 Excellent. So welcome back to this final part of the students presentation. We have three projects remaining. So let's start with the first one. So Alina and as me that can you share the screen please. So this is would be played 14, which is titled tackling quantum many of the systems with artificial neural networks. And it was a bit of ice. By Joe, but seen myself. So please Alina. Go ahead. Thank you. Can everyone see my screen. Yes, presentation. Yes. As me to. Yeah. So hello everyone. I'm a Smita and I've been working with Alina on project 14, which is basically tackling quantum many body systems with artificial neural networks, and we've been supervised by Dr Joe and Dr Isaac. So we'll start with an introduction of what is a quantum many body system. So it is basically a fundamental class of physical systems that involve a large number of interacting particles whose behavior and properties can be studied in detail using the first principles of quantum mechanics. For example, we have a superconducting magnets we have the nuclear many body problem where we try to understand the different forces and correlations between the nucleons, or for example take nano materials where we want to study the electron emission structure, etc. using quantum mechanics, and several different theories have already been proposed to study them. But, as of now only a small subsection of the properties have been effectively studied. What you see in the box is essentially a wave function of a simple one dimensional spin chain with n spins. It contains the probability amplitudes for each configuration of the constituent particles in in general they are complex value. So here for n particles, we would need to describe we would need to prescribe two to the power n amplitudes to completely describe the entire system. Therefore, studying systems that are mentioned above can become difficult because the Hilbert space that is associated. It increases exponentially with the increase in the number of particles and Schrodinger equation that you see there, it's becomes increasingly harder to solve for the energy eigenvalues. And therefore we try to make use of an artificial neural network to learn and solve for the ground state energy as we will discuss further. So, for the next slide we have artificial neural networks. It is basically an abstract implementation of biological neural network, where in a biological neural network the signals are received from the dendrites and bias is applied to them and it's then passed on to the cell body. And then it is sent down the action once enough signals are received at a particular time, and the output that is received at the action terminals then serves as an input to another neuron through the sign ups and and the process continues. So artificial neurons are also modeled in a similar way by accepting binary inputs and then applying weights to them and and checking whether the sum of these weighted inputs have reached a certain threshold, which is also called a step function. So that the neuron will decide to fire or not. And then a nonlinear activation function acts on this sum, which basically restricts the range of values that that the network and output. Next we discuss what is called a neural network quantum states which will be using throughout our presentation. So essentially whenever we try to find a ground state using variational approaches we propose a trial wave function as an insults and then use some mathematical operations and we see how close the answers is to the actual wave function. Carlo and Troyer for the first time proposed the use of a restricted Boltzmann machine which is a stochastic generative artificial neural network. And the proposed it as a variational many body wave function insults and they discuss it in this very interesting paper from science that was published in 2017. In the structure of the RBM. It has a visible layer and a hidden layer. The visible layer takes an, an in body spin configuration, essentially takes in discrete degrees of freedom as it as its input, and these might be bosonic occupation numbers or some other use as well. The weights wij are the network parameters that are updated using a feedback technique in the form of reinforcement learning across iterations to optimize the response of the restricted Boltzmann machine. For this particular problem. We have a single output, which is basically the probability amplitude to observe the particular spin configuration in a given basis. And as you can see that there are no intra layer connections between either the visible layer or the hidden layer. So the terms hi which can take values with only plus one and minus one they can be explicitly traced out. And then the final expression becomes independent of hi, which can then be written as a product of some function s so s is your many body spin configuration. And this function involves a hyperbolic cosine term that involves the hidden biases and the weights. And so now I'll hand over to Alina who will take you through the quantum model under study and the methods in the software that you used. Yes, so we're looking at the one dimensional transverse field icing model, which is a model of interacting spins one half on a chain. And additionally, we apply the transverse field in x direction. The Hamiltonian is then composed of two terms the J term describes the interaction between the spins where J denotes the interaction strength. And the gamma term describes the interaction of the spins with the transverse field where gamma denotes the strength of the magnetic field. So we have a diagram on the right, which describes the system in terms of magnetization versus the ratio of gamma over J. It is possible to see that at gamma over J equals one a face transition takes place there the system transitions from an ordered paramagnetic face to this sort of paramagnetic face and in the paramagnetic face two cases if J is bigger than zero the interaction is paramagnetic and the spin the majority of the spins point up and if J smaller than zero. This is the anti paramagnetic face and the most spins point down. So one dimensional transverse field icing model can be solved exactly with the Jordan Wigner transformation this transformation uses the fact that spins one half in one dimension, the hayflag fermions which enables us to rewrite the Hamiltonian in a way that makes it easier to solve especially for the transverse field, which is something as meter wall. Come back to. And next, we used a net cat, which is an open source software for simulating quantum many body systems with neural networks and it was developed by Khalil and several others. We either in built or custom built Hamiltonians and applying neural networks and able to us to find ground state properties, such as the ground state energy for a given system. And we're following a method previously used by Khalil and Troyer where we implemented a system using an in build Hamiltonian as well as comparing it to our results from a custom Hamiltonian on a non trivial lattice graph and use neural networks, specifically a restrict to Boltzmann machine as meter mentioned to try to find the ground state energy and other observables like the magnetization and the static correlation functions. And we run the optimization for 800 iterations of each iteration. The parameters are updated using metropolis Hastings sampling which proposes a new spin state with a certain transition probability in the diagram. And you can see the convergence of the ground state energy to the exact value for gamma j equals one comparing those results from the in build icing operator in black and the results from the custom built operator in red. And these results converge to the exact value the blue line obtained by exact diagonalization. And as you can see, after about 200 iterations they almost indistinguishable from the exact results. The ground state problem. It consists of finding a good approximation of the ground state energy. And the minimization of the ground state was achieved by using variational sampling to compare the machine learning results we use exact diagonalization to solve for the ground state energy. And the variational energy can be made nearly equal to the exact ground state energy for trial wave function that is a good approximation of the original ground state wave function. In the diagram you can see in that the restricted Boltzmann machine result for 20 spin system in comparison to the exact diagonalization result in blue. The ground state energy are plotted versus the iterations again in the machine learning process, and it is clear to see how the restricted Boltzmann machine result converges to the exact diagonalization result with increasing iterations. So based due to the fact that gum over J equals one, which is the point of phase transition, this convergence happens slower than it would above or below the ratio, since there are large fluctuations around the critical point. And, although exact diagonalization is limited to small system sizes, since the Hilbert space grows exponentially with the system size. The restricted Boltzmann machine can be used to simulate systems with sizes well beyond exact diagonalization. In our case the exact diagonalization broke down for about 25 spins. And I will now hand over to us meeting who will present and explain more about our results. Yeah, so now I'll take you through the error analysis, where we try to find the error in the ground state energy calculated using a restricted Boltzmann machine, relative to the energy that we found by using exact diagonalization. This is the energy, which is basically the expectation value of the Hamiltonian of the ground state it's the loss function that that needs to be minimized. And this is a function of the parameter alpha. So alpha is basically the ratio m by n, which is the number of hidden nodes in the restricted Boltzmann machine to the number of visible nodes, and the alpha dependence comes from the representation of the wave function that is present in our energy formula. From the graph, as you can see that as alpha increases the variational energy, E of n decreases and it eventually converges to the energy obtained using exact diagonalization with a very small range of error. And this error keeps on decreasing with increasing alpha. So one of this decrease also increases as the gamma over j ratio keeps on reducing as we plotted here for three different cases. And even though the error never really reaches zero as we would expect from any stochastic machine learning algorithm. Nevertheless, we do achieve a fair degree of accuracy with a modest number of hidden notes. However, does come with a computational cost and the cost is linearly dependent on alpha. So now we move to the next slide. Supervised learning. So, learning from data using machine learning algorithms can either be supervised or unsupervised. In a supervised case for quantum many body systems we have a technique called quantum state tomography that we don't cover in this presentation. We used supervised learning for our problem. So for a supervised learning problem, we would then need to provide the data, including an input vector x and an output label why the neural network would then is then trained to learn the neural network is then asked to learn. There is no noise, let me look at it. Sorry. Please carry on. Yeah, sure. So the neural network is then basically asked to learn a mapping from the vector x to the target label why. In our case the input is the spin basis and the output is the coefficient of the corresponding spin basis so it's basically the probability amplitude of finding a given state in a particular configuration. So we take the target state as the exact wave function that we obtained using exact diagonalization and the variational state is the one that needs to be optimized to reach the target state. The gradient is estimated by performing a Monte Carlo sampling of different overlaps overlap distributions. So the overlap here is between the target and the neural network quantum state wave function it's essentially the modulus of the expectation value. Alpha is updated using a stochastic gradient descent optimization according to the rule that is shown here where lambda is the learning rate. So lesser the value of the loss function, given in the formula which is essentially a negative logarithm more is the overlap and therefore better is the convergence of the of the trial wave function to the actual wave function. We plot the graph of overlap versus iterations for different learning rates and we see that more the learning rate the faster and better is the is the convergence to the target wave function. However, this is limited to many body systems where the exact wave function is known either analytically or using some reliable numerical approaches. So question necessarily arises, why would we use this when we already know the exact wave function. The answer would be that the optimized ground state can be used to predict excited states using supervised learning for systems whose excited states are numerically or analytically hard to find. And one such work in quantum chemistry has been done. And it has been described in the paper that's referenced here. Next we move on to one of the observables of the quantum ising model, which is the magnetization. The magnetization it's determined by the ground state expectation value of the Pauly X operator. As you can see the graph of MX versus gamma versus gamma over J. It's spotted using a restricted Boltzmann machine wave function and it's shown in black, and the exact diagonalization. It's shown in red and they are compared with the close from integral that was solved by 50 and this is shown in blue and we observe a very close agreement. So for gamma by J is equal to zero or essentially where gamma is very, very much less than J. The transverse field is very weak to the to the interacting spin term and therefore none of the spins are aligned and the transverse field is very high. We obtain an expectation value of zero. As the ratio increases we obtain a phase transition at close to one, and when gamma by J, like when gamma increases, much more than J, or the gamma by J ratio is much more than one, almost all the spins are aligned along the transverse field and hence we get to magnetization value almost equal In the next slide we also look at magnetization. In the next slide we also look at magnetization, but this is the longitudinal magnetization in the z direction which is referred to as MZ. So ideally for gamma less than one and on zero magnetization has been theoretically shown to exist in the z direction, while it becomes zero at the critical point, gamma is equal to J is equal to one, and further beyond as we venture into the paramagnetic disordered phase. Now MZ is way harder to learn, sorry way harder to calculate than MX due to the non-local nature of the spins given by the Jordan-Wigner transformation that was briefly described by Alina before, and possibly due to the absence of spontaneous symmetry breaking in the ordered state of a finite size system when we are very far off the thermodynamical limit. So to solve this we apply a small but a finite longitudinal field and the field strength is given by gamma one so that the z2 symmetry is broken. As you can see in the graph our results using a restricted Boltzmann machine are quite far off the analytical limit solution in red. However, we expect that if we keep on reducing the longitudinal field we might gradually converge to the exact solution. We tried to study another observable, which is the spin-spin correlation functions. So correlations functions essentially describe how two spins influence each other at a given distance apart. And specifically it's a measure of the probability as to what extent the spin at a lattice side i is aligned with the spin at another lattice side j. The longitudinal correlation function is given by the expectation value of the tensor product of the Pauli z operators at two different sides i and j. And similarly the transverse correlation function is given by the expectation value of the tensor product of the Pauli x operators. In this particular graph we plot the xx correlation functions with different neighbor interactions for modulus of gamma by j less than one. We keep the system size fixed here for 20 spins. So essentially n equals to one here denotes the nearest neighbor interactions and n equals to two denotes the next nearest neighbors and so on. We observe that the correlations are strongest at n equals to one which implies that the spin has the strongest influence on its nearest neighbors and reduces henceforth following an exponential decay for gamma by j ratios greater than or less than one essentially away from the critical point. And we see that it reaches a non-zero asymptote which is what we would expect from an ordered ferromagnetic state. And something that we would like to do further and visualizes is to calculate the correlation functions for the critical point for gamma is equal to j is equal to one. And see whether they algebraically decay following a parlor behavior or not as it is predicted theoretically. So that's something we haven't done but we look forward to and then Alina will take you through the final few slides of the presentation. Yes, so finally we compared different variational ansatzis for finding the ground state energy. And in the diagram you can see in red the mean field ansatz which is not very well suited for solving quantum systems, especially in this low dimension. You can see it is well off the other results. The mean field ansatz is known as an exact variational ansatz that is with a rounding error. And the variational parameters are the spin single spin wave functions psi. The mean field approach works best in high dimensions where spins have more neighbors because in low dimensions there are more fluctuations that need to be accounted for. And next from the top in blue is the Jastro ansatz it is a more correlated ansatz and in the short range version of it it entangles nearest and next to nearest neighbors and the parameters are J1 and J2 which have to be learned. We're using the restricted Boltzmann machine in green, and this is very close to the results obtained by a fully connected feed forward neural network with non-linear activation function and a summation layer, which is displayed in black. And finally our conclusion. Neural networks prove to be an innovative and helpful tool in many areas of science and here we have presented you a way to look at quantum many body systems with the help of neural networks. This is a compact representation of quantum many body states on neural networks and neural network quantum states allows an easier way of examining these quantum many body states and their properties such as the ground state energy, even for highly entangled states. This was shown in a paper by Khalil and Troy and this will buy disarmament. And we have looked at different variation ansatz and found that the restricted Boltzmann machine even though it is arguably one of the simplest neural networks can find quite accurate results for the ground state energy. So it will be interesting to look at more advanced neural networks and see what their advanced advances can offer in this area in the future. And additionally it will be interesting to see this method applied to other more complex models that are dynamic. So I may want you to frustrated this particular time dependent. An example is a paper by Valenti at our who use, amongst others have correlated restricted Boltzmann machine ansatz on the particular code model with periodic boundary conditions. And then there's Salma and his paper noted that quantum computing can provide a boost to machine learning techniques, for example when dealing with large amounts of data. One might want to use the quantum version of the fast Fourier transform, which is exponentially faster than the classical version. And is that quantum computers are intrinsically based on litany algebra, which can help solve a large number of matrix multiplications. Thank you very much for listening to this presentation. Thank you, Alina and as Mita for this. In my opinion very neat very clear presentation. I'd like to congratulate you for the work you have done. And also to Joe for the supervision of this project so let us. We have minutes for questions so again colleagues, please if you have any questions go ahead and ask directly or just write on the chat or raise your hand. Yeah I've got a, I've got a fairly general question. I think it's very interesting that you're looking at these. If you like machine learning techniques to to solve scientific problems. In the examples you gave you already knew the exact solution and so you really just testing whether or not the technique would work. What problems do you envisage where the exact solution, or even an asymptotic solution is not well known, and where these sorts of techniques can actually lead us to to new scientific results that we that we can't obtain anywhere else. Yeah, so what we try to look at this model basically because we had a test bed using exact results and as Alina mentioned in the conclusion for physical systems, there are frustrated models there are different. There are models that are based on different geometries in higher dimensions, maybe in two dimensional or three dimensional models that aren't exactly solved, but there have been quite a few numerical approaches that give an estimate of what its properties or what it might be. So that could be a place where, where we could apply neural networks, and also, I would say, for other realms of physics, maybe for particle physics where we try to separate, you know, where we have a lot of data that's collected and we try to separate a particular signal from a lot of different background noise and work has actually been done in a lot of these fields where they have successfully kind of separated a particular signal which is just actually very hard because the intermediate states kind of decay very fast and the products are similar, but it has been done using both machine learning and quantum computation so that might be another area. Yeah, there were also examples where they looked at excited states instead of ground states which can usually be sort of a bit easier but the excited states are not and then you can look at different levels basically. Okay, thank you very much. Thank you. More questions. Okay, so if not, shall we thank Asmita Lanina for this fantastic seminar. Thank you very much. Very good. We have to keep moving actually.