 Good. And so, please. So the last talk about neural quantum states. So here I won't get too much detail about the mean SR derivations and all those details about neural quantum states. I will give a comprehensive talk about the recent progress of neural quantum states and what lead us to do the mean SR method. First, first I want to talk about why we use neural networks to express quantum states. And as we know, the neural network has been applied to many machine learning problems. And I want to use these machine learning examples to show why is it also reasonable to apply neural networks on the quantum many body problems. So here is a first example about the image recognition. For example, if we want to use neural network to classify images, we can do it like this. We take some image as the input and the neural network takes it as the first layer and after many layers of overpass, the network can finally tell us some probability. Here we say the probability is, we define the probability to be the probability of this image being a cat. And of course, we can have some other input image and the neural network can give us some other probability. And if the neural networks will change, this probability will very well reflect the nature of the input image. So this is one example. And here I want to give another example about the game of goal. I guess you probably have heard of the alpha goal. So one thing we can do in neural network is that we can take the game status in alpha goal. We can take the game status in the game of goal and put it into the neural network. And then the neural network will tell us like how likely is the player going to win the game. And also if you are familiar with the game of goal, you may know that this game is a very long ranging tank of game. It's like when you put some pieces on one corner of the game, you can significantly influence the game in some other corners. So this is a bit similar to some other corner systems like quantum spin liquids. So based on these two applications, we can see that the neural network can really do well in learning some probability distributions in some traditional tasks. So it makes a lot of sense for us to apply this also to the quantum antibody problem. And the quantum antibody problem can be formulated as this. Here we take a spin one half system as an example. We can take the spin configuration sigma as the input and here you can see we take spin up as the black color and spin down as the white color. So it becomes really similar to the image recognition task or the game of goal. So it's not hard to imagine that the neural network can also do well in this problem and give us some nice output. Here we take the output as the wave function psi. And for some other spin configurations, it will give us some other wave function outputs. So in the end the whole corner state can be constructed in this way. And there is a difference in the quantum antibody problem. Here we can see the wave function is not always positive. It can also be negative. So it's quite different from traditional machine learning tasks because usually in machine learning we only learn a probability distribution but here we also need to learn the signs. And this is called the sign problem in quantum system. This is a very fundamental problem in quantum systems. And I will also talk about this problem later. So anyway, we can very well expect that the neural network can be applied to some simple quantum systems. Here is the famous paper by Joseph Collier and Matthias Troyer. And they use a very simple one layer restrictive bosomal machine. It's a very simple network to learn the Heisenberg to learn the Heisenberg antiferromagnetic in the square lattice. Here you can see they can reduce the original error to a level much less, it's not much less but it's better than the tensor network methods. So this is a very nice first attempt on the neural corner state. But this Heisenberg model doesn't have a sign problem, its sign structure can be exactly solved. So in this work, they don't solve the sign problem, the neural network just learn a probability. But anyway, it's a very good result and we can expect that the neural network may be able to also give us some interesting results in those complex system with the sign problem because those are interesting systems. In this Heisenberg model without the sign problem it can be solved by some other methods like quantum and color to a very high accuracy. So if we want to really use neural corner state to learn something beyond what we can have from the traditional methods, it's important to go to the frustrating model. So that's the main topic of neural corner state in the past five years and I will talk about what we do in the past five years and what's the difficulty and what lead us to the method of MISR. So the first attempt is on the frustrated J1, J2 model. In this model, it's a frustrating model with sign problem because of the J2 interaction and the first attempts are based on the convolutional neural network. So this is their original energy they obtained at the time. Here you can see the energy is not too bad. You can compare it to the ground state energy. It's accurate up to the third digit but you can also see that it's not as good as traditional methods like DMRG or traditional rational much color. So it doesn't give us something new. It doesn't outperform traditional method and probably because of the sign problem. And here are some other papers showing that maybe the sign problem is a very hard problem for neural corner state. Here's the paper. Basically what they did in this paper is that they chain the neural network in one small portion of the herbar space and they want to see whether the learned wave function can be generalized to the full herbar space. And you can see here that the generalization is not good for large J2, which means the large J2 means the system is strongly frustrated and the system has a non-trivial sign problem. So you can see the amplitudes for large J2 are not too bad. The accuracy can be around 90% or 80% but the signs were quite bad. The signs are basically zero accuracy, which means the signs learned by the neural network can't be generalized to the full herbar space. And although this is a work on the supervised learning it can also be very big problem for rational much color. Because in much color study what we can do is to generate some much color samples and the number of monocled samples is very small compared with the full herbar space. So anyway, we need to expect that the wave function learned by the neural quantum state can be generalized to the full herbar space. But this paper shows that it seems doesn't work. So this is a very severe problem for the neural quantum state. And on the other hand, we can also do some other tests to verify this conclusion. We can see here I show a result even by the simple sign rule limit. Here the simple sign rule is the martial sign rule. It's an approximate sign rule for neural quantum state in the square of 10 by 10 lattice. And this sign rule is an approximation. It's not exact. It can be obtained and it can be written down and applied manually. But it's not an accurate sign structure. So here you can see the CN energy is not as good as the simple sign rule limit, which means basically the sign structures in CN was basically wrong. It doesn't learn any sign structure. And that means the wave functions provided by the shallow CN was bad. It doesn't give us a reliable sign structure. It doesn't give us a reliable wave function. So this seems to suggest that the neural quantum state can't learn the sign structure but the question is is it a fundamental limit to the neural quantum state or did we make some mistakes? So the answer I want to give here is that we really made some mistakes. The problem here is usually we separate the amplitude and signs in neural network, but that's a bad choice. Here I show a traditional architecture for most neural quantum state applications. As we know, the neural network is good at learning the probability and so in the beginning, people just feel we can have an amplitude never only for the probability or a positive amplitude. And we have another sign there only for the sign structures. And if we do a product of them, we can expect that it will give us a good wave function. And you can see I'm also one of these people working on this direction and I work on this for many years but I gradually feel that this was a very bad choice. And I will give you a very simple example to illustrate why this is bad. Here I draw a function. It's a function of signs. It can have values minus one or plus one. So now I want to ask you if you need to extrapolate this function to the right, how would you extrapolate this function? You may think it seems to make sense to extrapolate this function in this way, right? It seems to make sense but actually the function should be extrapolated like this. So you may feel it doesn't make any sense. Why is this extrapolation more reasonable than the previous one? And now I will show you the full function with both amplitudes and signs. Here is the full function and with this function it's not hard to guess that the function should be extrapolated like this. So what I want to say here is that when we separate the amplitudes and signs, it will lose a lot of information in the wave function and it will break the underlying tendency of the wave function. So this is just a very illustrative example. It's a continuous case which is different from our wave function but anyway it can show that when we separate the amplitudes and signs we are making big mistakes. So I want to say this choice is wrong and so I want to say it's not a problem of machine intelligence. It's actually a problem of our human intelligence. We are letting the new never learn something which is impossible to learn. And also I want to mention that I didn't see any paper saying that it's definitely wrong to do this. It's just my personal understanding. So for the more across these years there are also a lot of progress on the design of neural networks. So in the beginning people thought maybe a problem in neural corner state is that we didn't do good standpoints because anyway in Markov chain much color what we can do is only correlated samples and we want to find a way to do uncorrelated samples and this is done by the autoregressing network and the recovering neural network. But gradually I found this is not the optimal choice because these networks will break the translational symmetry and due to their architecture it will be very hard to restore the translational symmetry. And then people found that it's actually very important to do a suitable symmetry projection and these two papers shows that if we do the symmetry projection in a good way we can actually obtain a very accurate refunction even though the network is very simple. And in these two years there are also a lot of progress on the design of network architecture. For example, the group convolutional neural network and the vision transformer. These are both very good architecture and in the end after so many years of efforts we are in a point that we want to see the neural network really produce something beyond traditional method. And the philosophy to give a final kick to make the neural network better than other methods is the simple philosophy. More is different. It's not only a thing in condensed metaphysics it's also a rule in deep learning. So we can very well imagine that if we have more and more parameters in our neural network it can these enormous neural connections can somehow capture those quantum information and quantum features in the quantum systems and we will be able to obtain more and more accurate quantities we want to measure. And the idea of deeper neural quantum states with more and more parameters actually have been proposed in the early years of neural quantum states. Here you can see in these two papers in 2017 and 18 we can see that they propose that in deep Boltzmann machines it can do much better. It can have much better expressive power compared with the one layer restricted Boltzmann machine. And also in 2022 there is also another paper showing that they spend quite a lot of efforts on training a very deep neural quantum state with around 10 to five parameters and they finally achieve a very nice energy in this prototypical model. So this energy you can see is not as good as traditional rational Monte Carlo but it can get better than the simple sign rule limit. So this also verifies my claim before this neural quantum state is actually able to learn the sign structure. And if we have a deep network with a lot of parameters and we change it in a suitable way it will be able to learn the sign structure. I think so. So we can expect that the neural network can become more and more accurate if we keep increasing the amount of parameters. But the problem here is that the training complexity of neural quantum state is actually very high and I will explain why this is a problem and then introduce our mean SR method. Here I won't give a very detailed of very formulated derivation. I will just do some hand waving derivation so that you can understand why is the optimization very difficult for neural quantum state and how to obtain the mean SR method. So here in neural quantum state what we can change is the parameters theta and what we want to change, what we want to tune is the refunction psi. And these two quantities are related by this partial derivative. On the other hand, the change of refunction we wanted to be given by this gradient descent in the Hilbert space. If your partial E partial psi will give us the gradient descent direction to minimize the energy. So this is the formula we have and I want to simplify it by introducing some notations. Here partial psi partial theta is Jacobian matrix. Here I give a name matrix A. And on the right hand side this is a matrix. Sorry, this is a vector which can be computed by rational Monte Carlo and I give you a name B. And this theta theta is the quantity we want to solve and I give a name X. So finally it becomes a very simple linear equation AX equal to B. And in order to solve this linear equation it's very important to know what's the shape of A and the shape I show here because A is a Jacobian matrix its shape should be the number of samples and S times the number of parameters and P. And for deep neural quantum states it's usually a good choice to have number of samples much less than the number of parameters. So we can see that this A is a very wide matrix and this is actually an under-determined linear equation. So to solve this linear equation there are some standard ways you can find in textbooks. Here the method I use is the pseudo inverse method which means we obtain the X vector which will give us the minimum residual error and also the length of this vector X should be minimized. This is the property of pseudo inverse this is the property of pseudo inverse. And here you can see there are two ways to obtain this pseudo inverse. The way in the middle is the SR method. In this method the important part is the matrix A dagger A and you can see the size of this matrix is MP times MP. So imagine we have a very deep network with like one million parameters then the shape of this matrix will be one million times one million. And to perform this matrix inverse it will be it will be very expensive. The complexity is proportional to MP cubed. But on the other hand we can also solve this equation by the right hand side equation. And then the matrix here has a shape of only an S times an S. So it's now independent of MP and the complexity is only proportional to MP. This method is our mean SR method and you can greatly reduce the complexity of doing mean SR, of doing neural corner state optimization. So now we have a method to change a very deep neural corner state. And now I want to show the results with benchmark model. Here the model is still the frustrated J1, J2 model. But in the beginning I want to show some results in the J2 equal to zero case. This result has been shown by Filippo in his talk and here you can see when we increase the number of parameters the relative error will reduce and finally it will achieve a level of 10 to minus seven and it's around 1000 times better than the best previous result. And on the other hand this is our recent results on the on the frustrated case. And in this work you can see this purple curve we can reach more than one million parameters and the original error is better than all other existing methods. And here is our original energy. You can see here that the original energy is better than the traditional method and it's very, very close to the ground state energy. Yeah, one part is that we use the mean SR method to change a deeper network. And on the other hand we also spend quite a lot of time to design a good network. If we use RPM then that's unlikely to work out because you can see in the left panel here when we increase the number of parameters the original energy doesn't reduce a lot. So we can expect that if we keep increasing the number of parameters, if we keep increasing the number of parameters it will somehow saturate. So one layer RPM is not enough but if you have more layers it's likely to work out. Sorry, sorry, we can chat later. Okay now, and we can, I've shown that we really have a method to all perform traditional method in the spin system. So I want to show some new physics based on this method. The application I want to show here is about the quantum spin liquids. So the first step is that we want to obtain the energy gaps of the quantum spin liquids. As we know as I have introduced in the J1, J2 frustrating model actually there is a quantum spin liquid phase as shown by this paper. And also in the triangular lattice there is also a quantum spin liquid phase in this J1, J2 model. So the existence of the quantum spin liquid phase has been verified by many papers. What we want to see here is whether this quantum spin liquid phase are gap or gapless. So what we did here is that we measured the energy gaps for different system sizes and we extrapolate to infinite system size. Here we do simulations on square lattice with 20 by 20 lattice and with up to 20 by 20 lattice, 20 by 20 size. And on triangular lattice we do up to 18 by 18 size. And you can see here that when we go to the thermodynamic limit at one over error equal to zero it becomes the energy gaps vanishes. So the neural quantum state methods suggest that in this instance there are gapless quantum spin liquid phase. What's more we can also measure the structure factors in this instance. So here is the spin structure in the case space. You can see there are some high peaks in the corner of the case space. That means we have a spin order in the system. But when we increase the value of J2 which means when we increase the frustration the spin order will be suppressed and you can see the peak of the corner becomes not so high. And also we can also measure the dimer structure over there and here you can see from the scale of the color map it's actually not very strong and when we increase the J2 frustration it doesn't increase a lot. So we can expect that for small J2 there is a spin order phase and for large J2 it's likely to be a featureless quantum spin liquid phase. This is also shown by this plot of correlation ratio. Basically what this quantity shows is that it's how strong are the peaks in the structure factor. So here we can see that in the spin correlation ratio that for small J2 it's strong and for large J2 it's weaker. And there is also a crossing point at around J2 equal to 0.06 which means it's likely to be a quantum phase transition point and for small J2 it's a spin order phase and for large J2 it's a quantum spin liquid phase. So finally I want to give a summary about my work. First across during these many years of studies on neural quantum states we gradually realized that the previous choice of the sign structure was not good and it's better not to separate the amplitudes and signs and gradually we learned better never designs from other paper and finally we proposed the mean SR method which allows us to change very deep neural quantum states. And finally based on all these improvements we are now able to change a very deep neural with one million parameters and obtain the state of the art accuracy and outperform existing method and then we can do some simulations on the quantum spin liquid to produce something interesting. And in the end I also want to talk about some outlooks of neural quantum states. So as I have said on the quantum spin liquid problem the neural network can be applied to obtain something different and this is a very good playground for the neural quantum state because now the DMRG method or the traditional VMC are not as good as or anyway and in our experiments they are not as good as neural quantum states. So we can expect that neural network can do something different in the system and also on the fermionic lattice the neural network can also make a difference. The fermionic lattice like the Harbor model basically the simplest model to explain what's happening in most condensed matter systems. So if we can really use neural network to learn what's happening in the fermionic lattice we will have much better understanding about the condensed matter systems like superconductivity. And what's more we can also apply the neural network on the quantum chemistry problem as has been introduced by the last talk. So in the quantum chemistry problem the neural network can learn the electron structures in the molecules and atoms and if you are interested in this problem I would suggest you to read this review paper on natural review chemistry. And what's more there's also another important problem about quantum dynamics because the traditional quantum dynamics methods are like basically exact diagonalization or tensor network method. So in this method there are quite a lot of problems like in tensor network methods the entanglement is limited but if we can apply the neural network on this problem we can really make a much better simulation on the quantum dynamics. But of course there are also still some technical details some technical difficulties in these problems and there are still a lot to do. So in order to solve all these problems including consulate liquid from only lattice quantum chemistry and quantum dynamics is very important to design a good neural network. So my idea is to encode the physical back wrongs like if your fermions into our neural network function and this will be, in my opinion this will be a very important next step and there are also some more directions that we can work on. Anyway, I can't list all the possible directions and the neural quantum state is a very open field that there may be a lot of other directions that we can also produce very interesting results and this will require the efforts of all of us. So thank you for your attention. And thank you. So thank you for the very nice talk. My question might be actually similar to the one that he was asking before but I'm not sure about that. I'm gonna try to reformulate. So can you please take the slide where you're doing this pseudo inverse thing? So my question is I can see that you have let's say computational gain by trading the number of parameters for the number of samples, right? Because in the regime that you're interested the number of parameters is very large so it's very much more difficult to do this thing on the left. But is the update in the same in the two cases if you fix an architecture and you're actually able to carry out both of the methods? Can I make myself clear? Sorry, you mean? So the S-R and the mean S-R if you fix an architecture for which you are able to do both cases in the end of the day you're gonna have the same update rule, right? Yeah, they are mathematical equivalent. Exactly, exactly. So it's a matter of computational efficiency, right? Yeah. If you hear about efficiency it's not about accuracy. Okay. Okay, thank you so much. Hi. Maybe I missed which gap are you computing in the triangular lattice? These two gaps I mark here, do you hear K and M are the K vector and this A1 is the rotation. But this is the triplet gap? Yeah. And A1 minus, what does it mean? A1 minus, minus means there is a, there is a total. This is the smallest gap on the six by six for instance for the triplet or not. On six by six? I'm not sure. I'm wondering if this is the lowest gap you have to compute for the antiferromagnetic state? I'm not sure. This is a gap for total spin equal to one. The smallest gap for total spin equal to one. For S equal to one. Okay, but so this, okay. So square means it's gap, it's gapless at pi pi. M means what? M usually means pi zero. It's in connection. It's pi pi. Pi pi. It's pi pi. Pi pi. Yeah. And this is for the triplet. Yeah. And can you compute gaps for the singlet sector? Yeah, well, I think we can actually increase paper. He shows some results, but I didn't compute. I just compute this gap. The second question is about the other slide if you can go to the next one. Structure factor, right? Yeah. Why the dimer structure breaks the symmetry? What kind of dimer-dimer correlation are you taking? This is because the dimer is chosen in the X direction. So it's dimer-dimer like that? Yeah, yeah, yeah. Dimer-dimer in the X direction. Other questions? Maybe you have a question. On the triangular lattice, there is no... Which lattice? In the triangular lattice. There is no unequivalent of Marshall sign rule. It's more complicated. But did you apply some prior on the sign structure in that case or just an unbiased simulation? I apply sign rule. This is the sign rule for small j2. You can see for small j2 there is also a spin order and the spin order can also show a sign rule and I apply the sign rule. Okay. It's like Marshall sign rule in square lattice. Thank you. You argued that taking different networks for amplitude and phase is bad. Do you think the same is also true if I have the same architecture for amplitude and phase but different output layers? Or do you think the only solution would be a complex network? I mean, this one. Okay, I think you have one network but for both amplitude and science, right? Yeah. Yeah, I think in this case, it can work. I'm not so sure but I think it's also better not to separate them but if it's a single network, that's possible to work. Okay, but in your opinion, it's the best to have a complex network. Yeah, or you have a real one but with plus or minus signs, that's also possible. Okay, thanks. Other questions? If not, we can say thanks to the speaker again.