 Thank you and good afternoon. Let me thank again the organizers for putting together this great workshop And also the school that we had last week. It's been fun having a couple of lectures, especially one on a Saturday morning And that I will never I will never forget So during this seminar, I will show you some Applications of the idea as we've seen the last week for those of you who didn't follow the lectures I'm still going to put some Introduction even though for sure it's not going to be very Dora, so Today's topic is about using machine learning techniques. So if you want artificial intelligence to study quantum many body problems The the problems I'm going to discuss today and the the the works I'm going to discuss today have been realized so first in collaboration with Matthias Royer who's now at the Microsoft research in Seattle So this paper here and then another work also pretty recent in collaboration with Roger Michael's group at perimeter and the University of Waterloo, Juan Carrasquilla, D-Wave and Williamo Mazzola at ETH Zurich so First of all, what is the problem we would like to solve and address with neural networks and What can we do about that? So the probably one of the most central problems in quantum physics Inferetical physics in general is the the quantum many body problem So this problem is easily stated. We've seen also already some instances of this problem already during this talk this conference and the idea is that imagine that you have a Hamiltonian which has a large number Microscopic degrees of freedom. For example, you can have electrons, neutralized on spins, whatever you prefer And this this microscopic degrees of freedom are strongly interacting for example You can have a cool of interactions or whatever your favorite interaction is So the quantum any body problem For example, one of its manifestations is the fact that it is very hard that typically both numerically and analytically to solve for the ground state of this this Hamiltonian and also to study the Quantum dynamics for example, if you want to solve the time-dependent shooting this equation So why is it the case? Well, if you turn to a classical computer and you want to solve this Complicated equation you want to find ground state for example, so the minimal thing that you have to do is to Let's say fix a set of complete set of many body states And you have to represent if you want on your computer store the amplitudes of the wave function at least if you want to find it So the point is that if you do a calculation, for example, if we take a spin a spin system Which which has n degrees of freedom. So the quantum numbers can be Go from 1 to n and they can be either plus or minus 1 in this case And you store again the amplitudes on your quad or your classical computer then you realize that Even if we use all the atoms available on our planet We could at most simulate let's say hundred spins So this problem is Very hard so really finding the ground set for example is extremely hard for a classical computer because the amount of Resources that we need to grow exponentially with the system side simply because the inverse space is 2 to the n So you need an exponentially large number of of amplitudes Now we don't want to to turn ourselves into a giant hard things can we want to solve this problem more efficiently there are techniques for example that Exploit other concepts for example quantum Monte Carlo techniques do not store if you want to the coefficients of the wave function Exactly the exponentially many coefficients of the wave function, but sample Efficiently from those from the wave function using a mapping onto an equivalent classical system So typically this is realized through the path integral mapping another philosophy that allows you to To some to solve efficiently for for a large class of quantum amiltonians Is what has been discussed by Norbert shook in his lectures and are for example magic water states So again the idea here is that you can write an efficient answers for the many body ground state for the many body state Which entangling which has all the only typically a very relatively no number of variational parameters and if your system is as a Low entanglement, so in particular if satisfies the air hello, then you can demonstrate that you can describe efficiently all those those states But both families have of methods which are very popular and are very powerful have several limitations which in practice Somehow limit a lot our ability to explore a lot of interesting problems in in many body physics in diverse domains Going from condensate matter to Ultra gold atoms so for example if you take quantum on the Gallup efforts They break so this mapping that I described before breaks for fermions or frustrated spin models So there is no way for example, so to access real-time dynamics in an efficient way So this means that these techniques suffer from this is a infamous time problem, which basically Does not allow you to study very important take the problems and most notably fermions and out of equilibrium dynamics on the other end Medics for the states so tensor networks in general are very efficient are extremely powerful at studying One-dimensional lattice geometries when you try to apply them when you apply them to other Systems for example in continuous space or in two dimensions There are some difficulties that emerge numerically and also the efficiency of these of these approaches is Much reduced with respect to one dimensional case So it's clear that it would be desirable to have some new approach So some alternative technique which would allow us somehow to take the best of these two words so on one hand the the ability of quantum Monte Carlo to Sample efficiently from the ground state or from the quantum state many boys it and on the other hand the ability of Medics for the states tensor networks to somehow compress the wave function using some Some regularity property of the of the many body state So how can we do that? Well during this talk? I will try to convince you that a good way of doing that I'm going to do so is by introducing a representation of the state which is based on artificial neural networks On machine learning ideas if you want and then I'm going to show you how we can use these Representation efficiently to find for example the ground state of some interesting quantum problems to study the unit dynamics And then towards the end I will also discuss an application of this representation to the problem of quantum state tomorrow So I'm going to discuss this towards the end now, so So all of these as I was mentioning is based on them on machine learning. I've discussed these things last week but just to recall a little bit what what the the fundamental stuff in this field that is Is it's important to recall first of all that to do machine learning? we need first of all a machine and Typically one of the machines that people have started considering and using in in in applications is a highly idealized version of the of the brain of human brain or of Even the animal brain. So it is a high idealized high idealized version of the brain We mentioned that our brain is nothing but the high dimensional function So something which takes a high dimensional input vector X1 X2 X5 in this case and which Outputs so these input signal if you want into another output vector, which in this case is F1 F2 So I have a high dimensional function which will depend on a lot of non-linear Synapses so a lot of connection which will transform this input signal into an output signal and in practice So mathematically adjust to all purposes. I can assume that I'm working in this case With with an artificial neural network, which is a function a high dimensional function of this input vector Depending on some parameters W which are somehow the parameters that I want to find to do my my my things So of course this this version of the brain is really idealized and from the mathematical point of view from the biological point of view It's also very bad in the sense that it's somehow Approximating the sphere the cow with the sphere and then going on right so it's it's like those Approximation as physical or mathematicians like to do but it's not very accurate But to any purpose we can use this idea of having a high dimensional function which mimics the brain functionality And what we can do is that then we can use this machine to the learning so the learning means that We fix a task and I'm going to show you some specific tasks in a moment And then I want to to find basically those weights I want to find the structure of the network that best does this task So the best realizes this task and this is achieved for a typically using a Sort of big data approach where I can generate or I can retrieve a large amount of amount of data And from those I can learn how to do this specific task that I'm interested in So this is the basic philosophy of machine learning Now of course, I mean, you know about the tasks that machine learning can do for example in the case of language translation, so in the Google Google translate is based actually machine learning So in that case my artificial neural network again is a high-dimensional function which take as an input a string So a high-dimensional input, which is a string and outputs another high-dimensional input Which is the translation in in English or in another language in world and these these things Basically, so the structure of the network then the weights in this network are obtained for example looking and Through a lot of pre-translated texts So the same thing for example happens for optical recognition of characters So you have a digit which is written on on a sheet of paper You have a lot of them and you can train your network to recognize one of those digits and Translate it into an actual number. You can ask pitch recognition autonomous driving and so on and so forth So these those are the cut most cutting edge and important applications of Neural net so how can we use those and how can you use machine learning to you to do quantum mechanics? so Basically the idea that we introduce is that we can represent the the many body state as an artificial neural network So this high-dimensional function is a brain highly specialized brain that I was discussing at the beginning So we can see it as an object whose task is to compute the amplitudes of the wave function So for example, I give you I give to the brain that is artificial brain a set of quantum numbers And then the task of this thing would be to compute the amplitudes of the wave function then What we show that is possible to do is that we can train those networks to to find for example the ground state To represent the general many body quantum states and also to find the dynamics So I'm going to tell you a little bit more about that in a second, but First of all in order to be more precise We need to specify at least what is the structure of the network that we are going to use to represent these quantum states and In our paper what we did is that we use the one of the let's say most straightforward networks If you come from a background of physics, which is called the restricted bots for machine So in this network you can imagine that your input variables So your visible layer as it's called in the jargon is connected to a lot of hidden Spins, so these are effectively hidden spins through a set of interactions effective interactions Which has to as to be determined in order to represent the quantum state. So an important Feature of these of these states is that Upon increasing the number of hidden spins so we can think about these hidden spins as somehow The neurons that you have in your brain are so your green matter So the more you have of those the smarter in a sense the way function can be and you can hope that you're increasing this number I mean actually it's not I hope it's based on some theorems You can represent basically any generic high-dimensional function and including a quantum wave function So in practice from a mathematical point of view this this means that we we've right to the amplitudes in this form as Basically the partition function of a system which contains so as you can see interactions between the spin variables this sigma so these hidden variables which are themselves spins and then Those interactions are parametrized by some weight interaction medics wij, which basically is Is the the most important part of my variational parameters? So I can interpret the weights of these neural networks as variational parameters for my from my quantum probe Then what I can do it again as I was mentioning is that I can increase or I can adjust this parameter alpha So basically the number of hidden variables over the number of physical variables in order to Achieve the accuracy that I that I'm looking for on some given problem So for example, how can I find the ground state? Well the basic idea that I detailed during last week's lecture is that What you can do is that you can sample so you can generate a lot of data from those variational wave functions So like you do in Monte Carlo, so in particular This is like variational Monte Carlo and then obtaining some feedback from the variational principle You can change the network parameters until you converge to the variational ground state so in particular I've shown you that Already that the expectation value of the Hamiltonian Which is the quantity you want to minimize if you want to find the ground state of your problem Can be obtained can be estimated the stochastically Generating a lot of samples which are distributed according to the amplitude squared of my wave function Okay So this is free from some problem in the sense that the quantity that I'm trying to sample from is by definition Positive and you can show that it is possible to obtain Stochastic averages not only of the energy but also of the gradient So this is detailed in the lecture notes that I discussed last week So in particular on this git lab Repository you can find the lecture notes I was discussing and also some detailed derivations of those estimators including the codes that we were Showing last week So this is just to mention again that also the energy gradient So the energy the gradient of the expectation value can be efficiently written as a Statistical expectation value over this probability distribution, but the details are in there now Once we have the gradient so we have a way to compute Rather efficiently both the expectation value of the energy and the gradient of the energy with respect to the variational Parameters, which in this case are just the network connections. So how can we proceed? So how can we optimize this quantity well the approach that we use in this case is to use an approach Which is due well that in the variational Monte Carlo community is known as a stochastic reconfiguration method which has been devised here in Trieste by Sandro Sorrella and co-workers So the basic idea is to to have an approximation of the of the of the metrics of the system Which is spanned by those variational vectors So of K is given by the log derivative of the of the wave function and One can also show that this thing is Somehow dual and it's completely equivalent to an imaginary time evolution. So what you can do So this approach is basically equivalent to taking your Hamiltonian and doing the variational Evolution in the variational subspace to find the ground state of your quantum Hamiltonian in the variational subspace The interesting thing that I find out when working on this is that this approach was developed But in a very different language in the in the machine learning community like more or less 10 years earlier by Amari So you can see that here we have the journal of chemical physics here We have the journal of neural computation. So completely orthogonal fields But this the more or less the very same method will have been developed in the in these other community And nobody knew about each other until basically last year so Now I can show you at work how this thing works For example, what I'm showing you here is the the value of the energy so the expectation value of the energy as a function of this iteration So these are these optimization algorithm is an iterative algorithm, which converts iteratively to the ground state and For example, this is the case of the of the Isenberg model So the one-dimensional Isenberg model in So on a chain And here you can see that the energy first of all converts when you increase the number of iterations to a minimum and This minimum Coincides in this case very nicely with the exact one. So this is a system. I believe of about 80 or 100 spins and the important thing is that you can see already here is that Well from this from this zoom of the final part is that when increase alpha So when increase the number of hidden and neurons over the hidden over the actual physical Spins, so when you somehow increase the gray matter in your brain The accuracy systematically improves. So for example, this is alpha equal to this is alpha equal four So you can see that you can systematically approach the exact ground state And you can see also that the accuracy is that one can reach are pretty high. So from these numbers here We can we can quantify a little bit better at those accuracy plotting the relative error on the on the energy on Few ground states for some models for example the 1d transverse field is in model You can see we can reach with alpha equal for some 10 to the minus 5 precision And you can also see something which is pretty interesting that if you take the 1d transverse field is in model And you go at the critical point. So which is h equal one. So the transverse field equal to 1. This is also the worst Point for the for the for the neural network Because it's it's a critical point. So somehow it's harder to learn for this machine But still you can see that using a relatively modest value of alpha we can manage to find Very high accuracy Then we can we can use the same approach to study for example again the 1d isomeric model where we can go to again to high precision and We improve for example some existing other variational Monte Carlo techniques like the just rule of Way function, which is an answer for the many body state into the we also managed to to find An improvement over some Existing the time results for example these entangled plug estates, which are also a variational state used with variational Monte Carlo some Existing pepsi states probably this number has been improved a little bit since this this original work But also in in this case in two dimensions. We see that increasing as far we managed to Even though with a slower convergence, but we managed to to find the ground state with relatively high precision so another important thing which is worth mentioning is that This representation of the ground state So here I think it's a 10 by 10 in this case We we can do more so the important thing that one of the interesting things that I wanted to mention here is that This representation since it is highly non linear So it's if you want the non linear decomposition of the way function coefficients It has to be contrasted for example with the linear decomposition that one does in the context of tensor networks So since this is non linear in principle can be more let's say compact and what we found is basically for example on those One the problems for the isomeric for example to reach the same accuracy we need more less even 10 to the 2 like 100 times less Parameters than the corresponding medics for the states so this points to the fact that somehow these networks are in a sense More compact representations of those states Now if you look at the those weights, so these are those effective interactions that appear in in the classical equivalent model of course Not always they they tell you something Physically relevant. So these are for example Some extract of the of the of the connections. So we have to think that this is a Matrix which were you for for each side you have an interaction with the other side So those basically are the effective interactions that you have in the neural network So in the case of for example the isomeric 1d you see that those interactions are pretty long-range So you don't see any local structure for the 1d icing instead We see some kind of local structure even at the critical point for the 2d isomer model We see again some locality in this code those coefficients. However In general we don't have yet a recipe like to understand from those numbers what the Features of the system are but I'm going to tell you more about this in a second Now how can we solve how can we use this approach also to solve for for unitary dynamics for example? so we've seen at the beginning that the two big open problems are For example say finding the ground state of interacting fermions say the other model most notably in two dimensions Or in continuous space some some models But also the also the very important open issue is how to solve for inter dynamics and find for example out of equilibrium properties of Interacting of interacting moments So we can do that too and the idea is to use an approach that We developed here during my PhD in 3s Describing this paper. So the main idea of this approach, which is Very similar to the time-dependent variational principle used in the context of Metrics for the states and then we came out more or less at the same time So the idea is that basically imagine that you have any birth space, which in this case is two dimensional So just to simplify the Notation and the plot actually so you have a given state at time t which is this green line psi of t Then you know that at time t plus delta t your time your state will evolve according to This infinitesimal evolution induced by the Hamiltonian, right? So you can you know that this is basically just a rotation in Hilbert space But on the other hand, you know that you can parameterize the rotation if you want of your state of your variational state Through basically the variational derivatives So those okay objects which are nothing but if you want the directions in Hilbert space that span this Rotation, so that's why also your rotation is restricted Then what what what what one can do is that one can minimize Basically this angle if you want this distance between this approximate and is This approximate and is exact state and when you do so you find an equation of motion for the variational parameters So I'm sorry for the confusion, but alpha now is the alpha of k is the set of variational parameters So it's not what I defined before but anyway, so imagine that you have your variational parameters that I call here alpha of k Then we showed I mean that you can what you need to do is that at each same step You can solve these these linear system basically which allows you to solve the optimal equation of motion So defined that if you want the optimal alpha dot so the time derivative of the variational parameters Which allow you to to best reproduce the unit dynamics. So and again, this can be done using Monte Carlo, so you can sample from size square And all of that is relatively straightforward to implement now We use this approach in conjunction with those neural network random states for example in 1d So here you can see that we started some quantum quenches So in this case we prepare for example the system in the ground state of my 1d Transfer field as a model say for one initial value of the transfer feed for or one alpha in this case And then we quench at time equal zero So we change Very rapidly the value of the interact of the of the of the transfer speed and observe the behavior of sigma x as a function of Time for example So again, this is those are results obtained at alpha equal four. So I fixed the number of Eden units and you can see that we can manage to we managed to reproduce also pretty nicely the Exact dynamics which can be computed in this case because it's an integrable model But also for non integrable models. Well at least for the isomer model even though it's integral We don't know how to compute I think the exact evolution of some quantities and still we managed to match some results obtained with with MPS so time-dependent medical states now So this was about the problem of let's say Trying to solve for the ground state or for the dynamics But once we have a let's say a compact and alternative representation of the one of the ground state or Regenerals quantum state we can use it also for other purposes So one of the purposes that we we introduced that we wanted to use is this approach for is The problem of quantum state tomography So the basic idea the basic problem in this case is that for example, imagine that you have a pure state So you have a quantum system which is described by pure state psi And then you can have access to some measurements in in those on those systems So it's a spin system. You can measure the spin in some basis for example Then the problem is that you would like to reconstruct the state of this quantum system Just from those measurements that you can do in the lab So standard approaches like Full quantum state tomography Require for example, basically an amount of measurement which scales exponentially with the system size Simply because you have to reconstruct all the exponentially many coefficients in your with function So so these are relatively ineffective because you cannot for example reconstruct more than eight or eight It's also already a very large number of spins of cubits that you can reconstruct with standard approaches So the idea that we use is that is to do the so-called unsupervised learning So the idea here is that we do not want to determine the variational drowns it because we don't know the Hamiltonian and we don't know some Exact properties of the system what we can measure from we can sample from the wave function using some measurements So what we can do is basically we can do the measurements in the lab in some given basis For example, we can measure in the sigma z basis sigma x on some other alternative basis And then using only basically that this information reconstructing the Square models of the wave function in those bases using Machine learning techniques. We are able to to reconstruct both the phase and the the amplitude of the of the wave function So in particular we found we've demonstrated this approach for the that First for a simple w state, which is the state that has been mentioned also in other cases So just for reference here if you do full tomography It needs it takes about one million measurements to reconstruct the wave function of eight spins with machine learning techniques We can substantially reduce this number. For example, we can do 80 spins So 80 cubits with 10 to the 3 10 to the 4 measurements. So it's substantial improvement over Let's say brute force tomography and this is the overlap with the with the ground with the target state that I'm plotting here as a number of samples Another thing that we can do is that we can reconstruct again also the phases of the wave function This is an example for the W state with random phases here. We managed to reconstruct So these those are the exact phases for some relatively small system. Those are the rbm reconstructed phases Also, we managed to get a pretty good reconstruction one of the applications, which I believe is also particularly interesting is to reconstruct some quantum systems From which you can measure in the experiment some quantities, but not some other quantities So let me mention the case of the Okay, so we need the unit dynamics We can discuss this later if you want But this is the one of the most interesting application that I believe could be realized especially with cold atoms So the idea is that For example for a system of bosons just using in-situ images of the densities We and not a lot of measurements of those of those density We can reconstruct the many body wave function of possibly a rather large system and then So in silico on the computer. So using this reconstructed wave function. We can measure So let's say in a post-processing phase Quantities which are not directly accessible in the experiment. For example the the entanglement entropy So the entanglement of is very hard to measure in the experiment of cold atoms. So we can do that indirect now legend, let me just flash the some Very quickly some properties of those neural network quantum states. So this representation of the many body state in terms of Artificial neural networks which people have started studying after our work For example, it has been shown already that they allow for an exact description of many interesting topological phases for example, they allow to describe 1d symmetry protected state or the toric code in two dimensions with a number of neurons, which is very relatively Is a polynomial large? But they also allow to describe efficiently a class of states like a chiral p-wave states Which are instead not necessarily very efficiently described by other approaches Another important thing which has been shown in this paper is that this theorem So basically if you have your weights in the wave function Then and you call are the range of the weights in your wave function You can show that the entanglement entropy is basically bounded by the distance. So basically the correlation between Somehow the support of those weights interactions that effective interactions that you have in the in the mathematics So this means that in practice it is very easy and efficient to satisfy the volume law in those with those states Which also has the somehow suggests that we could use those kind of approaches to study efficiently critical states in in one or two Another thing is that there is also a connection with Modest process and answer networks at least well, it's relatively straightforward to show that a general MPS state corresponds to an MPS with an exponentially large bond dimension Where this exponentially large part comes from the fact that you might have long range entanglement and long range weights w so it comes if you want from the previous theorem The inverse mapping is more subtle So, I mean there are people that conjecture that in 1d those states might be more or less completely equivalent to MPS, but this is only at the level of conjecture and we don't have a proof So the mathematical reason is that proving this for nonlinear states. It's very hard So the other thing which I believe is also particularly interesting is that there are strong Representability theorems for Boltzmann machines So if we take an network which cost is which is done of two layers So not like the one with only one layer that I showed you before but with another further layer Then you can show that these such networks can represent efficiently were efficiently I mean with a polynomial in large number of of Parameters basically any physical state so gap the gapless or anything you can think of so the important is just the end tone and is local and So these theorems in particular show that any quantum state exactly of n qubits is generated by quantum Speaking of the tea that can present excited by sparse deep Boltzmann machine with order n times t neurons So if you have a gapless system this t typically grows like this is the number of spins But otherwise it's a constant depth circuit if you have a gap system now, let me just Recap what what I've shown you today So I've shown you that we've introduced a new class of many body states, which is based on artificial neural networks with those things you can do several Several applications already for example, you can find the ground state You can all do quantum state tomography. There's also an application that we are doing that I didn't discuss which which Namely you can try to describe Quantum circuits so you can try to approximate classically quantum circuits And then I've shown you that there is a lot of also work in the context of quantum information But there's a lot more to be done. So there's also my call to the community For example trying to understand why and how what are the limitations of those states? But already we know that some universal states based on Machine's exist. We know the connection with tensor networks, but this should be improved Etc. Etc. So the the other things that we'd like to do are for example extending those things to fermions even though This is almost completed now by now and Also using deep networks, which is one very important methodological step to be done and taking also advantage of this Theorem that I described in the in the last part of my own metal. So, thank you