 We can continue. Already a quarter of three. Whenever you. Sure, let's, let's go ahead. Let me share my screen. Can you see my screen. Yes. Okay. So. Okay, now, so I'm going to go ahead and so it's recurrent networks for quantum anybody, but if it is, I'm going to give you two examples of research that I've done using RNNs one for quantum state reconstruction and then one for variation of annealing, which is that idea that I just explained roughly or not roughly, but like a closely related to the question I just answered, which is so simulated annealing is a technique to solve a combinatorial optimization. It turns out that what you do is simply use Markov chain Monte Carlo or Monte Carlo on this problem Hamiltonians and then what you do is you slowly decrease the temperature and, and then you hope that at the end, you find the ground state of the classical Hamiltonian, which is the solution to the problem. But for challenging problems, this dynamics of the Monte Carlo, the Markov chain is very slow. So there's a chance that we can make progress with these models. And I'm going to give you an example of that. So let me start with the quantum state reconstruction, which is a little more involved. So, oh, and before this slide is in the wrong place, but so I wanted to mention that last week we posted this paper, this neural networks in quantum anybody physics hands on tutorial. Okay. And so we describe many of these techniques in that paper. Even the risk, the record neural network. But then the cool thing is we provided code that you can play with is very simple code, but it allows you to get started with all these ideas and so on. And so I encourage you to check this out has all sorts of examples variation Monte Carlo quantum state reconstruction. And so yes, if you're interested in you want to learn how to code this things, then this is a good starting point. So anyway, so let me go ahead with the quantum state reconstruction. So what is learning a quantum state and what is quantum state tomography. So it's says the following so quantum say tomography is the process of reconstructing the quantum state of device by measurements. So you take a quantum system. You measure it many, many times. And then you try to infer what the quantum state is. Okay. So this is the gold standard for verification and benchmarking of quantum devices. And it is useful to characterize, for instance, optical signals. It's useful to diagnose and detect errors in quantum state preparation. For instance, states produced by a quantum computer. It can be used to detect entanglement and many more things. So the idea is that we need to go beyond standard quantum state tomography reconstructions, because there has been recent progress in controlling very large quantum device quantum systems. There's also availability of arbitrary measurements that are performed with relatively high accuracy. And so the bottleneck becomes progress in the estimation of these quantum states. And they associated core cursive dimensionality. So this quantum states, when we try to represent them in the classical computer, the, if you want, they entail using exponential resources in time and memory. Okay, so, so we need to go beyond standard quantum state tomography, which uses all exponential resources. And for the future of quantum simulations and the benchmarking of quantum computers and so on, and quantum simulator. So I have here trapped ions on top. Google processor D wave is a quantum computer quantum annealer. And here I have cold atoms. So these are just examples of devices you may want to say characterize through quantum state tomography. So this is like the pace of growth. This is already outdated, but this is 51 atoms, quantum simulator. 53. This is 1800. So this is, I'm very proud of this. I participated in this one. And quantum chemistry simulators and so on. There's lots of exciting and quantum simulations, mostly that are becoming available and grow. And so can we find ways or can we did the device tools to benchmark this state preparations. So what are the ingredients, quantum system that you can prepare repeatedly. So because, as you know, when you produce a quantum state or as you may have heard of it. When you produce a quantum state you measure you destroy the quantum state so you have to be able to repeat the same experiment many, many times. And then you need some availability of measurements that you can apply so set the measurements. And then you need a training procedure and a model. Okay, so a training procedure is, you have a model for the quantum state, either a full representation of the density matrix, or a matrix product state or a matrix product operator, or even a neural network. And then a training procedure that if you want fits the measurements to this model, okay. And then at the end you need a certification, kind of like a putting a stamp on the model you train, which is, for instance, computation of the fidelity of the reconstruction with respect to an ideal state that you are trying to prepare in your quantum computer and so on. So a typical tomography protocol prepares many copies of the state and they're measured in multiple ways. Finally, the outcomes of those measurements are fit and produce an estimate of the quantum state raw start, okay, that's the fitting part and that's roughly how it goes. So here's one example of how you do this in practice. So this is called maximum likelihood estimation because it's the same spirit of what I was discussing. So it requires computing some probability and you use this my likelihood function that I discussed to fit a model. So what is this model? It's a physical density matrix in its most general form. So meaning that it scales, like the representation scale exponentially with the size of the system. But then what you do is you assume that the measurements are independent, which they are, and then you compute the probability of observing these outcomes in the experiment and you maximize this probability. This is the so-called maximum likelihood principle that we discussed. And then you fit the density matrix through the data, okay. It has some issues, which is exponential scaling in the parameterization and in the handling. So it's the time scaling of the processing of the algorithm is exponential, but this is the most reliable tool in the sense that it's the most general tool, but scales poorly. So you cannot apply it beyond a handful of qubits, say 10 or 12 qubits, I think at most, as far as I know. And so you cannot apply it for this large quantum simulator, okay. Then the question is how to make quantum state tomography efficient. And there are multiple ideas out there. So the most, one of the most interesting ones is introduce a parameterization of the quantum state with good scaling, but non-trivial structure. Like for instance, you can use a matrix product state of, or a matrix product operator and then just do a reconstruction using this as your model for the quantum state. And this is, these two are like one of the most powerful techniques is based on a matrix product states and matrix product operators. However, there are other approaches that we came up with which is introduce a parameterization of the quantum state with good scaling. So it follows that same trend about using said restricted Boltzmann machines or like neural networks in general, right. So this paper which we co-authored a few years ago, we used a restricted Boltzmann machine and we performed quantum state tomography, okay. And this was extended later by Giacomo Torlay and Roger Melco and they were able to introduce if you want a density operator that was written in terms of a neural network. So this is the reference three works for pure states with good structure and it has good scaling in terms of the resources you need to use. And then this latent space purification approach handles mixed states, but has fast scaling. So it doesn't solve all the problems. So today I want to tell you about an approach where we, so instead of parameterizing the quantum state directly, we parameterized the measurement statistics of a measurement, which is given basically by a born boot and to parameterize that measurement statistics or the probability distribution of the measurements, we use the RNN, okay, which is the model that we discussed earlier today. And then we use this idea to learn synthetic states. So basically numerically generated experiments, mimicking experimental data. So that's the idea of this whole approach. So let me tell you how to do this, but before that let me tell you this. So it works for pure and mixed states with structure meaning that it works well for quantum states that are well represented by a recurrent neural network and it has good scaling in terms of the resources and so on as long as this structure in the quantum state. Now, which states can we represent with an RNN? Still a little bit of an open question, but since RNNs are also universal function approximators, there's hope that as you make the RNN powerful or more and more powerful, that you're able to capture more and more the quantum states that are interesting and so on. And the evidence that we have is that this RNN can represent a reasonable and non-trivial quantum states. So I know I'm going to show you examples of that. So let me remind you the setting. So it's, so we have a large quantum device. We want to know if it's working as intended. So we think this device can produce some non-trivial quantum states and we want to certify that the system works in some simple cases, right? So we have this device, we ask the device to produce some quantum state like a matrix product state and so we want to benchmark this preparation. This is useful in the near term because as quantum computers become stronger and stronger, they're going to be producing quantum states that we cannot represent classically. That's the hope with quantum computing. So there's no hope that this is going to be working forever, but that we're going to be using it to benchmark near term quantum computers or quantum devices. So let me discuss a little bit the theory behind what we're doing. So quantum states measurements and probability distributions. So a quantum state is traditionally described by a density matrix, which describes the statistical state of a quantum system in quantum mechanics. Everything we can possibly know about a quantum state is encoded in the density matrix. So what is this density matrix? It's a positive semi-definite Hermitian matrix of trace one acting on the Hilbert space. This family forms a convex set, convex set meaning that all possible quantum states form a convex set and for one qubit is the block sphere. So it's this sphere that we have here. In high dimensions, the shape of this convex set is not known, but it's known to be convex and basically be similar to a sphere like a deformed sphere. So that's the traditional approach, but can we represent quantum states with just probability? That's what I want to do. And because I'm going to be using the RNN in its simplest form to represent it. So, and the idea is you can do this through measurements. That's why this is so natural for tomography, right? So and measurements are described by positive, by some operators M, right? We call them positive operator value measures. And there, if you want descriptions of what you do in an experiment when you go and measure the quantum state, okay? So there are this, these POVMs are just that represent mathematical representations of what you do when you measure a quantum device, right? There are collections of positive semi-definite matrices with a measurement outcome index A, okay? I, for instance, A could be if I measure along, like the spin along the Z direction, then you get either up or down. So this index A is that index. Is it, is it, is my measurement up or down if I measure, say, along the Z direction. And this operator sum up to the identity, okay? So you go to the lab, you prepare the experiment so that you measure M and your device gives you A, either up or down, for instance. So what is the relation between experiments and measurements? Is this expression here, it's called Born rule because Born basically came up with it, okay? So you have a quantum state, you're preparing the lab row. You have a measurement apparatus that you use, you measure it, and then the probability that you observe A in the experiment is given by trace of row and still operate the measurement operator, the POVM element, M, A, and that gives you this probability, okay? That's a probability that you observe either up or down, for instance, in the lab, okay? That's, that's, if you want a fundamental link between quantum theory, which is the description of the quantum state row and the description of the measurement and what you see in the experiment, which is some probabilistic outcome of the measurement. Okay, so now I'm going to use the so-called informationally complete POVM, so informationally complete measurements. So, and what are these informationally complete measurements? They're basically a measurement or a set of measurements such that if you measure the quantum state with those, with data apparatus, then you get all the information about the quantum state. And you can think of this as the following. Suppose you have like your quantum state is a, I don't know, a three-dimensional object and then you, you want to understand that object, right? Like, so for you to do that, you have to take this three-dimensional shape and you have to see it from different directions for you to be able to see it entirely, right? Like to characterize its shape, okay? So informationally complete measurements are that thing, right? They are like observing the quantum state from all possible directions such that you can determine what it is, right? Determine the quantum state. So I like this analogy, right? It's like observing an object from multiple directions. For instance, measurement that is not informationally complete for a 3D object is, I don't know, observing only along the x direction. That doesn't tell you what happens behind or above and so on, right? That's the idea. So informationally complete measurements, a set of measurements that allow you to see the quantum state in all possible directions such that you can throw a cartoon of this quantum state. That's how I think about this or the simplest way, but a simple way to think about it. So informationally complete means that the measurements of this 6PA, which is beyond the observations, contains all the information about the quantum state. It also means the following. If you have a Hilbert space, then you can span it with this set of operators, right? Meaning that you can write any operator or as a linear combination of this operator's MA. The final meaning that I want to highlight is that the relation between rho and the probability distribution peak can be inverted. So we go from rho to peak. If you have informationally complete measurements, typically you can invert you. So you can write rho in terms of peak, the probability distribution, so that peak becomes the quantum state if you want, okay? And that's what we exploit here. We exploit informationally complete measurements and we represent the quantum state in terms of this probability distribution pA. So this is the inversion part. I don't want to go through the math, but this is the following. Modern rule tells us that you can go from the quantum state and the measurement to the probability distribution. If you have an informationally complete set of measurements, you can go the other way around, which is you can write rho in terms of the probability distribution p, okay? That's the key element. And then the expression is this here and what I have here in this representation is a tensor network representation of this expression, which is telling me that what we're doing here is factorizing rho in terms of probability distribution that is complicated and a set of simple product of tensors that are very simple. They're factorized, right? So all the complexity and potential interactability of the quantum state gets pushed into the probability distribution because that's what makes this distribution not be, say, for instance, a mean field. So all the interactions and all the entanglement gets pushed into the probability distribution and that's what we exploit. So what we do or the insight we had was we can create a representation of the quantum state in terms of a probability distribution over an informationally complete set of measurements M, okay? And what we do is we use recurrent neural networks to represent this distribution p. And we know recurrent neural networks are very powerful. So that's what we use, right? So because they, first, they allow for exact sampling. They have a tractable density. We can compute the probability of configuration A and we can use maximum like the estimation, for instance, to learn the RNN. If we have a data set of experimental outcomes, that's the idea. And so what we do or what we have here is a model for the density matrix, raw model, that is an RNN model here, which I call language model is they're typically using in language processing. And then you attach a set of simple tensors that tiny, okay? And that's our model for the density matrix for the quantum state. So let me recap what we do or what we would do in the lab. And we have done it actually in the recent paper, but so what we do is this. We prepare a quantum state repeatedly on a quantum device. Here's Google's processor or IPMs quantum computer. We perform this measurement, this informationally complete measurement. This gives us a large collection of measurement outcomes. So it's a big data set. Then we fit the recurrent neural network using maximum like an estimation exactly what I explained in the first part of today's discussion in the morning or in my morning. Then we invert this density matrix. And I said, I use quote unquote, this is only a formal thing, inverting this exactly is exponentially difficult. So we don't do it, but we can do it in practice. If you want, if we want, for instance, compute them say correlation functions over the quantum state and so on. Or if we want to compute say fidelity is a classical fidelity and so on. And then we perform some sort of certification, either through fidelity, classical fidelity, or measuring correlate like correlation functions that are relevant for the system you're interested in. Okay. So that's what we did. So let me, let me give you one example of reconstructing numerically generated quantum states. So, so here is so called pure G8 set states. So it's the so called cat state too. It's a superposition of all spins are 0000 plus all spins are 11111. Okay, so that's this cat state. The density matrix is this one here. And then we also introduce a model of noise because so this is a superposition of the quantum state, but we also want to explore whether we can represent a mixed states. Right. And so we use a model and noise model. And this noise model is the following. So with probability P. We apply an error on the quantum state where we with probability one that we apply either Sigma X Sigma Y or Sigma Z on the state. And with probability one minus P we do not. So that if probability P of introducing an error is zero, then we are back to the pure state. But if we make P large, then we have a completely mixed state. That's the idea. So can we reconstruct this quantum states using a recurrent neural network? That's the question. And here's like learning all. So this is two qubits. And we play measurements for different values of noise. And then we train this model. So in here I'm using a restricted boson machine first as a first example. And this is the KL divergence, which is basically the difference or the distance. And it's not exactly a distance, but a divergence between the model distribution and the exact probability distribution of the measurement outcome. And this is as a function of training as we train these models using maximum likelihood principle for maximum likely estimation. What we see is that as we train the model, this distance or this divergence goes to zero for the different values of noise is zero all the way through is one. Okay. And so we successfully train this model. What we see is that training the distribution is harder for low values of noise, but easy for large values of noise, which makes sense because when you have high values of noise, this is basically a constant distribution. It's a completely random, completely flat distribution. So it's very easy to learn. There's nothing to learn. It's just one parameter. So this is the easiest to learn. So it goes to zero faster. So it goes to zero, whereas for low values of noise, it takes some effort, right? So this is the KL. This is for the fidelity, which is basically the overlap between the two distributions, which goes to one as you train. And then finally, we have the quantum fidelity, which is the distance between the quantum states itself. And it follows the same trend, right? It goes to one as you train the models, meaning that everything is working. However, this was only for two qubits, and it was difficult to scale it to more than three or four qubits. So we said, how about we try something else? And we tried the RNN. And for the RNN, we were able to go up to 80 or 90, I think 100 qubits, which was pretty surprising. So we were very happy. And we thought this was very strong result. So the RNN is capable of representing this quantum states in this form, like using probability. And what we have here is a function, like a classical fidelity as a function of the number of experimental outcomes that we use in the reconstruction. And what we see is that for small values of noise, then you see that as you add more and more data, you get a better and better reconstruction, right? The higher and higher fidelity. And what we also see is that for high values of noise, you need less and less data for the same reason that learning a flat distribution is cheap. It's easy, right? You need fewer data points. The interesting thing is that for you to get a high classical fidelity, say for a value of 0.95, you only need a number of samples and star that scales roughly linearly with the size of the system N. So this is the number of qubits. So in that sense, this approach is, I mean, it has some scalability properties that are mild. In the sense that if you're trying to use this so-called classical fidelity, you can achieve high reconstruction accuracy with moderate resources. And this is in part due to the fact that we're using this RNN, which is very powerful, and that we can represent many important distributions with it. Now we moved on to ground states of local Hamiltonian. So this was all for the GHF, but then we wanted to explore if we wanted to, if we could actually use ground states of many body Hamiltonians. And so this is for 50 spins, we use DMRG matrix product state, and we find that we can also reconstruct this ground states pretty accurately. So this is the orange curve is synthetic data. So coming from density matrix randomization group, expectation value of sigma z, sigma x as a function of the site. And what we see is that our reconstruction matches like the correlation functions of the reconstruction match pretty well the synthetic data. Okay. And this is for two body correlation functions, sigma one and sigma i along the z direction. And again, we see a good agreement. Then we have a little bit more complicated model, which is the Heisenberg model on the triangular lattice, which I mean, it's an interesting model in that it has a complicated sign structure, right? Like this is a model whose function is not known, and we know it has a complicated sign structure. And we wanted to see if we could represent that ground state with only probability, which is what we're doing. And so that's why we picked this example. So this is on us eight by eight lattice, and these are correlation functions of the synthetic state, namely the ground state of the Heisenberg model on the triangular lattice. And these are the reconstructions. And it seems to be working really well. So yeah, so that's it. So using the RNN, so it's kind of to conclude, so using the RNN was important for the success of the method because of its tractable likelihood and the fact that we can exactly sample them. Okay. As I highlighted. If there are any questions. Now it's a good time to discuss. No questions, so maybe I can go ahead. Okay. So that was it for reconstructing quantum states. Now let me tell you about something newer. So this was a question. So can you explain why we can get exact sampling? Oh, yeah, so the, this is from Kazuki. So can we, how can we get exact sampling is because this RNN is constructed by parameterizing all the conditionals in a probability distribution, P sigma one, two, three, two N. So what we do is we, the model parameterizes each of the conditionals, meaning that we parameterize P of sigma one, and then P of sigma two condition on sigma one, sigma three condition on sigma one and sigma two and so on. So if we have all a specification of all those conditionals, then you can use the conditional at one to get a sample of P sigma of sigma one, then you can feed that into the conditional for sigma two, and you can sample that conditional. And then you take sigma one and sigma two, and then you compute the conditional on sigma three, and so on until you exhaust all N spins. And so since you have access to all those conditionals, which is through by construction, then this is an exact sample of the model, and then that's why you can, just because of the construction of the model, okay? Which I think, yeah, it's very important. Okay, so if there are no other questions, oh, there's one more. So this is from Robertson. How does the measurement MA comes in the example of calculating ground states of Hamiltonians? Yeah, so that's a good question. So it comes in the following form. So what we do is we prepare, so we imagine we're preparing this quantum state, right? In a device. And then we measure the, we measure this device or this quantum state, that is the ground state of the Hamiltonian, and then we collect the statistics, and then we reconstruct the quantum state. So we use basically the measurement outcomes of those measurement operators M to train, okay? The RNN model in these examples too. I hope that answers the question. Then let me go ahead with one more, a pollinario. So can Fisher information be used as an alternative measure for UVM? That I don't know. I'll have to think about it. It could be, but I'll have to think about it. All right, let me go ahead. Okay, so let me go ahead with the next. So this is new from a couple of weeks ago. So it's variational neural annealing. So let me introduce this idea of combinatorial optimization. So many, many important challenges in science and technology can be cast as optimization problems, right? So there are famous optimization problems that are, that we use, like, and motivate this, but say traveling salesman problem, nurse scheduling problem, vehicle rounding problems, spacecraft scheduling, circuit design, discovery of the heat bosons. All those can be recast or the data analysis of this experiments or this problems can be recast as an optimization, okay? And these are computationally very difficult problems. It turns out that many of these problems can be formulated as finding the ground state of a classical easing Hamiltonian that I call H target, okay? So this is the expression. So this sigma i sigma j variables are just basically plus or minus one. And this is extremely general. Like many, many, many problems can be cast as finding the ground state of this. The ground state of this Hamiltonian is finding the spin configuration sigma i that minimizes the energy or this ground state is energy or this Hamiltonian, okay? But finding these solutions is extremely hard for some problems, right? So there are heuristic methods to do it. And one of them that I really like is called simulated annealing. So what is simulated annealing? So it kind of like inspired by old technique. So it mirrors the analogous annealing process in material sign and metal region where a crystalline solid is heated. So you warm up a piece of metal like a sword and then you slowly cool it down. And as you cool it down, this piece of metal finds its lowest energy and most structure is stable crystal arrangement, which makes the material really durable and hard. And this is actually used in that when people build weapons and swords, okay? And so people took in the 80s, they took inspiration from this metallurgic technique to devise a way and approach to solve combinatorial problems of this form, basically finding the ground state of an easy Hamilton. And so what they did was they defined this simulated annealing because it's not real annealing. You don't heat up your computer or anything. You basically simulate annealing and explore this optimization problems, energy landscape by a gradual decrease in thermal fluctuations. But these thermal fluctuations are generated by Monte Carlo by the Metropolis Hastings algorithm. So basically what you do is you take this Hamiltonian and you simulate it using Monte Carlo, okay? And then what you do is you slowly cool down the temperature of the simulation, okay? So you decrease the temperature little by little until you reach the temperature to zero and then at zero temperature, you should be seeing only configurations that are consistent with the ground state of the target Hamiltonian, okay? That's the idea with simulated annealing. It provides a fundamental connection between thermodynamics and the behavior of physical system with complex optimization problems. So I think this is very appealing and a very beautiful algorithm. The problem with simulated annealing is sampling the Wolfman distribution using Markov chain Monte Carlo or Metropolis Hastings algorithm becomes very slow for hard optimization problems as you cool down, as you make this temperature too low. And this is because the autocorrelation of the Markov chain becomes very large, okay? So finding solutions to this problem becomes very expensive because you have to wait for a very, very long time, okay? And this is a schematic cartoon of what happens, right? So what you do is, so this is a simplex. This is the space of probability distributions. And so what you do is you start at very high temperature somewhere here, and if you did annealing at a very, very slow speed, you would follow a path that connects infinite temperature with zero temperature. So this is if you want, is the exact Wolfman distribution at all those steps, and then you solve the problem exactly, which like the solution is either, like for instance, for a degenerate problem, you find this configuration or this configuration. So that's if you do it very slowly. However, if you do simulated annealing, which is you do a Markov chain Monte Carlo, this Markov chain would go out of equilibrium, okay? So that I represent here, and then you get stuck somewhere here, and you end up with some approximate solutions, maybe good, maybe bad, okay? Now that's what happens with simulated annealing. Now what we're proposing is here's one idea. Replace annealing and approximately sampling the exact Wolfman distribution by annealing an approximate distribution that is closer to the Wolfman distribution like an RNN that can be sample efficient. So there's no auto correlation, okay? That's the idea. And then this may or may not lead to better solutions. So that's why this method is heuristic, but what we find is that there's approximation to the Wolfman distribution that we compute with the RNN is a better approximation to the exact Wolfman distribution. And so it means that you can get better solutions than simulated annealing by using the RNN because of the fact that you can get exact samples without auto correlation, okay? So that's the idea. And this was posted on the archive a few weeks ago. And so let me show you one example of solving optimization problems with this technique. But before doing that, let me tell you how we train this RNN for how we open my thing. So how to train this RNN so that it mimics the Wolfman distribution. That's what we want to do and then anneal in temperature. So we use a time-dependent free energy and then the nice thing is we'll have an energy part. This is the target Hamiltonian. We have a time-dependent temperature, which is kind of like this temperature that allows us to go from high temperature to low temperature and then the entropy. That's going to be our cost function, the thing that we optimize. And then we start at high temperature P0 and use a linear schedule function E such that as we make T small T larger and larger all the way from zero to one, we end up at the ground state of the. And then what we do is we optimize this free energy. Okay. So what is the algorithm we perform a warm up at high temperature where we make the RNN match the high temperature distribution. Then we do small time steps. And we retrain the model at each temperature. Okay. And we use the variational parameters from the previous step on the model at the next time step P plus delta P. Okay. Then at the end of the annealing process, this distribution given by the RNN is expected to assign high probability to the configurations that solve the optimization problem. So that's the strategy. And so this is this are the results. So let me highlight, for instance, figure a. So this is for a spin glass program with. I think with 100 spin, it's called the sharing for sharing the Patrick model. And it's a fully connected spin glass. So it's deemed to be challenging. And what we find is that as we make the number of annealing steps, meaning the time we take from high temperature to low temperature, what we see is that the residual energy, right? Meaning the energy, the excess energy with respect to the exact solution to the problem, goes to very small numbers, basically around 10 to the minus six for variational simulated annealing. So basically for our technique, which is this thing in blue. And it does so at a faster rate than traditional simulated annealing, which is this red curve here. And even simulated quantum annealing, which is also a powerful method inspired by quantum, by quantum annealing. Actually, and we find that this technique is like our technique is significantly better if you put a lot of annealing steps. So this is for the sharing tone here, here, Patrick model. And this is for so-called, we share planted ensemble. It's also a fully connected spin glass. This one is very interesting and very difficult to solve. But we also find that our method, if we allow enough annealing steps, we find solutions that are orders of magnitude more accurate than simulated annealing, as well as simulated quantum annealing. And this is again one more example of the we share planted ensemble, where we also find, I mean, we find better solutions, but not as accurate as in this two other examples. And with that, let me conclude and take a few questions. So we introduced a formulation of the quantum state that is closer to statistical theory, because we represented the quantum state in terms of probability. We use that representation to reconstruct quantum states of increasingly large sizes. And so it provides a way to approximately reconstruct this quantum states. And then for the final part, we introduce a variational formulation of simulated annealing that produces very accurate solutions to spin glass problems that may have applications in all these areas. And just to conclude, this personal belief that there are a lot of opportunities at the intersection between physics, quantum physics and machine learning. And with that, let me take questions. Thank you, Juan. So yes, you have two questions already. Okay. Hi, dear, is it possible to get these lights of this courses? Yeah, I can send my slides to the organizers and then. We will post it in the web page of the event. Okay. And then there's a question by you, Seth. Could we use variational annealing to construct states ensembles in a quantum, in case of quantum steering. I don't, I'm not familiar with quantum steering. So can you explain it to me? Maybe we can allow you. Good evening. Thank you, Juan, for the, for the presentation. The quantum steering is a correlation between the non-locality and the entanglement. And it's used, it's used to, to secure the communication, the quantum communication. I am understand, and there's, and there's, could you understand me or I need to clarify? I think I want to understand what's, what is it that you want to do? I'm not sure I understand. So what the case, when we have the case of entanglement, we have two trustable parties. If we, for example, Bob and Alice, but in the case of steering, we have, we don't trust one of them. So Bob sent their states and, or Alice sent their states and Bob need, need to check them and need to construct them, to construct the state. And maybe we, we use, we use, in the normal case, we use the semi-defined programmation, if you, if to optimize the state, but Alice is in the tubal. So could we replace the state, the semi-con, semi-defined programming by, by the variational method? Yeah, so I think I now understand. So I think that as long as you, you can formulate this problem as, as a combinatorial optimization or as finding the, the solution as finding the ground state of a classical Hamiltonian, then you can use it. Thank you very much. Yeah, the question is, can you reformulate it that way? And if yes, then you can. Thank you. Okay. There's a question from John Carlo. So can you study metastable states, local minimum and free energy barriers and characterize the free energy landscape with, with this variation? So I think that's a good question. I think yes, right? Like you can try to do that. The problem is, or the problem that I see is you may, you may, so the training, so the optimization of the free energy may fail. Okay. And there's, there's no easy way to check for that. But I, I, we, with my student, we think there's a way to solve this issue of like, like a drop mode dropping, which is basically missing some of the local minimum. But I think the method itself doesn't guarantee that you will find all the modes of the distribution or that it's going to explore all the, like the entire landscape. But I know people have tried this, like this, perhaps this sit style of approaching this problem using machine learning. So there's, I think there's hope and there's potential to approach this type of problems with this techniques with annealing. Thanks. Okay. There is a question also in the chat. Could you share references for answers for spring last solution with additional essay. Let me in the chat. I did. I didn't hear very well, but let me. Okay. Let me read it again. Could you share reference and source. So the references. Let me share again. So the reference we have is, is this. So archive 2101, 154. And the source we have not released it yet, but we will do it. Okay. However, this is very easy to code. So if you go to the handsome there's an iron and already there. And you can change. So this is basically a loop over, like this simulated variation of simulator annealing is just a loop over different temperatures. When you try to optimize the free energy, which is easy to compute. And so there's plenty of code that is available to do this. And even in this handsome tutorial, there's already code to do it. But, but we'll eventually release our code in here for spin glasses. Any more questions. Yes. There is another question in the Q&A. Yeah, so Robertson asked, I'm not sure if I missed it, but how do you verify results thrown out by the algorithm, for instance, in the spin glass problem. So, and this is a good question. So for the spin glass problem is so for the sharing don't care Patrick we use the so-called spin glass server. So there's a server, I think this is in Germany, you look at where you just give them the problem Hamiltonian and they give you the answer. They use some heuristics and they, they tell you, yes, the algorithm. This is the answer or like they tell you we cannot solve it either because it's too big or there's no heuristic to do it. And so for the sharing don't care Patrick model, we use the spin glass server and for the Wischer ensemble. So the Wischer ensemble is very interesting. It's a fully connected spin glass problem with so-called planted solutions and planted solutions. What that means is that the solution is known to you by construction. So it's kind of like finding needle in the haystack. Where you know what the needle looks like. It's just that the needle is surrounded by strokes by a big bunch of like haystack. And so the energy of the straw, the haystack, they're very, very similar really close in energy to the energy of the needle. Okay. But you know what the needle is. So you know the solution. So you don't. So the way you verify the solution given by the VCA, but the variational simulated annealing is because you know the exact solution. You're just trying to find it in a say rough landscape of similar states surrounding it. Thank you. But what if we don't have that known solution? Then you just hope for the best. Yeah. Many of these problems have no, no known solution. I mean, if we did then there wouldn't, there wouldn't be problems. Like they wouldn't be optimization problems. You just knew the solution then. So the solution is that for many of these problems, what you do is you try and you find the best energy you can and then you take it. And then, yeah, that's how, how it works for some of these problems. There are bounds for some algorithms. So some algorithms can tell you you're this far. From the real ground state, but not all of them. So in our case, this doesn't tell you. It doesn't give you any guarantee. It's not your list. Okay. If there are no more questions, I think we can finish here. We have plenty of time. So people want to ask you yet. But if not. Thank you again, Juan. Yeah. Thank you very much. It was a pleasure. Next time you came here. And that's all. Well, I don't know. Do you want to say something for concluding this series or machine learning? Yeah, I hope it was very useful for all the participants. It was definitely useful for me. Much in the subject. And so, yeah, thanks to Juan and to all the previous speakers and tutors. Yeah, thanks for having me and it was fun. And hopefully we'll see each other in the future. I learned a lot. I learned a lot. So thank you very much. Thank you. Thanks to everyone. So thank you. Let's stop here. Yeah. Bye.