Everyone, welcome to the session talking about quantum s pad algorithms as a means of harnessing quantum advantage in Niscara. So, before going ahead with the session, I'd like to explain how the whole session will be structured. So, there will be two components to the whole session. One would be the presentation deck, and the other would be the notes deck. The presentation deck would serve, as you can say, major bullet points or the main points which I'll be covering in this whole session. And the notes would contain an in detail explanation of the main points which I talk about in the presentation. So the two things will go hand in hand, and I think both of them will be provided to you after the whole session, so you can go through both of them and use them as a computing material to each other. So without any further delay, let's get started. So here we have the presentation deck. And today, as I mentioned, that we'll be talking about quantum and SpAd algorithms as a means of harnessing quantum advantage in Nescara or quantum computing. So even before we get started with the whole presentation, it's important that I talk about what Nescara is, because I'll be using this whole chronology of Nescara a lot in this whole presentation. So basically, Nisq, as we call it, is noise intermediate scale quantum, or you would basically say and has of quantum computing, in which the computers which we have right now, the hardware which we have right now, are prone to error. They are not that much scalable, and they have their own limitations. As we'll talk about going ahead with the presentation. So the first thing is, again, always is the introduction in which I'll emphasize why even we are talking about quantum computing and what is the potential of quantum computing when we talk about solving large scale problems, then I'll emphasize on limitations of quantum current hardware. In sense, I'll talk more in detail about NisQ era of quantum computing. Then we'll talk about some concepts of quantum mechanics that we particularly exploit when we are talking about quantum inspired algorithms. And. Yeah, I think before further delay, I will now I'll switch to the notes part of the whole presentation and we can get started. Yeah. So, as mentioned, even before we get started, it's important to talk about why even are we talking about this whole. Why even are we having this session, to be very honest. Right. So that would be the capacity of. Yeah, give me a second. That would be the capacity of the quantum computers. Computers to solve complex problems. So I think there already would have been a lot of sessions which will be particularly highlighting this aspect of quantum computing before you get to this session. But I'll try to keep it brief just so that, just for context, so that we have the continuity of explanation. So, see, in quantum computing, we have a concept called qubits, right? And just to put it very, in a very abstract sense, we know that the qubits, whenever we add one qubit to our, let's say we have some n qubits here, like we have some n qubits which we are using to solve a problem. Let's say the problem is px, right? So if I'm adding one qubit to this whole system, I'm basically doubling my computing capacity, or I'm doubling the potential of this system to solve any kind of problem, right? That is how it goes. I think that is what you must have heard by now from previous presentations or by your own knowledge of this domain. So this is the core concept, this is one of the core concept of why we even want to go towards the quantum computing side. Some problems, like you can say your traveling salesman problem, your traveling salesman problem, and then you have problems like, you can say topology optimization problem, you have your genetic, you can say your usual combinatorial optimization problem, I would say combinatorial optimization problem. So these problems don't have a particular solution, right? So they work on an approximate algorithms. So there's no better way to solve these problems rather than doing it on a quantum computer. So this in some sense, is where we see the potential of quantum computers to be able to solve complex problems. But again, now, this would be a sort of like a basic of why even do we go towards the quantum computing side? I means I think I didn't go much in depth for this particular section. And the reason for that being that I think it would be already being covered by other sessions, because I already saw the lineup. So they are like particular sessions which talk about exactly this aspect of quantum computing. Now I'll come to the more on the side of the things which actually rates sense, which is actually important in sense of what we are trying to achieve here. So that would be the limitations, limitations of current hardware. Let me switch to a different pen. See, the very first number, very first limitation is how we put it, is scalability. So whenever we are talking about quantum computers, right, whenever we are talking about problem solving, any kind of problems. So an aspect of scalability is one of the most important aspect, because let's be honest here, right, whenever you are trying to solve an industrial case problem, the number of variables or the number of potential candidates, which you have to consider as a potential solution to a problem, they increase exponentially. Right. Let's take a very simple example here. Like, say, I have to say, just basically find this traveling salesman problem, but instead of a city, I have to solve it for a whole country, right? So it would make sense to you, right, that now the number of possible solutions or the number of possible path that that salesman can take, they increase exponentially. Right? So whenever, basically we're running one more point or one more possible endpoint or access point to the whole TSP problem, we are potentially increasing. We are doubling the number of possible candidates, right? So it's easy to see that. And scalability is an important aspect whenever we are talking about any kind of problem. But that is one of the major limitation of current hardware, because you can see there are multiple architectures which we have in our quantum computing domain right now. We have like, annealing architecture. We'll talk about annealing in details later. We have gate based architecture. Gate based architecture. You have your topological qubits, logical qubits. So these are some of the major architectures which we have in quantum computing. All of them, they could be like superconducting qubits. You have your diamond head qubits and your electron spin qubits. All of them are derivative of this major architecture here. I'm talking about the architecture of hardware, not exactly type of qubit which we use, right? So now, considering these architectures, there are a certain hard limits or the number of qubits which we have available, right? And as we'll proceed, we'll also talk about your quantum error correction as a sort of like, what do you call, like extra weightage, which we need to carry when we are never, we are trying to solve anything on a quantum computer, right? So coming back to this. So scalability becomes an important aspect because, see, due to certain aspects like quantum error correction, we are already losing some qubits. So by some chance, say we had total n qubits available as given by any hardware vendor, hardware being the quantum hardware. And out of these n capital n qubits, some small n is usable, while some small m would have to be reserved for certain aspects. As we'll talk in detail has, we go ahead, like, there are aspects like quantum error correction, ancillary qubits, support qubits. So those are not usable qubits. So we are saying that even if we have this capital n number of qubits, not every qubit of every qubit is usable. That just means that we are further reducing the scalability. And this value of capital n is already not that large. We are talking about 120 usable qubits. That is like more or less the upper limit, which we have right now. And 120 possible qubits is not that much as, I mean, you have to take my word for it, because if I go in detail and you try to explain what different kind of algorithms would require different kind of number of qubits, then it comes out to be a large number. Any algorithms which you will develop right now on quantum hardware or quantum architecture will have certain limits. And those limits are very hard, and those limits do not scale well, when we are talking about scaling the whole problem into something which can be used directly in the industry. Right. The second aspect, or let's say one of the second issues which we have in the whole quantum computing thing, is noisy qubits. Okay? So even if we say in the previously I mentioned that say we have n qubits, right? But I said that out of this n qubits, only small n are usable, right? But even the small n are not high quality qubit. They are not usually. Okay, when I say quality. So, see, when I use a qubit, like any other thing or any other particle is just, what do you call, like, say, if this whole box was my quantum system, then this qubit is a particle in that system, right? So in ideal situation, we would want this particle to be able to retain its property for lawless amount of time, right. We would want this particle to be able to maintain its state, as we call it, in the quantum realm, for the largest amount of time. But as I'll talk about that, there are certain, you can say, losses, which occurs, right? So this particle is not able to maintain its state. So these, what we call are known as noisy qubits, because over time, due to decoheration interference or through outside interference of material, they lose their potential state, right. What would mean is that, say, somehow, if this was a quantum particle, this was a state of a qubit. Like this has some information which is important, right? But if I were to leave this qubit, even in an isolated system, isolated system here being a quantum hardware or a quantum system, the state would be lost over time, and this would actually release back, come back to its original state, which is zero. Zero being your usual qubit, zero or one. So this is usually called decoherence. Decoherence. And what this leads to is that this leads to loss of information. So basically, you cannot rely on a qubit to be able to maintain its state. And other aspect to this whole noisy qubit thing is one would be, as I mentioned, coherence. Coherence. The other is the error propagation. Error propagation. So what error propagation basically means is, I'll take an example, example of a gate based architecture. So in rates based architecture, what we usually have is we have a qubit, which usually starts as zero and it passes through gates. Whatever gate is X-Y-Z Hadamard or whatever gate it is, it passes through the gates. So, yeah, for this talk, I'm assuming that you have the basic understanding of quantum computing. So that is something which sort of assumed in the beginning. So again, this would be the x gate, and then say we have some different gates, say y gate, and then we had some different gates, h gate. So every time a qubit passes through a gate in mathematical sense, it's very easy for us to know, right? All of us at this point, if you have read about basics of quantum computing, you already know that every qubit is basically a matrix, right? Say I'm having a simple matrix of zero and one, if this is basically a one. So I mean, when I say I pass it through a gate, I basically means that I multiply it with the matrix associated with that rates. In the rates based architecture, at least, right? We get some solution state, and which is the state at the end of this whole, let's say this whole socket, right? But in actuality, in actual physical terms, in hardware term, we know that we use different possible principles. Like we use microwaves, we use certain, what do you call photonic radiations, right? And we use certain aspect, certain actual physical, physical properties of these particles to store the state of the particle, basically to change it, like to modulate it, right? So basically what we are saying is that after, say, if I started with psi one, wait, just give me a second. Yeah. So if I started with psi one state, so what I'm saying that after passing it to every gate, I'm changing it to some state, psi two, and I'm changing it to same state, psi three, right. And so on. But in the course of this action, because I said that the qubit, which we have right now are not of high quality, we sort of lose some data, we lose some state of that particle. So some error gets propagated, like say some e one error is propagated the first time we pass it through a gate. Some e two error is propagated the second time we pass it through a gate. And basically this, and goes on. So basically what happens is that now this e two component will be a sum of e one plus sum plus e two. Basically the error of the previous gate plus the error of this gate. So what I'm trying to imply here is that there's a propagation of error as we go on to pass the qubit to gates. What this means, and what this actually means in terms of consequences sense, is that we have a limit to the number of rates, number of gates, or circuit depth, as we call it in quantum computing. So the circuit depth is limited, and limitation of the circuit depth would mean that you cannot apply certain algorithms because certain algorithms require huge sockets for larger number of qubits. Because, see, usually one thing which I think you got out of intuition is whenever we are talking about scalability aspect, we are basically talking about number of qubits. But it's also important to understand that when you increase the number of qubit, you are in some sense also increasing the circuit depth because more qubits require more gates, right? I mean, in sense of the circuit depth. So circuit depth and number of qubits both becomes an equal problem when you're talking about the scalability aspect of current hardware. Okay, then I'm talking about errors, right? So it's important to know that there are error correction code, quantum error correction code, or QEC, as we call in short terms. But these error correcting code, error correcting codes right now are not of high quality. This is something of a very high research priority right now. There's a lot of research being done in this exact domain. But again, what does this mean in physical sense? Right? So in physical sense, it means that even if we are able to get capital n user qubits, we will only be able to use small n number of qubits, because we will have to give this some m qubits for these quantum error correction codes. This m can sometimes be large if you really want a good higher fidelity results, right. Then you'll have to make this m large, which would mean that this small n would be small. So basically you'll be limited by the number of usable qubits which you have for solving any kind of problem. So this becomes an issue. And again, you can say this is more on the, these were like the major ones, which have actually have physical implications, but there are other smaller aspects as well. You're talking about calibration issues, right? How do you calibrate calibration issues? Like how do you calibrate these qubits? That's again, that becomes an issue. And then you're also talking about the variability as a factor. The thing is that, as I stated, in ideal world, all the qubits should behave identically. Right? But in physical sense, it's important to understand that. And qubit is basically a particle, right? That follows certain principles, as you must have heard, just for context, that any qubit is any kind of particle that has two possible end states, which being zero or one. We are just naming it zero one. It could be anything. It could be plus, it could be minus, as you must have heard. So we have different bases, right? This is the computational rates. We have the planar bases. We have the polar base. There are different kind of bases, which we have. But any particle which has two possible end states and has a possibility of being existing in a state, say psi, such that it could be a combination of both. Sorry. Zero plus beta one is basically a qubit. So, qubit is a concept. It's not actually a physical thing. And physical particle can behave like a qubit, so that physical particle, again, will have its own limitations. But how do we ensure that? Say we have different particles, but all those particles behave exactly the same? Because I can assure you that. Okay, I tell you that, okay, this particle is a qubit. And I'll tell you. Okay, basically, this p one particle, I tell you, is a qubit, and this p two particle also behaves like a qubit. But how do I make sure that these two are calibrated? They don't show any kind of variability. Right. That, again, becomes an circumstance, sort of like an issue on the hardware side of things, to be honest. Like, this doesn't have a lot of limitation directly, but it becomes an issue when you're talking about from a more in depth hardware perspective. Right? Again, one of the other important issue which you'll face when you talk about limitation is qubit connectivity. This is one of the issues which you won't directly have issues with. This won't become an issue to you, like in the initial phases of, what do you call your planning, your algorithm. But it becomes an issue later. So what happens is that, see, because, as I said, every quantum hardware is basically a quantum system. So if I say that I had qubits, I told you, right, every particle, a qubit is basically a particle that follows certain principles. So, say these are the particles. And I have already proved by some physics method or physical method, that they fulfill the properties of a qubit. Okay? So, now, given that this system, this is a closed system. So this is a closed system. Now, I already stated that this system is already existing quantum hardware. But the issue becomes here is that, see, there is a limitation of the connectivity between the qubits. Like say this is the kind of connectivity which we follow and say this is connected to this and this is connected to this. So this is the kind of connectivity which we have. I think if you're interested in this, you can just Google connectivity graph of any, say quantum hardware, famous being the IBM basic. The IBM provides the quantum topography. Or you can say this particular connectivity map of all their hardware. So you can go look it up. So let's say this was some quantum hardware, q one. This is the connectivity map of that hardware. So this is particle p one and this is a particle p two. So let's particle, this is particle p three. So you can easily see that there is a direct path from p one to p three. So now, okay, now I'm trying to say run three qubit algorithm. So I am basically taking p one, p two and p three. And all of them start with zero. Because initialize, we initialize it with zero. So it's easy to see that. Okay, now say I have to use a c zero gate, right, which is a two qubit gate. And I want to make the control qubit at p one and basically the not gate at p two. It's easy to see that. Let's make it p three and this would be p two. Yeah. So it's easy to see here that if I want to perform this simple operation, it is possible, as my connectivity map tells me, that there is a direct connection between p one and p three. Particular p three qubit. Right? But what if I now need to form a c zero gate between p one and p two, p one and p two. It's easy to see that there is no direct path between the two. I mean, I'll have to connect. Basically there is a path, I mean there's a path, but it goes to p three and say p four and p five and p six and so on. Right? But there is no direct path. There is no one step path from p one and p two. So p one and p two are not directly connected. This becomes a limitation. Okay. I mean, there are ways to do it, right? If I didn't tell you. So this is not even if I want to, this operation, it's not a wrong operation. It is possible. We basically use swap rates now. So what basically swap gate is just for context is in simple notation term, we just represent it through two x's. This is basically a swap rates, but in actuality a sortnet swap gate. Is basically three c zero gates. So it's like this and this. So you can basically swap gate has a whole matrix associated with it. And if you try, if you multiply the matrices for three c zero gate, tensor product them, you also get the same matrix. So basically, in theory, we just use this notation. But in hardware sense, this is what happens in the background. What this swap actually means is now, if I want to go from p one to p two, I'll basically be swapping this p three with p four and swapping this p four with this p five and p five with p six and so on until I'm able to swap p two with p three. So basically, I'll swap this gate with this, right? So it's easy to say here that what I'm saying right now is that for every swap, I'm basically adding three c zero gates, and every c zero rates is basically two single qubit gates, right? One is control and one is c zero. One is the not gate. So basically, what I'm saying is, for every step, I'm having six gates more to the circuit. So for this kind of operation, let's say, which we had 1234-5678 so it's basically eight into six, like 48 gates just to do a single operation, single operation, single logical operation. But 48 physical operations needed to be done in order to do that, right? So it's easy to see that this affects the whole scalability aspect of our hardware. So again, connectivity becomes an issue when you're talking about the broad and broader sense, the limitation of niscara devices. So these were some of the major limitations of, you can say, your niscara devices and why even we are considering these are some factors, why we do not basically run all the algorithms which we have right now on quantum hardware. These are the things which are stopping us. These are major research points. I'll be honest with you. I'll not say that this is something which will affect us for a longer time. Like, people are diligently working on these things, particularly to improve the state of the current hardware. But this is what I say, what is what I mean when I say that we are in sort of like a Nisq era of quantum computing, where we are limited by the current hardware. So just last put it, I talked about Nisq. So now I think every point, every single wording of this whole abbreviation should make sense to you, right? Nisq, noisy. We talked about noisy qubit intermediate, because we are in an intermediate stage right now scale, because the scalability is a huge issue right now. Mean we do not have scalable hardware, quantum, because again, we are talking about quantum hardware. So why not? So this list should make sense to you. I spend a lot of time on this because it's important that you understand what are the limitations. Why are we even talking about this? Why are we having this session? Why are we even talking about quantum has bad algorithms when we already have algorithms for quantum hardware? So now comes now, coming to the main subject, quantum s pad algorithms. Algorithms. Okay, so what are quantum AspAd algorithms? So, let's go with the wording. First, we have quantum again, which would means quantum mechanics principles. Inspired, meaning that we are inspired by the principles. We are not exactly. We are not able to simulate them in some sense. Sorry, we don't have that hardware of that kind. We are not doing it exactly on a quantum mechanics realm. Right. But we are simulating those properties. So inspired would means that we are limitations simulating those properties. Simulating and algorithms, again, is a series of steps. So, basically, when we say quantum has bad algorithms, we are talking about the algorithms which simulate some quantum mechanics principles to get performance benefit. I mean, it wouldn't make sense, right, if you don't get any kind of performance benefit. So we are talking about some performance benefits that go along with the algorithms. So, coming back to the presentation now, so, again, we talked about, in an introduction, we talked about the potential of quantum computing to solve complex problems. We talked about niscara, basically, the whole Nisq thing, and we introduced the concepts. Now we'll go in detail in the whole concept thing. Again, this is something which we have already covered, basically, understanding the niscara of quantum computing. It's important to understand the challenges and the limitations which we face with the current hardware. And again, we could talk about, you can see basically the third point, which means leveraging the use of quantum AsPAD algorithms to still get some advantage, but not exactly be limited by the hardware of current time. As you will see, that basically, what, when we say, when we talk about quantum aspired algorithms, which we are basically saying, as I stated, that we are simulating the properties of quantum computer, we are simulating some quantum mechanics principles. Those principles I'll talk about in the next few seconds. So, those simulating certain properties of quantum computing to get benefit. Right. So, this simulation happens from classical hardware and by lot of work done by our ancestors. Classical hardware is at its very best right now, in some sense. We do have great hardware. We have gpus, we have cpus, we have GPU, CPU, collaborations. Right. So we have hardware that is classical hardware that can scale. Again, one thing which I would particularly point out in no sense, by simulating the property of quantum mechanics, will we be able to get the whole benefit or the full potential. We won't be able to achieve the full potential that these algorithms have. These algorithms are basically, in the end, are meant to run on a quantum hardware. But because the hardware is still in the budding stage, we are just simulating them on a classical hardware and still getting some performance benefit. So, basically, the thing of essence here is that scalability is possible on classical hardware, even if we are not exactly gift. It's like you can select a trough, has sort of a trade off. On a real quantum hardware. We expect huge performance bonus, but scalability is an issue. But in quantum inspired, the trade off is that performance might not be that big as compared to a pure quantum hardware, but the scalability is there. We are potentially able to scale them in order to solve, you can say, industrial problems. So that is where the essence of quantum inspired comes into play. Okay, again, we have talked about what quantum inspired algorithms are. I sort of talked about the motivation. Why even are we considering quantum inspired? We are considering quantum inspired for its scalability aspect and still being able to deliver some performance benefit. Because, again, we are here because some kind of performance benefit is what actually a performance boost is actually what we are behind. So, again, now I'll talk about the properties. I've always been saying that we use certain properties of quantum mechanics and this and that. What exact properties am I talking about? The very first is parallelism, or means, as I like to call it, parallelism or superposition. Superposition. So even before getting, or getting in exact, detailed discussion of his concept, I'll take a very simple example. So let's say I have a coin, right? And for any coin, we have two possible end states, heads and tails. Now, let's say that this is a qubit. For the sake of this discussion, I'll assume that say this, my coin is a qubit. Now, whenever I flip this coin and I say it falls into my wrist, right? It falls into my wrist. And now I close my wrist. So until I open my wrist, there is a 50%. I means this, heads and tails. There's a 50% chance that it could be has. And there's a 50% chance that it could either be tails. Right? So we can say this particular coin right now is in a parallel position. It is a parallelism, is in a state of parallelism. It's in a state of superposition, it's equally likely to be heads. It's equally likely to be tails. Or if, for the sake of consideration, as I previously mentioned, that I'm considering, this coin is basically a qubit. And we know that I talked in very slight details that any qubit, basically, should follow this principle. That it should be able to have a state which should be defined with the help of this alpha and beta component, which I think you guys should know that the square of this alpha and the square of this beta is basically the probability of this particular state, psi, after measurement, to either gives heads or to either give tails. Right? So, I mean, this is some basic understanding, which, as previously mentioned, I'm sort of assuming that you guys have this kind of understanding. If you don't, don't worry, you can just record this. You can just pause this session and you can go back and just type introduction to quantum computing. You might even find some of my videos back there. But, yeah, my point here being that say, okay, now, the sake of this. Coming back to this discussion, for the sake of discussion, as I said, I'm assuming that the coin is sort of like a qubit. So in theory, it should follow this principle as well. Right? This coin should also have a state, right? And as a state, and as I previously mentioned, that a coin has two possible end states, heads or tails. So, basically, the state of a coin should be. We should be able to define it in this sense as alpha heads plus beta tails. Where the square of alpha and the square of beta defines the probability of the coin being either heads or tails, respectively. So, because as I previously, in our case, I'm saying that the coin is in my fist, right? And my fist is closed. So by common sense, we can say the value of this alpha and this beta with a square of these values are both zero five. Right? Because, as I said, I mean, right now it's in a 50 50 state, right? So it's zero five, right? Square. I mean, the value of the square is zero five. Mean the actual value would be somewhere along one by under root two. Right? Because one by under root two, the square would be one by two, which would be 0.5. So basically, alpha and beta, to be precise, would be one by under root two. And the square of these values is one by two. Or 0.5 or 50%. Right. This is what we call superposition. Now, how exactly does this come into play? In actual, you can say, quantum computing aspect. So what happens is that. See, okay, now, there were only two possible states. But now, in some sense, I'm saying that this one single coin, c one which has a state, psi one, when it is in the wrist, right. It is a combination of heads and tails. Or basically, it is a computational of two states existing simultaneously until we open the wrist. So, just for context, the opening my fist. Opening fist, basically, is what we call measurement, which, again, is, again, is a concept which you must have heard when you were studying introduction to quantum computing. So, basically, measurement is at the end. Measurement is nothing. But after I open my wrist, it should always be either heads and tails. After I open my wrist, it cannot be in that in between state. Right? Now, it has to decide. Now, it has to either be heads. It has to either be tails. So at end, that finite, that discrete state which the whole state, which the quantum particle has to fall into, this is in some sense called measurement. I mean, measurement in actuality is a bit complex. It's just a complex concept. But if I talk about it, then I have to talk about a computational basis and then talk about how exactly the best example I can give a measurement is like you're talking this cat, the whole experiment. Until you open the box, the cat is both dead and alive, right? But has you open it and you observe the particle, you observe the cat. It has to either be dead or be alive. Then it cannot be in a state in between. Right? Exactly. Like a coin. As soon as I open my wrist, as soon as I observe my coin. Now, it has to be either heads or it has to be tails. Exactly. Now, in actual theories, in qubit, which has two possible states, like zero or one, as soon as I perform the measurement, or I observe that particle, it has to be either one or zero. Now, it cannot be in that in between state. But in quantum computing, what we do is we take benefit of the principle or the possibility of this particle to exist in state of in between, say, zero and one, we take the advantage of this while it's still in superposition. And what we do is like, see, this I'm talking about in pure quantum computing sense right now. So what we do is we modulate this values of alphas and betas. And by doing that, we are basically modulating the final result, which we'll get when we perform measurement of this psi, of this state. So, let's, for the sake of convenience, like, very simple example, if we had a black box, and we need to find the solution. This black box is basically a function f x. And we need to find the minimum of this black box or minima of this function, right? And for the sake of this convenience, let's say that the minima was zero, right? Basically the value of x being zero is what gives the minimum value of this function, f x. The sake of this convenience. And we are using only one qubit. We are using only one qubit, says zero. So this right now is in a state where alpha is one and beta is zero, right. I should just make it one. I think it would be better. So it would be alpha is zero and beta is one. Right. For the particle, following basically this whole aspect. So now that we know this, now we know that the solution in our case, for the sake of has convenience, we know that at the end the alpha should become one and beta should become zero for the particle after measurement to give this discrete value zero. Right? Now, as an algorithm designer, in this case, we know what we need to do. But usually we have a fitness function which tells us how close we are to a solution. We have some function, let's say we have some finite function which tells it how close we are to the solution, okay? Or some methodology by which we keep telling the algorithm that, okay, you are approaching to the solution, you are getting farther from the solution. So make changes accordingly. So let's say that here, because we already know, and we are an observer, so we know we need to, at the end, make this alpha to be one. So now what I'll do is I'll make an algorithm, the particle. After, let's say I'm using an Hadamad gate, after I put it into a superposition, I'll now perform certain operation which will make sure that after those operation, this alpha falls into alpha becomes one and beta becomes zero. In this case, because we had one qubit. So there was no sense of using parallelism. But now let's say we had n qubits. So now when we are talking about using n qubits, we are basically talking about two to three n potential states existing simultaneously. When the n was one, basically we had two states existing simultaneously, right? Which was zero and one. So we were basically checking the values of zero and one both for the black box, right? And we saw after, because now in this case we already know. Like we just saw that, okay, this has to be zero. So we make alpha one. But let's just say we did not know that. So now in some sense, we are basically using the two values simultaneously for checking the value of the black box. This is exactly what happens in dutch algorithm or Dutch Zosk algorithms. If you're taking multiple qubits and for single qubit, it's called dutch algorithms. So you can look it up. I mean, it's a pure quantum algorithm. It's dutch algorithm. Okay. This is once, in some sense, what supervision is. So, superlation is basically using two to the power n possible states simultaneously for a given function, f x, to find whether. Which of them is the closer to the solution or which of them is a solution sometimes. So that would be even the Grover search algorithm takes advantage of this, which is comparatively a bit popular algorithm. So that also takes advantage of quantum superposition. Again, now, this principle is, we can say one of the most important principle which we make use in quantum has bad algorithms. So if you talk about. I'm talking from a very simulation sense here, what we are basically doing is we are changing this alpha and beta parameters, right? For any particular particle, we are just a state, or say we have the particle psi one. These are a function of these two particles, alpha and beta. So if you modulate this alpha and beta, in some sense, we can simulate this quantum property. Right? So initially, what this mean is that initially put it something random and over time, keep on modulating the properties of alphas and betas and accordance to the closer you reach to the solution state. So this is, in some sense, how we simulate quantum parallelism in quantum inspired algorithms. And this is some sort of like an important concept. And again, the second concept, which we usually have in quantum inspired realm. Let me just change my color. Yeah. Quantum tunneling, this is more. If quantum superposition was more on the general concept. Quantum tunneling is more prone or focused towards your quantum annealing concepts. Your quantum annealing is basically, as I previously mentioned, is a type of architecture which we use, I think annealing in general, I think you do have an idea, right? Annealing is basically like, over time. Like if you heat a metal, over time, it cools down, right? So every particle in this whole universe has a tendency to be in a minimum state of energy. So if I draw the energy versus time graph, if I draw this, so every particle will eventually reach that state of global minimum or global minimum energy, right? That's a property. So they will basically utilize this property to, what do you call, like, simulate, to basically find the minima, global minima of certain functions. Right? Fx. So basically, when I say Fx, it doesn't mean that this is only limited to your function optimization. In theory, it said that any kind of problem which you are trying to solve, it is always possible to map that problem onto a function. And so every problem, in the end just becomes an optimization problem if you think about it, right? So this Fxfx is basically the problem which we are trying to solve. Okay? Now, let's say the landscape of this function f x over x is, say, along this line, it changes like this, right? So as I said, annealing works on a simple concept, classical annealing, I'm talking about. It will keep going down. It will keep going down until it reaches this point. Let's just give me a second. Okay, this was how it was. But you can see that this is a local minima, and this right here is the global minimum. Right? Now, here is a limitation of classical leveraging. Now, if it needs to reach this global minimum, it also needs to go over this whole steep hill, right? Until it reaches again this valley of this whole, you can say landscape, right? But here, whereas classical would depend on randomness or would depend on sort of like a miracle to just cross this whole trope. And in quantum, we have a concept of tunneling. What exactly is tunneling? I'll give you a brief about it. So what happens is that, okay, this is, interestingly enough, quantum tunneling was discovered while we're trying to improve a classical hardware. So what happens is, has, per Moore's law, you already know that the number of transistors which we have on an integrated circuit doubles every decade, every three to four years, right? So that means if you're doubling the number of transistors, that would be. There are two ways of doing it, right? Make the integrate IC chip larger or make the transistor smaller, right. So again, it's obviously not possible to make the IC larger because we want our devices to be smaller, not bigger, right? So we make our transistor smaller. So, see, again, there is a limit to which we can reduce the size of a transistor. After a certain limit, the quantum mechanics principles comes into play, and they start interfering with our results, and we don't get the idealistic results. So when that happens, the scientists started to notice that there is this phenomenon which happens of quantum tunneling, that the transistors become so small that electrons just passes through them and they don't have to interact with them, they just pass through them. So this is what they call tunneling. So in our respect, what happens is that in quantum annealing, there's a possibility that the quantum particle, the particle which we have in annealing, they can just tunnel through this whole steep and directly reach this point. And then by usual annealing, we will be able to find the global minimum. This is sort of like the concept of quantum tunneling. And this is the concept which we used when we talk about quantum inspired annealing. Quantum inspired annealing algorithms, which I think you do have a session on again. So, again, I won't go in bit of detail about this, but this is another concept of important concept which we use when talking about when designing, you would say, quantum annealing algorithms. Okay, so these were the two majors. One, I mean, we do have concepts of quantum entanglement, but those are sort of like pure quantum concepts. And it's very hard, or it's close to impossible to sort of get those on, simulate them on classical hardware. So these are very advantageous if you have quantum hardware. But in quantum inspired world, it becomes a bit harder to simulate these. So we don't particularly rely on these things. Okay. So these were some of the basic or important principles of quantum mechanics, which we try to simulate whenever we are trying to work on designing quantum as pad algorithms. So we just cover this slide in the previous slide where I talked about the principles which we usually employ when we are talking about designing quantum inspired algorithms. So this slide talks about some of the major, you can say, quantum inspired algorithm, which we do have in operation. Surprisingly enough, not many people know this, but algorithms like QAOA or quantum approximate optimization algorithms, your variational quantum eigen solver, and even your quantum neural networks, you can say all of them, are in some sense, example of your quantum inspired algorithms. So how does that come into play? Is that, see, what happens usually, is that these algorithms have, basically what they do is they map the problem. We have a problem statement, and they map the problem onto a function. Right. And what they try to do is they try to optimize that function. And while optimizing that function, as I previously mentioned, they try to use the principles of quantum parallelism to improve the result. Again, these algorithms, as I like to call them, are also quantum ready. That would mean that you can run these algorithms on actual quantum hardware as well. So what they basically, you can say, assure you, or they try to guarantee you, is that in future, when you will have better hardware, you should be able to just take these algorithms from here and just put it on a real quantum hardware and get all the benefits of quantum computers as well. Okay, now I'll talk a bit detailed about some of the use cases which we have of these quantum computing, quantum inspired algorithms. And one of my personal favorite being quantum inspired optimization algorithms, which is sort of like my specialty. Sort of like what I particularly work on. Yeah. So, again, some of the let me just switch my color some algorithms. So one of the most famous and common algorithms are your Qio algorithms. Quantum inspired optimization, as I call it, basically optimization algorithms, again, which is something which is more of my expertise. This is where my expertise lies. So, in Qio, we have evolutionary algorithms, we have annealing algorithms. So these would be the two type of algorithms which we usually have in quantum inspired optimization. In evolutionary algorithm, you can have a number of algorithms, like QGA, quantum inspired genetic algorithm, basically quantum inspired particle sum optimization, quantum inspired wolf optimization algorithms. In annealing, you'll have your usual annealing algorithms. I'll not go in detail about each and every algorithm, but I'll try to give you a glimpse of how they function. So, basically, any kind of optimization problem is basically optimizing a given function. And this fx is basically the mapping of a problem, px. So every function has a landscape. Let's say this was the landscape of this function. And within this landscape, there is one global minima, right? There could be multiple local minima, said this is also local minimum. This is a local minimum. This is a local minimum. There's a local minimum. I'm talking about. This is like a 3d projection, just viewed from above. But in actual sense, you can also look at it like this, say this could also be the landscape. And you can see here that this should be the global minimum. So now what happens is that when we are talking about your classical optimization algorithm, so they usually follow a very first, I'll start with simple gradient based algorithms, like your stochastic gradient based algorithm, or gradient based algorithm. They start with some, say some point x. So this is some point x. So these algorithms then follow this gradient of the landscape. So in our case, it's easy to see that this algorithms is likely to get stuck in this local minima, because it won't be possible for the gradient based algorithm to make the jump to this side, right? We also have stochastic based algorithms which can make this jump, but again, they too have the limitations. And for complex, say, non convex, what do you call landscapes, they fail. So that is where evolutionary algorithms come into play. Evolutionary algorithm. See, I'm talking about classical evolutionary algorithm right now. So those do solve these problems. That is why. So if you ever work with genetic algorithms, then you must have heard it that genetic algorithms can, in theory, solve any kind of problem. Given that, first thing is you can map that problem onto a fitness function, fx. And second is that you have infinite computational resources. Those are very computationally expensive, your genetic algorithms and stuff. So how the genetic algorithm, or any evolutionary algorithm in general works, is that they prepare candidates c one, c two, c three, c four. And they prepare candidates and they test every candidate on this function fx, right? So this candidate would basically mean maybe some random points on the landscape. Some random points on the landscape. So they depend on the theory that out of all these candidates, one would perform the best. In our case, this is the best performing one, right? Let's name it, c three. So c three is now the best, and the worst is, say, in our case, it's obvious to see that, okay, this was c one, then c one was the worst performing, then c one would get eliminated. It just follows the basic darwinian theory of evolution that fittest of the survival of the fittest, right? So c three is the best, c one is the worst. Now, we perform certain operation depending on what kind of evolutionary algorithm we are working with. So in order to ensure that more candidates in the next generation are towards c three, right? So basically, we keep on repeating this process until we finally reach this point x or our global minimum. This is how your evolutionary algorithm works, right? So where does quantum come into play? Is this that. See, as I said, that the biggest bottleneck here is the computational resources for genetic algorithms or for evolutionary algorithms. So, if you remember, I talked about the principle of quantum superposition. So what that would mean is that now that now I can test multiple possible, for any psi estate, we know that there are two possible parameters, alpha and beta. So if I modulate this alpha and beta by using some, I won't go in details, but I start using some operations, then I should be able to check multiple solution, multiple candidates on this landscape simultaneously in single operation. So, basically, I used to produce this advantage of quantum superposition, or quantum expand superposition to improve, to reduce the number of generation it would take me to converge, or basically, I would improve the coherence rate and I would reduce the computational resources required. Because I'm actually saying that even if I use less number of candidates or, say, small n, then also I am assuring that I can still converge to the global minimum. So that is, in some sense, how we look at quantum inspired algorithms. From a practical point of view, we are talking about scalability, right, and still getting some performance boost over the classical hardware. And that is one of the reasons why we are even going towards these kind of algorithms. So I personally, just for the context, I have worked on QGA quantum genetic algorithms. That is qpso quantum inspired sampling algorithm. Quantum inspired sampling algorithms. I have worked on some of other quantum annealing algorithms and everything. So, okay, just for quantum and annealing, we use the same concept, but because annealing, we are talking about reducing the minimum state of energy. So we convert our function fx into a hamiltonian. What is a hamiltonian? You can look it up. Hamiltonian is basically an energy function which we try to minimize. So some of the practical use cases, which I have myself worked on by using this evolutionary algorithms are some of the, one of the huge industrial use cases. There could be airfoil optimization, optimization, there would be topology optimization, and much more. So these are some of the problems which I have personally worked on. And if you look it up, these are very industrial and very high scalability problems. They are computationally very expensive. The function fx associated with this function fx associated with both of them, it is very expensive. It's not easy to evaluate this fx multiple times continuously. So it becomes very important. The less the number of times I need to calculate this fx, the better my algorithms is. So I was able to get best results and better results for these two kinds of problems. So these are sort of like the use cases which I have worked on. There are a tremendous number of use cases. So, as I previously said, right, that every problem px, which takes argument as x can be, if you can map it onto a function f x, then you basically need to perform function optimization over this function, right? You just need to improve the fitness or improve the value of this f x function. So it becomes a function optimization problem. So basically, any combinatorial optimization problem can be mapped to function fx, which can be solved using quantum inspired optimization algorithms. So that is, in some sense, I'm talking about it from a very abstract blueprint point of view. That is, in some sense, how we use quantum inspired algorithms to perform certain industrial or practical applications. So, again, this is what I particularly highlighted in the given, you can say given rates that, what are the benefits of quantum and Spad algorithms that they perform certain, they give you performance, performance, certain performance boost over your classical algorithms or classical counterparts, while still being able to maintain, in some sense, scalability, right? But again, there are some limitations of quantum inspired as well, because, see, as I said, that we are not using the full potential of your quantum computers, right? You're still sort of simulating the properties. So whenever we talk about simulating the properties, it's important to know that we are restricting the performance which we can get. So this restriction, depending in some sense, in actuality, depends on you. How much performance can you give up for scalability, and how much performance do you actually need to be honest in the problems which I mentioned, like the airfoil problem, topology problem, even a slight performance benefit or a slight performance boost means a lot, because it's such a sensitive industry. And the function which I'm trying to optimize is so computationally expensive that even if I'm able to reduce it by ten generations, the convergence rate, that means a lot. So that totally depends on the kind of problem or the kind of use case you are targeting. Okay, so, as I talked, considering all of this into perspective, considering taking this into perspective, it's easy to see that quantum inspired algorithms provide an immediate solution of harnessing the advantages of quantum computers or the principles of quantum mechanics, and yet not being directly dependent on the hardware side and limitations of hardware. So they do provide you a sort of like an upkey opportunity to dive deep into the whole quantum computing aspect in sense of practicality and sense of application basis, and yet not being directly dependent on your hardware limitations again. But this, again is sort of like, what do you call, like a backup plan sometimes like an immediate plan of action, until eventually we get the hardware of high quality, which is where we actually run our quantum algorithms and get the so called quantum advantage, as we call it, the quantum supremacy, as we like to. I mean, we don't use this term anymore, so I might get some backlash from using the term quantum supermassive, but in some sense, I'm talking about the pure quantum advantage. That is something which we can only do after we have proper quantum hardware. But for the time being. For the time being. And until that happens, quantum inspired algorithms are a good way of still harnessing the advantages of quantum mechanics principle by simulating them on a quantum hardware and getting, you can say, some kind of performance benefits for your use cases or your applications. So, yeah, this is sort of like the whole session. I had to keep it a bit short and did not dive deep in depth with a lot of concepts. So in the future, I might has well do different sessions on that. Or if you're interested, you can reach out to me in person and you can ask me any questions which you have in depth question or detailed question, or if you want to have a conversation about detailed discussion about some of the concepts or some of the use cases or application, which I pointed out, but could not discuss in detail because of the shortage of time and everything. So I'm always welcome, is always welcome from my side. You can always reach out to me that has pretty much it for the whole session. I hope you learned something new, because I think. One of the problems which I used to face, I'm used to hearing is that from a commercial and industrial point of view, is that if this is not scalable, if it is not that scalable, why are we even going with it? Why we are investing a time with this whole thing? Well, my answer to that is quantum inspired algorithms. Exactly the immediate answer. Like, I'm still using the principles of quantum mechanics. I'm still using my knowledge of quantum mechanics and quantum computing and quantum information, but I am simulating those in some sense on classical hardware. So I'm still able to get some advantage and still able to maintain some scalability. So quantum inspired is sort of like one of my favorite things, because I get to enjoy quantum computing, and yet I don't have to worry too much about scalability. It is a bit complex. It does get a bit complex, because sometimes that trade off, you'll have to decide on that trade off, and it's not always good, and it's always about rainbows. Right? You have some limitations to this quantum inspired aspect as well. But overall, as I said, that it's one of the potential perfect methodologies of harnessing the advantages of quantum computing in this career. So, yeah, that would be it. And thank you for your time and patience. And again, as I stated that, I'll add another slide after this, and the deck, which I'll share with you guys, which will contain my contact info so that you can reach out to me and you can ask any questions you have. If you want any detailed, if you want to get a conversation about detailed discussion of any of the topics, I'm always here and feel free to reach out to me. So thank you.