 discussion. So now the floor is open for whoever has a question. For example, Robinson was about to ask a question. Yeah. Yes. So my question goes to Inak. And if you check your slides, you use the original Monte Carlo. The question I want is that how do you choose your travel function? How do you choose your travel function? Very good question. So basically, you need to choose a travel function that is expressive enough, and that can be efficiently computed, right? And you can sample very efficiently from it. So we choose neural network because they are universal approximators. Here's the reason. One of the reasons is that now there is a zoo of neural networks and that you can choose, right? And the first one that was used in the context of quantum simulations was receptors on chains. But we didn't choose that because of the need of devising either a mark of chain to simulate it, right? And since you are simulating glassy systems, I expect that when you enter into the glassy phase, I mean, most of your proposal to flip a spin are going to be rejected. So we chose to have a notoregressive model which generates like samples exactly without any kind of correlation function. That was the main motivation behind it. And as I motivated, they are powerful to encode correlation like when we see machine translation or speech recognition. We see the same thing for spin systems. So both, I didn't show results on language spin model, but we consider that spin model like the Sherry-Turkey-Patrick spin model, which is a leasing glass, right? And that has, which is believed to be anti-complete. And we found also quite good results using the RNA code function. So it's mostly the expectability and the ability to compute the efficient, right? And for the variational classical annealing, we need to compute the entropy. And for the entropy, you really need to have a probability distribution. But if you choose a non-zap like the RBM that is not directly normalized or CNN, I mean, it will be held to compute the entropy. So this was like one of the motivations of choosing this kind of authoritative neural network. I hope that answers your question. Yes, yes. It's okay. Thank you. Thank you for your lecture. Okay. Please have a question. It's okay. Who has a question? Can you hear your vision so that we can see you? Okay. Okay. Yes. Yeah. So my question is, to the first speaker of the day, we talked about the quantum thermodynamics. So I mentioned some few things about adiabatic short-code. So the question is, I think that adiabatic short-code necessary in use non-adiabatic long way. And so if this is true, a full running machine may include both adiabatic and non-adiabatic process. So my question is, how to minimize the energy costs during non-adiabatic long way in adiabatic short-code? Okay. Yeah. Thank you. Yeah. The idea is, when you start to run your engine, so as you mentioned, in the long time limit, then you still have non-adiabatic and then yeah, the adiabatic contribution. So but this is minimal. And the idea is, you want to engineer your machine in a such a way that using the short-code protocol, so the short-code to adiabaticity, what it does is that it suppresses the non-adiabatic contribution. So it reduces the entropy production rate. And thereby, once you reduce this, then you can have a better performance. But in a very short time, this is not useful because the cost that you use to generate this kind of protocol will outweigh the work friction that you have in your system in a short-time process. So but intermediate, it gives you some advantage in a very long time. Two of them, it doesn't matter whether you do it adiabatically or non-adiabatic. So but what happens if you want to do it very fast, then if you want to be better, you have to put some control in your system. Okay. So what happens if in the system we have a source of decay, so that maybe the overall Mediterranean becomes a non-emission of the system? You still have some way to construct short-code for non-emission operator. So but for heat engine, I haven't looked at this to know, okay, how does this influence? But you can construct this short-code to adiabatic system because basically what you need is that you have your eigenstates, your solution of your system, and then with this eigenstate solution, you use it with each time derivative to construct this short-code to adiabatic city protocol. Okay. Okay. Okay. Thank you. But everything that, particularly, you may need to increase some dispassion. But Ali mentioned that question. If the system, maybe you have non-harmonic oscillator, what will happen to an harmonic system? So thank you. You have a non-harmonic oscillator. The only challenge you have is just how do you construct, yeah, the cost of the protocol would be more challenging. So like at moment, we try to have a system like you have a hybrid of oscillator and two-level system like the Jen Scamming's model. Yeah. So in this situation, we can construct a short-code. So but now what we have to find a parameter range where you can actually run your engine. So once we are, yeah, we have already like some area, we can do this and then calculate the costs and then you can see perform engine cycle. So like having non-harmonicity or non-linear system is not a problem for the control. I think it's more of a problem. Maybe if you are thinking of implementation, yeah, how you interpret these like in experiments. Yeah. Okay. Thank you. Thank you. So are there other questions? So I would like to ask. Okay, go ahead and see. Go ahead and then I have a comment. Yeah. I would like to ask one or two questions to the last two speakers. So more or less you presented two new methods to obtain the ground-state energy for a spin system or less. And the last speaker actually mentioned the possibility of, not the possibility that what about the system for, what about more complex Hamiltonian. So I'm going to take you a little bit out of your field. And if you are thinking about a molecular system like either big molecule or smaller big molecules or materials, how efficiently do you think the method you just presented are going to be able to obtain the ground-state energy of those systems? I know that there have been a few papers in the past three years where those ideas of machine learning and those type of development have been applied to molecular system. But what about the method that you just presented today? Let me start. Yeah. So first of all, I mean, the method that I presented is to find the ground-state typically of classical systems. But in the annealing, you can stop somewhere. I mean, you have to stop somewhere and then you can find the ground-state of a quantum system. So for a molecular system, what usually people try to do is that, I mean, maybe it's better to walk into a formulation of the Hamiltonian that is in the second quantized form. So if you have a molecular system, you write one term in first quantization, and then you basically write it in second quantization. There are some techniques like the private Kitev method that allow you to map it into spin systems. And when you have the formulation of the Hamiltonian in this kind of spin system, I mean, you can basically find the ground-state. You don't even need annealing to find the ground-state. I mean, you just use Vaishwan Matikalo. And I think in the paper I mentioned by Muhammad, I don't, I think he used, he had to refine the ground-state of the hydrogen atom by doing this kind of mapping, if I remember well. And yeah, so it's pretty possible to find it. Yeah, I think maybe answer Glenn's question that Matikalo is also for quantum circuits to find it. If not, QA, maybe Vaishwan or quantum, I can solve it on a great quantum computer. Typically, they do the same mapping, right? They go from the first quantization formation of the Hamiltonian to spin configuration, to a spin formulation, and then they basically simulate the system. Maybe it can also, what is to say? Yeah, so just add a few comments. So thank you, so what the state is actually correct indeed. And I'll just comment that there have been recently some advances in bringing the, transforming the classic optimization problem of optimizing for a molecule, which is quantum into classical, into a classical problem, of which then one can apply annealing. So this kind of approach now it's actually studied, but again, as this or as optimizing directly the problem with the quantum Hamiltonian, the performance of current available devices, the devices are not yet compatible with the classical optimization algorithms. So this is a war we are actually trying to push forward both simultaneously, let's say various group, both theory and experiments such that it's possible actually to have competitive solvers that are using this kind of algorithm, because ideally one would hope to exploit the quantum speed up or use a quantum speed up arising from this kind of devices. Okay. Anne, you had a question. So Glenn, how many years do you think are we from this reality? Yeah, okay. Is it decades or years basically? This is a question. That's a good question. And I think the answer, it depends. There are decades from having the ability to perform a full general computation where you give me a problem and I solve it. And I think we are years and we are not far from having a solution for Taylor problems. So that there are, there could be some problems which can be solved if more efficiently with the quantum device. For example, yeah, for example, like the Google group showed that there is proposed the kind of problem where the machine was solving that problem more efficiently than classical algorithm. Previously, Fari proposed the quantum algorithm that could help it form classical algorithms in solving some kind of linear equation. Then again, the classical algorithms overtook it again. But still, when we are talking about single Taylor problem, on some problem, there is competition. When we talk about a general purpose solver like a computer, there is, I think we are pretty far from that. But I see. Okay. Can I, thanks for your answers. Steve, can I just follow up with Estelle? Yeah, Estelle, I know we had discussed this many times before, but I wanted to see if you have given it more thought. Of course, in the problem of polymer physics, I'd say in biological systems, this idea of the challenge of detecting the low energy structures has been, people have been thinking and working on this for decades. Do you see a possibility of applying these types of techniques that you're working on to those classes or problems? Or do you really see them more as doing more fundamental ideas on proof of concept models and toy model Hamiltonians? Do you really see a possibility of making the bridge? Yeah, I see actually the possibility of making a bridge. I mean, what I am going for is to solve real-world optimization problems. Now we are kind of investigating a couple of real-world optimization points. This technique, things like what's for your optimizations and stuff like that. Now for the protein, I believe that there's the lattice formulation of protein folding and stuff like that. So I believe that iron end will actually be quite good to try on it because I think somebody mentioned about what happens when you have degenerate growth and stuff like that. So with the autoregressive sampling, the moment where you are able to capture all the modes of the distribution in the weights and the biases of your iron ends, basically can sample the degenerate growth stage without any problem. So you won't have the problem of having to cross a barrier, like the color simulation. But if you can capture the mode, of course. So for a protein, it will be like capturing different conformation of the protein. And for sure, it will be pretty much interesting to see, to test that kind of problem of protein folding. Yeah, okay. Cool. Can I ask a question? Yeah, yeah. Seth? Hi, yeah, my name is Seth. So maybe this is more to maybe a stale also. So you mentioned that in your wake, you considered only Hamiltonians, which has not got the sine problem or wave functions with no sine problem. But in general, for quantum systems, you can have, it comes with a complex wave function. So in that case, I know mostly the Monte Carlo techniques don't work. So are there techniques maybe in your field that tries to address these kind of problems? Yeah, that was an excellent question, Seth. So because at 20 minutes, I had to constrain my talk, not to talk about the non, I mean, problems with sand problems. So if you can pair with me, I can share a couple of slides where we have these also. Is it okay? Sure. I mean, I'm okay with it, but yeah, sure. All right. So if I go back to my PowerPoint and I, sorry, sharing will stop. And then I do this and I come here. So yeah, this is basically a way in which you can capture. So if you check Mohammed's paper, it basically writes, because I told you VMC is not like other quantum metals, right? It doesn't have an intrinsic sine problem. So this is the way you can represent the amplitude of the wave function of the so-called stochastic Hamiltonians and Hamiltonians that doesn't have a sine problem. The amplitude of the wave function is positive. But if you want to have a complex wave function like you mentioned, so you can have basically out of your iron and cell, one that produce like positive amplitude and one that produce some sort of phase, right? And so you can use the soft sine layer that produces the phases and then you can have actually a complex wave function. So they try it in the paper on some sine to vertical prime. So in this case, you will learn this, the sine continuously, but if it's a complex with any kind of phase, you can learn the phase as well. So let's see. So this is how you estimate the energy version of Monte Carlo. Since the samples are drawn according to absolute value square of the probability distribution, this is always positive by construction. But the local quantity depend not on the probability, but on actually the way of the wave function. So this is where you need to have either the sine or the correct phase of the problem. So we tested that on non-stochastic Hamiltonian. So basically you just have to put a plus here. If you put a plus here, you basically have sine problem in the granted wave function. And we found that by looking at the variational energy respect to this parameter, we can capture it exactly. So we also did annealing on those kind of problems. And this kind of dragger and Hamiltonian where you have this kind of couplings. So these two couplings have sine problems. And we find that basically you can have the residual energy going down as a number of annealing steps. This is very interesting because actually this paper is a paper by D-Wave, where they actually try to implement non-stochastic Hamiltonian because of basically the claim of this paper saying that you can have an exponential enhancement in quantum annealing by using non-stochastic Hamiltonian. But in this paper, they can only do two tricks. I mean it's pretty hard to implement this kind of YY coupling, but in a method, this is like a 64-spin simulation. This is like the first simulation of the non-stochastic Hamiltonian up to 64-spin. You can't do that in any other kind of Monte Carlo method. So yeah, you can basically use this formulation to simulate both complex Hamiltonians and Hamiltonians of sine problems. I hope that answers your question. Okay, yeah, yeah, that's interesting. Yeah, I will look at the paper. Thank you. So there was also a question in the chat for the speakers. And it was from Omoroluakinojo asked about how to get excited states from the method you described. So you mentioned you're interested in the ground state. What about the excited state? Do you have an idea? It's the question for who is the question? Any in one of you actually. Okay, okay. So obtaining an excited state actually with the method I discussed now. So with annealing evolution, it's quite, I'll say directly, it's more challenging because you're not protected by gaps with the ground states as much. Usually ground states are further than from the other states. So the evolution can be faster. So technically, the scheme I described can be used with also with unexcited states. So the adiabatic theorem still works, but the performance of the algorithm will be worse. So yes, so that's why it's usually used for ground states. So there's no way of controlling that you get precisely an excited state rather than the game. If you slow down, you have to, as I said, when the gap is when during the evolution, there are places where the gaps are becoming smaller, you have to slow down. So if you slow down sufficiently, so you take a sufficient amount of time, at least with the direct annealing evolution, you can obtain, you can also obtain and start from an excited state. You can also remain in the excited state and obtain an excited state. However, the gaps you will encounter are most probably smaller than the ones you would find when doing the same process for the ground states. So you would have to have a slower dynamics and a slower algorithm. Whereas for the variational algorithms such as QAOA, I must say that there I'm not, at the moment, I'm not aware of any application of the algorithm to find excited states, because there you have to minimize the energy variationally and for excited states a bit more challenging. I'm not aware of that, but maybe Estelle can add something if she knows. I think for variational Monte Carlo, people have used other cost functions that could somehow project out the ground state. I'm not also very much familiar with it, but in quantum chemistry usually they, I mean, they do variational Monte Carlo first and then they do something like differential Monte Carlo to have more exact results. And then I think in differential Monte Carlo also there's a way of projecting out the ground state and then obtaining the excited state. But I'm not familiar with that. I am not familiar with that at all. Are there any more questions? If not, I would like to thank all the speakers for this very interesting session. We will end. Steve, I'd like to just, I want, you know, if we were at ICTP we would have taken a picture near the sea. But obviously we cannot take a picture near the sea, but instead we can take a virtual picture. So I'd like you all to turn on your cameras and I will take three pictures. And just, just so that, you know, we can say we took a picture. So everyone turn on your camera. Smile. Smile. Okay, hold on. Now wait for a couple of everyone, turn on your cameras. Not everyone is turning the cameras on. Okay, here we go. I didn't put a finger in his screen shot. Yeah, this is more fun. Come on. Okay, great. Everyone smile. Thank you. Very good. Keep smiling. Keep smiling. For some reason, this might, the second group of people who don't have all their cameras off, but fine. Anyway, all right. Thank you very much. Great. So thanks again, Steve, for hosting the session. Thanks to all of you for joining. And we'll see you all tomorrow at 10 a.m. Central European Time.