 today. I'm glad to introduce to you Sophia Valecosa. Sophia started her career as an experimental particle physicist, analyzing data by experiments at particle colliders in the US and Europe. Obtaining her PhD in 2007 from the University of Geneva, she moved to the Israel Institute of Technology as a postdoc for five years, and then returned to Switzerland as a fellow of the Swiss National Science Foundation. In 2015, she joined CERN, the European Organization for Nuclear Research, where she first worked on the simulation infrastructure that is required to model modern particle detectors. Since 2018, she is a researcher at the CERN Open Lab, which is CERN's initiative for collaboration with industry. She is in charge of coordinating CERN's quantum computing initiative and is also responsible for the deep learning research program at Open Lab. In her talk today, she will combine these two topics, speak about benefits that quantum computing can bring to machine learning, and outline possible applications in particle physics. With that, Sophia, over to you. Thank you. Thanks for this nice introduction. It doesn't feel like me somehow, but it's correct. Well, thanks for the invitation. I'm very happy to come and talk about this topic. I'm very passionate about it. I think it's very interesting and exciting times about quantum computing and machine learning, so it's only natural to find the time to work on the tool at the same time. What I really want to do today is to give you some examples of what we've already been doing, because it's a very, very new field, but at the same time that a lot of investigations are going, tests, prototypes, studies that people with a lot of enthusiasm actually have been trying. I will start first with some general introduction. This will not be an introduction on quantum computing per se, but I will really try to stress some general concepts about what do we mean actually first when we discuss, when we talk about quantum machine learning. What do we expect from this field? What are the challenges related? What is it that we already know, and what is it that we don't know yet? Then, of course, I will go through some examples. I'm not sure we'll manage to cover everything, but I might be in case to take questions on this or also I have additional slides as additional material. What do we mean when we talk about quantum machine learning? In very general terms, we are talking about interaction between machine learning approach to data processing and quantum computing. Now, this can go to waste, because what we can do is to use machine learning to improve the way we do quantum computing, but we can also use quantum computing to improve machine learning. We will see machine learning algorithms in general. We will see what I mean by improve, of course, because this is a very general term. When we go this direction, when we are talking about using quantum computing to improve machine learning, what we mean is that we can interpret quantum computers such as differentiable, such as types of differentiable computers, computing, and so circuits that can be trained by minimizing a cost function that will depend on our data. That's where the analogy and the similarity with machine learning comes. Now, why do we want to do that? Is it possible to get quantum advantage? What are the conditions to get quantum advantage? Is it something that is going to be related to the kind of problem that we want to solve and to the status of quantum hardware? Because let's not forget that quantum computers are a reality today, but the limitations of what we call near-term devices are still very strong and drive the kind of algorithms that can be developed at the stage. In practice, when we talk about quantum machine learning, we want to use quantum effects in order to modify our approach to machine learning. What does it mean? There are a few things that we need to take into account that we need to think about if we want to use quantum algorithms to do machine learning. First of all, the problem of data. How do we do data loading? Now, in most cases, in most of the applications you will see today and that are in general being developed today, quantum machine learning algorithms are being used to analyze classical data. Of course, this raises the problem of properly loading, properly representing classical data into quantum states, quantum circles. That is also, of course, the possibility of or the idea of using quantum machine learning algorithms to analyze directly quantum data and that could probably be a direction in which more easily we would see an advantage with respect to classical machine learning, but that is not something that you will see here today. There are other typical problems related, not problems, but the features related to machine learning that we need to take into account. One of the most important things is the idea of non-linearities. Those are aspects that are extremely important in any machine learning algorithm. In most machine learning algorithms, it really makes a difference to give them the power that we needed to actually represent properly a probability distribution to recognize, to classify our training data. Now, quantum computers, quantum circuits are not so easy, are not so friendly with non-linearities. It is very difficult to represent in most cases, or the most general case, non-linear operations on quantum circuits. There are ways to go around this problem, but it's something that does not come straightforward. Then, of course, there is the problem of convergence. In general, convergence is a problem that exists also for classical methods. It has been observed already a few years ago. If you can see my pointer, I put a reference here. It was one of the first studies that discussed the problem of barren plateaus and vanishing gradients in a quantum environment. What was observed already at that time is that quantum machine learning models suffer a problem of vanishing gradients in a way that is harder than in the classical case. In general, the probability of ending up with vanishing gradients increases exponentially with the number of qubits. This is something that also needs to be taken into account when designing the structure and the architectures of quantum networks. Now, it's a general problem. There are, again, ways around it. There are tricks that can be used that are choices that can be made, for example, in the topology of the circuits also that are used to represent the networks. There have been many studies that have tried to characterize better this behavior. You see an example here in the two archive papers that I've read below. One example is really study the statistical properties of the Hilbert space in order to try and understand and quantify somehow, somehow, sorry, the representational power of the network and their trainability. Their capability to probe large space and to reach a global minimum, sorry. You see an example here. This is a way, for example, of measuring the trainability of a model by using the Fisher information metrics, and in particular, its eigenvalues. The idea, it's very interesting because it can be applied both at the quantum and classical case and can be used to draw direct parallel or comparison between the two approaches. The idea is that you can connect. It is possible to draw a connection between the spectrum of the Fisher information metrics and the probability of ending up in and the invariant plateau. So the level of degeneracy, sorry, my English is limited, in the Hilbert space. What you see is a classical behavior here. In most cases, this is true in general. The Fisher information spectrum is degenerate. Most of the eigenvalues are collapsed at zero, and there are only a few very outliers in the distribution. In the case of quantum models and quantum neural network, the situation is not so bad. The deeper the network and, in fact, the trainability of this network is measured to be better. This was a test that was done fairly recently. This paper was published last year and compared the performance of a quantum classifier to a classical one on a binary classification problem on the IRIS data set, if I'm not mistaken. Now, this is general, and what I mentioned so far are very much just examples of the kind of studies that are needed in order to better understand the behavior of those networks. Now, there are a few points that start to be like common, if not like a, I wouldn't say, common ways of doing things, very well tested, let's say, behavior. In terms of quantum machine learning implementations, one of those is the use of variational algorithms. Now, variational algorithms are not used simply or only for quantum machine learning. They are a more general class of algorithms that can be used to solve different problems, in general, different optimization problems, and they are very, they are an advantage, let's say, with respect to near-term devices because, in general, they can be implemented using a smaller number of qubits. They can be implemented in with a shallower architecture, and they are more robust to noise. The idea behind this is that they are indeed some sort of hybrid algorithms in the sense that the optimization happens on the classical computers. So, there you see immediately the downside that is advantage in going this direction. It's the fact that, indeed, you might be somehow reducing the possibility of, you know, a pure advantage because in any case, you need to rely on a measurement out of your circuit and on a classical optimizer in order to minimize your cost function. Here you see how the typical structure of a variational algorithm for machine learning problem would be that are processed by the input circuit. So, a set of gates that is used to embed the classical data here into the quantum state, you would have then the real variational part as a circuit that is based on gates that are parameterized by a set of parameters that are then optimized classically. So, in general, in the most general case, I would say in the simplest case, the embedding part, so the first part of the circuit is not trainable. It is, however, becoming more and more, I would say, evident that the way the choices that are made in order to embed the classical data into quantum states are very relevant to the final performance of the variational algorithm. And so, more and more attention is being paid on how those inputs, so those embedding circuits are designed. And in most cases, so they are also being designed as trainable algorithms. So, in order to improve the classification, for example, capabilities of the variational circuit as a whole. One thing that needs to be taken into account, and this, if you want, is still related to classical computation, is that all of those studies are happening right now on simulators. At least, most of the development work happens on simulators, and simulation of quantum circuits is extremely expensive in terms of classical computational resources. Usually, all those kind of simulations are memory bound. Whenever we increase, after a certain limit, increasing the number of qubits, even by just one unit, can lead to doubling the resources, the needs in terms of memory. So, this can become, at some point, a bottleneck when we need to study those models. And you will actually see, I will find it out. Another approach is the use of kernel methods. We just had very recently a nice seminar at Tern, this was last week, by Maria Schultz. She is one of the top experts on quantum machine learning, I would say, today. So, kernel methods, it's another approach. It's not mostly regarded as alternative to the variational approach, because in this case, the idea is to draw this parallel between classical kernel methods and embedding in quantum computing. So, the idea is really to use the embedding circuits that are used to load classical data into quantum states as kernels, as a way to define kernels that then can be used to probe the quantum universe space, exactly in the same way. A kernel in the classical machine learning model would be used to probe feature space. Now, if you are interested in this kind of approach, well, there is a lot of literature. I pointed here one of the latest works, but there's more. I have additional references in the slide, and I will show them. You can have a look. The idea behind this is really the fact that the advantage behind this is really the fact that when you use a kernel to train your models, you are sure that you are going to have access to convex losses. Now, this brings as a big advantage, especially if you consider what I mentioned before about the problem of vanishing gradients and baron plateaus. The minimum that you are going to find is going to be a global minimum. While normally with a variational training, there is no way to make sure a priori that the algorithm is going to convert to a global point. Now, the disadvantage is that in order to the way the kernel, because of the way the kernel-based training works, it is necessary to calculate distances between pairs in the training data. If the training data set is large, then this can become very expensive in terms of computations. On the other hand, in this case, this would be an disadvantage with respect to a variational algorithm, which, however, on its side has the disadvantage of needing some kind of prior knowledge on how to design the answers that are being probed. Again, the two methods are being explored, I would say, at the same time. It is really unclear at this stage which one, if one is absolutely better than the other. In order to, before I show you an example, how do we define, let's discuss the advantage a little bit, just to make sure that the concept of quantum advantage is clear in the case of quantum machine learning. This is very important, I think, when, especially now, that there is so much research trying to prove that quantum algorithm can bring supremacy or advantage depending on a problem. Now, in the case of quantum machine learning, the problem I would say is really not so easy to define because of many reasons, including the fact that sometimes there are no direct or simple physical, classical benchmarks that we can compare to. But it's also a matter of definition. I mean, we can define quantum advantage with respect to a machine learning model along different lines. We can have advantage in terms of runtime speed up, of course, yes, this is something that can come from, for example, from a simple fact of accelerating primitive calculations. Anything that has to do with algebra, linear algebra can be accelerated by quantum computing. So we could expect to have a speed up in terms of pure runtime. There can be an advantage from what's called a sample complexity. So how much data is indeed needed to train to convergence a model? And there are indications that quantum machine learning models can converge using a smaller, relatively smaller number of events with respect to classical models. And that is the whole problem of a representational power because quantum machine learning models are defined on a Hilbert space on a higher dimensional space. It is reasonable to assume that their representational power could be stronger, bigger than in classical cases. Now, there are no proof, no direct or let's say, there are no absolute proofs. There are actually measurement or specific cases for specific models in which this has been observed. But again, it's still some and you have an example here in the paper here. There is still no, I would say, there is still research ongoing in this direction. And I would like to stress that there are some models for which the representational power is a very important concept. Anything that has to do with generative models, so the idea of learning this underlying distribution from your training data set and being able to represent it properly over a support space that is as big as expected. And this is something that for which quantum model could indeed bring advantages. And then of course, the concept of advantage is modified and somehow needs to take into account everything that has to do with practical implementation, right? So first of all, the limitation of modern hardware, so near-term devices are noisy. They suffer a coherence problem. The deeper, let's say, the circuit, the higher the probability of having errors at the end of your measurement, random errors that need to be taken into account with specific error mitigation techniques that might require additional qubits. Then that is the old discussion about the data preparation step. And I'm not even talking about what you saw before, so the software level, the way in which we can build the data embedding circuit. I'm really talking about the engineering problem related to the number of channels that can be used to load the data on the device itself. So all of those, again, practical aspects can completely change the initial, let's say, estimation of an advantage based on the concept of theoretical complexity, computational complexity. So we tried to look into examples that we considered a controlled environment. So what we wanted to do with our work was really to take examples from our standard problems in data reconstruction or data analysis and use those as baseline in order to understand what are the challenges related in trying to replace those models with quantum models. And the idea is really to try and start some sort of systematization of the approach. First, by building prototypes, then by trying to characterize better the performances and the qualities of more like theoretical aspects of the algorithms. And then learn how to build new algorithms without really going through the translation of a classical algorithm directly into a quantum one, which might not give you the best possible architecture. So that said, you see here an example of quantum machine learning applications. And I have replaced the specific high energy physics jargon with a more generic one, because those are examples that for me represent very generic problems. So when we test quantum support vector machines to classify Higgs bosons, we are trying to solve a generic classification problem. When we wrote a quantum graph neural network to do particle tracking, what we are actually doing is trying to do some pattern recognition. The same way we've been studying how to do data embedding, how to optimize data embedding in order to improve the capabilities of our GNM. We've used the quantum multiple machines to build a setup for reinforcement learning. Now the example that is actually mentioned here uses the WSCG data monitoring data from the specific sites that are used by the Alice Grid experiment. But this is actually a very nice project that is now ongoing in collaboration with our accelerators department. And we have colleagues that are using this method to optimize the beam in one specific linear accelerator. And then you will see an example of a quantum generative of the second network, a couple of them actually five times, started as an example on detector simulation. But again, it's an example of a generative model. So I plan, I'm sure I'll not have time to talk about everything. So I plan to talk about graph neural network and quantum guns and say something about the embedding. For the support vector machine example, there was a seminar that was given recently by Professor Saoulin. She's one of the, she's the principal investigator in one of the projects, there were more than one project that were devoted to SVM. We have one that is still ongoing ourselves. And you can find the link here. Okay, so let's start with the pattern recognition problem. So the idea of using quantum graph neural networks. The problem in this case is the problem that's actually solved classically by the extra track X project. So some years ago already, the idea of using classical deep learning methods to do to do particle tracking was explored by different, different initiatives. And at that time, it was called the head track project, a project that surveyed several, several different approaches, convolutional neural network, variational encoders were also used to do that recurrent network were tested to do particle tracking and graph neural networks. So what we use as our classical benchmark is the graph model. This is based on the idea that the data in the tracker, in the tracking detector can be interpreted as a graph of connected kids. And then the problem, the tracking problem then becomes in, you know, classifying the connection, the edges in the graph that optimize the trajectory, the particle trajectory. The model, the extra track X model is built as a set of actually different networks. It's a set of two classifiers, one is called the node network and another one the edge network, because each one of them is actually classifying the quality, if you want, of either the single hits and the detectors or the hedge, the segments that connect them. I don't give you the details of the model, but it's a well known model. It works very well. This is one of the first versions actually of the model. You can see with which purity and efficiency it actually could reconstruct the correct tracks. You see here the generic structure, which is built, as I said, as it, maybe I didn't say that, the f-track GNN is built as a chain of edge networks and node networks that are iteratively run one after the other until the last edge network reaches the classification accuracy that is required. So for implementation in the quantum circuit, we looked into hierarchical quantum classifiers. Those are very interesting models, for many reasons actually. So one of them is the fact that a classical counterpart for those models exists and it has also been started in applications in jet-tugging problem with very nice results. So three tensor networks were initially designed as quantum-inspired models, but they were purely classical and of course their quantum counterpart is, an example is represented here for qubits, their specificity is the topology of the entanglement gates. So this typical hierarchical structure that is entangling the external qubits first and reducing the final dimension, the deeper you move along the network. Merographs are other similar models. They are characterized by some, they have additional ensembling with respect to TTMs and they are better suited if the data set is expected to have longer range correlations. Now TTMs are also interesting because well, they are one of the models that have been architectures in general that is being proposed to reduce the problem of vanishing gradients at the same time that have been measurements. Okay, yes, I put the measurement reference here that have observed robustness against noise, that as I mentioned is very important for near-term devices. So with these results, well, we recreate basically the same structure, the same iterative structure that is typical of the X-Attract model. What we did however was to reduce the size of the classical edge and node network in order to do the comparison that you see here. So the orange points are three different quantum classifiers. They all have a single hidden dimension. It means that basically they have a structure that is like this that you can repeat it many, many times depending on that you can increase the number of qubits that you use as input and then your circuit will become deeper. In our case, the input is a set of two points. It's basically the three coordinates of XYZ for two points that define the edges of a segment. So what we did then was to compare the performance. This is the area under the rocker that is measured for the three different quantum classifiers compared to the original classical model, not the original, to the simplified classical model in which we have reduced the depth of the network. And you can see that, well the scale here was from 0.68 to 0.84. I don't know if it's readable, but you can see that we don't do better than the classical model. And we get close to classical models that have a similar number of parameters than the quantum one. So we are still working now in trying to make tests with deeper segments. The problem is that we have basically, the bottleneck in this case is the simulation time, the time it takes to train those models, to simulate all the circuits. Now, the other thing you can do, so if you want to improve the performance of your network, you can, of course, try to improve the topology of the variational circuits that you are training. So in particular, in this case, it would be our classifiers. What you can also do is work on the data embedding, trying to see if there is an alternative representation, an alternative feature space in which the classification problem becomes simpler. Now, for this specific tracking problem, already at the classical level, it was observed that it was possible to improve the performance of the model by designing and by representing the data in a different embedded space. And this was done by training a multi-layer perceptron using a ninja loss. But what was done exactly, this is basically the same thing we do, or very similar to what we do in our study. So if you look at the plot on the left, here, the leftmost plot, you will see that how an event would look in the original XY space. So you would have a set of points that you need to select and classify according to the correct track. And this is done selecting a seed, so an initial starting point. Now, if you transform those points in an embedded space using a fully connected layer that is trained in order to maximize the distances, this is, in fact, something very similar to an SVM, if you want, where the hinge loss is used to optimize this distance. You will see that the sampled points or cluster are very close to the original seed, and all the others are instead very, very far. If you do that, and you select samples according to the distance indian by the space, and you transform it back, then you will see that the true hits, the hits that were selected, actually do align according to the trajectory that was, the true trajectory, the real track. Now, this improved the result at the classical level. We decided to do the same at the quantum level with an hybrid circuit, basically. So what we did was to combine a multi-layer perceptron with a quantum circuit. And, well, that is also actually an additional fully connected layer that is used only to tune the final dimensionality of the output. And what we looked into was to understand how changing topology in terms of entanglement, for example, of the quantum circuit would change the performance of the networks. So this is some work that was started actually this summer by students. So I mentioned we wanted to understand how different quantum circuits would, what effect would the topology of different quantum circuits would have on the performance of the embedding. So how do we define those quantities that are too big, too important things that you need to, that you can quantify about your, you can describe the performance of your circuit. One is the expressibility and another one is the entanglement. So to measure those, we used the measurement, the definitions that were proposed by this work. This is work from a group at the University of Waterloo in Canada. Now, the idea of the expressibility is inspect the capability of your net of your circuit to probe the full, the Hilbert space. How much of the Hilbert space you can cover with the circuits. Now, for example, here you have the block sphere that represents easily the Hilbert space for a single qubit. Now, if you build a circuit, which is just made of identity gates, you will, and you initialize your qubit, you will not move out of it. The expressibility is minimal. And because you cannot probe the the sphere at all. If you just make it slightly more complex with a simple rotation and another other amount of gate, then you are capable of already probing a one dimensional line around your block sphere. If you add an additional rotation along the other axis, then you can see that you are start to probe more and more of your sphere. And this is just a generic example. Now, you can measure this, this expressibility in different ways. And we used a simple Kovac library difference or distance between the fidelity measured on our circuits with respect to a stochastic circuit. The second quantities that it's important is the entanglement. It somehow measures the capability of your circuit of recognizing or reproducing quantum correlation. So in principle, let's say that the property of the entanglement will tell you or it's what will make it so that your quantum circuit that your that your quantum circuit can recognize correlations that you could not represent with your classical model. This is what I want you to say. I'm sorry to kind of a complex sentence. Also to measure the entanglement, there are many different ways. The major wallock entanglement is a possible way of doing that. And this is basically what we use. In the study, we started with a certain set of circuits. Those were circuits that were proposed by the original work at Waterloo. As examples of circuits that would have very different entanglement capabilities and different expressibilities. So basically, well, the circuits are numbered here. And well, you can see that there are different number of parameters. Not necessarily the number of parameters means that there will be better capabilities either in terms of expressibility or entanglement or both. What it means in most cases, though, is that there will be a big difference in terms of training time. Now, if you look at the entanglement capabilities, the higher the better, the expressibility goes the other way. I just pointed out because to make it easier. And now you see that those circuits all have a different kind of entanglement. In some cases, you have something here that is just very simple. So where just the two neighboring qubits are entangled, you have cases in which you have all the qubits that are all to all kind of entanglement. You have cases in which you have something very similar to the hierarchical classifiers. So we introduced those different circuits. This is the same table that you saw in the previous slide. I just reported here for simplicity. And we try to see how closer to, if changing the topology of the circuit would change somehow the performance of the network. So how fast or how well the network could converge. Well, first of all, you see that the training turned out to be rather unstable, especially for some of those circuits, in particular circuit 5, which is the one here that had the largest number of parameters seem to be less stable in training. You see that we have larger error. Circuit with the lower, so the best expressibility, which is the circuit 14, is also the one that seems to converge faster and be more stable. And just after that, circuit 11 also seems to work fine. Now, those are results that, I mean, those are preliminary results. If you look here carefully, we, those, this is the behavior of the circuit after just one epoch of training after only 80 batches of training, which is a very short time. But again, when we talk about training time simulation for this kind of circuit, we're talking about basically data. So using or speeding up, this connects to what I said at the beginning, speeding up simulation of those, those circuit is very, very important to this kind of study. Oh, I wanted to mention that there is a whole. So what we're working on now is to, we've observed that one way in which you can easily modify your total entanglement and expressibility for the, for your circuit is of course to repeat. So to, to increase the depth of your circuit by, by repeating it as, as like stacking layers one after the other. And, and this is basically what we're doing now to see how and increasing the depth of the circuit or if increasing the depth of the circuit is a direction is something that can make the train more stable or not. Now, that is not necessarily the best test to take the best solution. If you're talking about near term devices, because as I mentioned at the beginning, when you do optimization for hardware, you'll need to try and keep your, your circuit as shallow as possible. I think I'm almost out of time. I didn't realize I'm, I'm sorry, I should put my alarm. I will just very briefly discuss, take five minutes to talk to you about generative models. Now we have developed generative models. There are many prototypes that have developed gun and variational rot encoders to do simulation, classical models. We have our own prototype that actually performs very well for the simulation of a high granularity color electromagnetic colorimeter. And we decided to use that as benchmark to test the possibility of implementing quantum gun. As the first test, we looked into hybrid classical quantum models that was proposed by IBM a some time ago. Now, in order to do that, we had to simplify our colorimeter output in order to fit to a number of qubits that would be suitable. We did that and we managed to reproduce with the quantum gun model images that look very, very realistic, average image that looked very realistic with a nicely stable training. Now this was just a test that proved that we could indeed design a quantum generator that would be able to, to encode the probability distribution on the quantum circuit. At the same time, this model is cannot be used as a real generative model in the sense that cannot be used to do sampling. We worked with Cambridge Quantum Computing in order to improve on this model. And this is work that is ongoing today, is still ongoing. We basically designed a quantum generator that is split into two steps in which we have an initial step that learns an average distribution and a second step that then is capable of sampling events. Now, this is something that we tested on a small subset of the data and we managed to reproduce the average images and also a subset of the generated samples. I wanted to stress this is the case both for the IVM model and this model, this double generator model, that when we compare the number of parameters that the generator network uses with respect to the classical discriminator that is used to train in the adversarial schema, the number of parameters is very, very different. Okay, we are talking about order tens of parameters in the case of the quantum generator, we're talking about order thousands of parameters in the case of the classical discriminator. We have also introduced another model. This uses a different approach, it's called, it's based on photonic hardware. So you can, the advantage in this case is that you can do directly, direct representation of continuous variables. And in these cases, we both, we tested two different architecture, an hybrid model and a fully quantum model. And I'm showing you here the results of our, well, first of all, our hybrid model works very well. In this case, we used only 264 parameters for our generator network and what is it, about 44,000 parameters for the discriminator for the classical, for the classical gun. If you, we also try to measure to understand if we could say that the hybrid model learned faster. Now, the conclusion at this point is that we reached convergence by only 100 epochs of training using the hybrid model. And after 1000 epochs of training using the classical gun, you can see the two, the way the loss functions change here. It's a first hint that the hybrid model in which you have a quantum generator has indeed some, some higher capability, but this cannot be still, cannot be considered for now yet, any, any, you know, real proof of, of, of advantage. There is more work that we're doing in, in replacing the classical discriminator with a quantum one, but the fully quantum model still doesn't really work very well. We do see mod collapse. If you have experience with classical gun, you will know that it's a rather unstable model to train. And it's very common to end up with the, with the mod collapse problems and with the generator is just producing very specific modes without, you know, covering the full space space. And this is exactly what you've served with the quantum gun. Okay. So the conclusion, the summary more, more like it. You, you probably know that CERN has started recently, it's quantum technology initiative. Now this goes beyond quantum computing. It involves work on quantum sensing theory, quantum communication as well. But quantum computing represents a pretty big part of it. And, and the idea is really to try and understand what are the opportunities in quantum computing for our fields and also how to actually interact. How do we, how do we build collaborations between our field and industry and, and find this in other, in other fields. More specifically, quantum machine learning are among the, the models that we tested the more, I would say so fast, so far. I think there's many reasons beyond that. Probably in some sense, it's maybe because there is a lot of overlap between the, the communities, the people that have been doing machine learning before and not doing quantum machine learning now. It's also true that machine learning models at the advantage are so, I should say, promising. There are so many problems that you can solve with machine learning or with deep learning that if quantum computing could bring an advantage at this level, then, you know, the advantage would prop it immediately to a lot of different fields. So I think it's a very, very promising, promising kind of problem. Of course, the results, there is still a lot of work to do. The results so far are interesting, but there is a lot of work that needs to be done not only at a specific level of the applications, but really at a more general level as far as understanding quantum properties is, is, is concerning. So while those are the additional references for, for a few things, I will share the slide. So no problem. Thanks. And sorry for being late. Wonderful. No, you were not late at all. Thanks a lot for this great talk, Zafia. Questions? Anybody wants to ask a question? Please raise your hand or type them in the chat window. Oh, the chat. Let me see because I can, I will, I can, I was able to open it. Okay. Maybe let me go first. I have a couple of questions, actually one sort of technical one and one of more general on the technical side, you said you hinted at the beginning that non-linear it is very difficult to construct in a quantum system. So how do you actually do it? Yeah. So there are different ways in the case, in the standard case of, you know, the qubits representation. If you're using kernels, if you're using kernel models, you somehow go around it because you are assuming that you can do a transformation. Yes. But when you do your transformation in the quantum, in the quantum space, you, so in the quantum space, your model will be a, as a linear model, but you, you take care of the non-linearity at the level of the embedding, basically when you do the transformation. There are, there is also the idea, in fact, I think there is also the idea of, oh sorry, the other way, approximating, yes, activation functions with, in such, of using approximated activation functions, for example, doing a series development, no, how you say in English, sorry, yes, expansion, expansion in such a way that, that you, you can actually implement it on the quantum, quantum circuit. Now, the problem in that case is that the performance of the model might change. It's not that there are only very simple models in which you can, you can replace a real activation function with a series of, with a series expansion. Another way, so is using, for example, this, this quantum, these continuous variables. So in the continuous variable case, you actually add, sorry, I have, I want, here. So in the continuous variable representation, so what, what you do is to represent your states as, as Q-mode, as, as, for example, in the Fox space as the number of, of photons in your state. Now, in this representation, you actually have gates that are not necessarily linear gates. For example, the Kerr gates is a typical gate that could be used to introduce nonlinearity. This is, in fact, another, another advantage of using this approach, the continuous variable approach. From a practical point of view, it's one of the few cases in which we, we still don't have hardware, we can test our models home. So we need a little bit, Xamadu is the company that is working on this approach, and they have hardware that is available, that is accessible via cloud infrastructure. But, and we talked to them actually, so, but, but they are still missing, it is still not possible to run machine learning models there. They're still missing bits in the, in the software stack, but it will come soon. Yeah, I see. Thanks. And then the other question that I had, since you spoke at the end about, you know, quantum computing, revolutionizing machine learning, because machine learning itself is very widely used. And then if you can improve on the methodology of that, that could potentially be very valuable. Maybe more in the shorter terms, I've heard in the general theoretical computer science community, that it's actually not been quantum algorithms per se, that, you know, made big short term improvements, but actually quantum inspired classical algorithms, that people have taken away lessons by thinking about quantum algorithms, and then taking them back to the classical world. Do you see something of that sort happening also in your field or in your area? Yeah, yeah, yeah. I think this is a very good point. And well, there's the, there are, in our case, so I mentioned those three kinds of networks. So that is very nice work that was done at the University of Tadova on digit tagging, and they use quantum inspired. So they use classical three kinds of networks to solve the, to solve the, to solve this problem, digit tagging problem. Beyond that, I know that they're, and beyond the quantum machine learning, I know there are actually a lot of applications, quite a few applications that can run on, you know, quantum inspired hardware, like the Fujitsu or others, two of them, I forgot a second. Bevy is hardware that is designed to solve simulated annealing classical, and then a lot of the problems that you would solve, for example, on the D-Wave hardware, on the quantum annealer, you could also solve on quantum inspired. Yeah, so there's the two things that is the algorithmic side, but that is also your work on the, on the hardware, from the hardware point of view. Yeah, that makes sense. And then maybe, lastly, looking into the future, you know, people, since now quantum computers are ramping up, and the number of qubits that, programmable qubits that become available is also increasing, people are somehow ranking open problems in science, in terms of how soon people would be able to put them onto a real quantum computer. So assuming that we have programmable quantum qubits available, how many qubits would you need realistically to be able to do some of your problems, or to be able to show some of your examples actually running natively on a quantum computer? I mean this, the generative example that I showed, for which we have very nice, very nice results, maybe it's not the simplest, but, but I mean here, we are talking about, we are talking about for, for order for, for qubits to represent, or even here, you can see this is done already in two, in a two-dimensional form, okay. So we are using six qubits to represent 64 pixels. Now in the real case, this problem would actually be, well it would be three-dimensional first, it would be 25 by 25 by 25 pixels, I'll let you do the math. Okay. So even assuming, even assuming an exponential advantage from the point of view of, you know, moving from pixels to qubits, which is, it's not the case actually, because to do that, you need to use specific data representation, for example, amplitude them, amplitude the encoding, which doesn't necessarily bring you, which is not, for example, as easy as process, which requires deeper, deeper circuits to actually be implemented on real hardware and so on and so forth. So for me, it's hard to say in the sense that I would, a problem like this would probably need thousands or at least hundreds of qubits to be solved. Also because it would also depend on the topology of the qubits, it would depend on their quality. If we need to add error mitigation techniques, so additional parts of the circuits that take care of that, then you need even more. Yeah, I think I was talking about logical qubits, so just, you know, in the very simplest case. I was just curious about the order of magnitude. Okay, now I don't like to give those numbers because, you know, things change very rapidly and then in a week maybe got updated. Right, thanks anyways. Okay, anybody else have any questions? I lost my chat. Okay, if there are none then I would like to thank you very much again for this great talk. I did learn a lot and that's it for today. That concludes the seminar. Talk to you again next week. Bye-bye.