to do instead of the chalkboard is something called a low density parity checkout.
So because these are called parity checks and low density implies that they have support only on few qubits. So you're not asking a question. Are is the product of these five qubits? Of these million qubits the same? But you only have to ask it about like a finite set. And how do you do this? The most well-studied example is probably something called the surface code, where you arrange qubit on such a checkerboard pattern here.
And okay, this has some size L So this works for a for any, any size of rich. This is a grade of size five five sample. And then okay, you put the cubits on the, on the, on the, on the vertices here, right on these little suckers. And then for each face you have an associated check. Right. And you have to see that the faces are. Well, they are. It's a checkerboard, right? So they are in this case, they are blue and red.
And the blue checks correspond to measuring basically a Z check on the on the on the corners of the face. Whereas the red things correspond to measuring an X operator on the corners of the face. And now this. Okay, you well, hopefully you see that at least in the set up you will at most measure for buddy operator. So that's much better for all experimental colleagues. And also it will protect for an arbitrary sizeable protects l have rounded down.
Right. So this is the floor function, this angle of brackets it will protect against any error acting on at most l have cubits. So you build this off size five. You can, for example correct two errors. All right. So. Yes. So this looks much more tractable and deep. The suspended. So this is a fantastic experiment done very recently, actually.
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, the quantum devices left that I mentioned before, they have actually built a three by three version of this. 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. That's fantastic. So I can show you this. And you see how these little stars are basically the qubits.
There's a few more. I would probably not have the time to fully explain why, 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. And so this is really a chip. These are aluminium wires. Basically they get superconducting, this is how you perform the computation. And then all these coloured lines are microwave electronics that you use to control this chip.
Okay, Google has much more money than academia, so they have even the two of these things. And they have built a size three one and a size five one. And this is a plot, so we don't have a picture because they're not academia, unfortunately. So they don't give you pictures anymore that chips of the newer models.
So we don't know how this device looks in practice, but they have on their on their block, they actually did publish this this nice plot where they compare the performance of the small and the bigger court. Right now, told you all. Maybe now going back to the coffee example. Right you want if you repeat more, you want to be better at correcting and then you see, okay, this is so they built this chip and they have actually built a bunch of chips and each chip has a distance.
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. Right. And each of these dots, this thing in this plot is a single device. 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. 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.
If quantum error correction is supposed to work and you see their very best device is a bit better if you scale it up. So they are really just scratching the surface of quantum error correction being being possible and why it actually can can get worse if you make it bigger. 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. And do you have to fight that? And now the question is maybe connecting.
So I promise in the abstract, I believe, to connect quantum out correction to what my original speciality is. That is. His metaphysics, and that is known as metaphysics. Come in if you want to model the statistics of these devices, because now you can ask how many errors are actually too much. So how good as do your constituents have to be? Two for error correction to work.
And formally the statement is that there's a threshold, so you have some form of the logical parameter describing, you know, noise that is pro and it has to be smaller than some critical value. And actually then the statement is, okay, if you make your call bigger and bigger and you are below the threshold, so your your computer is good enough, 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.
Right. So as you scale up your your surface code to you on the left, right, your cat gets sharper and sharper. Right. And this is really a quantum state that you are stabilising, right? So the cat is both that in the life you. But if you are not good enough, then actually you make it worse by encoding. Right. And this. Some of you have background comments made up of these. This might remind you of the behaviour at a phase transition. And this is because this is a phase transition.
And in fact, okay, this is a very, very complicated process to model. But remarkably, you can map it Exactly. And certain assumptions on a condensed matter system and a very old one. 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. 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.
And this is a remarkably because a random analysing model is a very, very old model of condensed matter physics and has been devised by by Philip Byrne and is an of Princeton, one of the giants of the field, to describe the interactions of magnetic impurities in magnetic iron in metal alloys. Right. And this model remarkably describes this phase transition as well. The form the statement is, okay, you draw a phase diagram of this model that describes like magnetic impurities.
And this magnetic impurities can become basically they are magnetic and is a known magnetic phase. There's a pheromone phase and there's a known pheromone phase. And then it turns out if you compute the phase some of this model, right, it has to exist as a disorder strength. So you have like some dirt in the model and then it's temperature.
And basically the statement is that the largest amount of noise that your court can tolerate is actually exactly equal to the largest extent of this or that phase that I indicated in RET, you projected onto the x axis, which is it disorder in this in this model? This is a remarkable connection where you can learn something new about this quantum system, something that exactly to a very old model, the system used in condensed matter physics.
Maybe now, in my last slides, I want to tell you a bit about what I do. So. Um. So. What is the active research question in kinematic correction is the. 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. So they are complicated beast. They look mesmerising as I like them. Like these patterns as well. But really, what you should remember is this is basically your circuit layout.
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. And then the statement is, okay, of course this is harder to build. You also have to define some checks. I don't want to go into too much detail in this, but again, they are local and there's some description based on the graph and they are harder to build because they are not naturally embedded. Right. It gets very crowded to the edge if you go to the edges of this plot.
But if it is built, it has a much reduced overhead. So this connects maybe to one of the previous questions that we had. The topic, quote has a lot of all that a single qubits will be encoded into this grid, right? If you have a ten by ten grid, right. 100 atoms, for example, it will be one logical qubit.
Effectively. The problem is this if you do it by this, not on this gregori, but on this geometry, what you will get is if you make it bigger and bigger, you will not only protect information better. But you will also get more cubits and call it. So the information gets better and you encode more. But this one is quite maybe some of, you know, the code series called a finite rate cut.
And of course, in principle, this can be built. So this is a beautiful picture of an artificial, hyperbolic lettuce also made from superconducting resonators in the laboratory of Andrew Hauck in Princeton from this publication.
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, for example, is now you can also claim the again for these quote model, the error correction process can be mapped to some model of condensed matter physics and indeed you can in maps on a very complicated model that lives in in curved space.
So on a on a negatively curved space, these models that maybe in the past had been only studied, but mostly studied actually by and by high energy physicists because the universe kind of is a negatively curved. Well it's supposed to be maybe negative because surface and and turns out okay we could map it.
Exactly. And and this mapping kind of then allows us to calculate for these kind of codes the optimal information, theoretically optimal performance that you can get at a certain size and a certain strength of noise. And remarkably, also, this actually yielded some new insights into the statistical mechanics of these of these models. So you can not only learn basically about new systems.
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. Okay. And was that I reached my summary. So hopefully I'll convince the quantum of corrections in you. Possible, which is surprising. And Alice introduced the Threshold Theorem, which says, okay, QC, the constituents have to be good enough. The experimental state of the field is. We are just scratching the surface, maybe off the threshold.
And I've also told you about some current research that that I am doing just to connect with the team. Oh, thank you so much. I want.