 Okay, so thanks a lot and let me also thank the organizers for the opportunity to speak about this work here My name is Roberto Verdel. I am a postdoc here at ICTP in the group of Marcelo Dalmonte and today I'm going to tell you about Some of our recent work where we explored the use of data-driven approaches to extract physically relevant information of many body problems and most of what I'm going to show you today is actually based on our recent publication on PRB that you can check out here and that was the result of a Really nice collaboration with all the people that you see mentioned here So let me begin by giving you some kind of Broke motivation to the topic of my talk and this really comes from the field of quantum simulation And I should say from all the amazing developments in this field Which has really opened a new door when it comes to the study of non-equilibrium quantum dynamics and the key aspects of Quantum simulators to make this realization possible is the capability to you know Prove quantum matter having control and Observation capabilities at the individual quantum level So what that means is that with current technologies and also thanks to the developments in imaging techniques Nowadays we can take essentially photographs of our quantum many body state Illustrated here from this figure taken from this recent experiment at work where what you see is three examples of such snapshots and in this case of linear chain of trap ions Where the ions represent the qubits and this simulator and what you see there each bright spot Essentially corresponds to a spin pointing down while the dark Squares is a spin pointing up Now this is really amazing because this kind of snapshots contain lots of information of your many body state of interest There are however a couple of remarks that one should make here the first one is that The number of degrees of freedom that you can probe nowadays with current quantum simulators is already quite large In the sense that the dimension that you can explore by a number of these Realizations is a still very small compared to the dimension of the full Hilbert space So that is to say that an approach like full quantum state tomography is just out of the game for a system like that and On the other hand by having these kind of single site resolution to the system That also makes them very special compared to other many body kind of systems like solid state Materials where of course we don't have this level of resolution So that brings us to the question as to what's the best way in which we can extract information out of these many body wave function is snapshots and Of course, perhaps the first thing that one could try is To do what we already know how to do that is for example Extracting these kind of few point correlators know like a local density to point correlation function and so on and This is an approach that is of course based on Physical assumptions and it works very well to describe very interesting physics in many situations and there are plenty of Also experimental works where such an approach has been carried out However from a data science perspective if you want What we are doing here is a kind of dimensional reduction the many body wave function snapshots on that side are really large objects and By only structing this kind of few point correlators We are reducing the dimensionality of the problem But in a way this could be a non-controlled dimensional reduction because we are not sure whether What we are throwing away is actually or could contain let's say physically relevant information of the system There are some other reasons why we would like to go beyond this kind of description and here I'm just showing you three examples of what I think represents quite well this point so the first one is systems for which we know actually that The physics is described in terms of no local correlations The best example here is systems that feature topological phases of matter where a local order parameter will not Tell you about the whole story of the system Another example of it in more broad terms is when you study non-equilibrium dynamics in a genetic interacting system It is not always obvious which quantities are the best in order to describe such such dynamics So here is another example where Utilizing the whole information content if you like of these many body wave function as snapshots could be of interest Finally, I think it's also clear with that if you want to benchmark the workings of our quantum device Probably you will need to go beyond computing these relatively simple observables that I was showing you before So all of these tells us that it would be highly Decidable to have a new theoretical framework That's really leverages on all the information content that these wave function snapshots contain So today, of course, I'm not going to Tell you the final answer to all the questions that I posed before Instead, I'm going to show you our effort Towards building such a theoretical framework if you like some kind of baby steps towards that final goal and the approach that we are putting forward here relies on the following realization that If you think about these wave function snapshots, they are simply data objects in fact There are classical data objects and therefore it's kind of natural to think of using data science tools to analyze the you know, what as I was saying the full information content of such objects and Importantly these tools that come from the field of data science are designed such that you don't need to do any physical assumption To analyze the data So here's an schematic representation of the kind of approach that I'm going to tell you about today So our starting point our collections of these wave function snapshots and at this point One should note that this could come really from an experiment from a quantum simulation or they could be Sample if you like also from a classical simulation so here what I will have in mind is that these Data sets are matrices Where the number of rows are the different? Realizations that we take the different snapshots and then the number of columns correspond to a different degrees of freedom that we have in the system As a second step, we will do this kind of what I call here exploratory data analysis with the idea of discover You know the hidden structure in the data without making any physical assumption and hopefully What you will learn from it will allow you to tell To say something about the physics of the system, which is the last step in this diagram And I should say it here that of course there are many works that in a way are complementary to the approach that I was showing you just now and Of course, I apologize if I didn't include all the relevant words in this respect so Now here's the outline for the rest of the talk I'm going to start by giving you some kind of little introduction to two of the tools that we use and then I will show you some examples where we apply these tools to extract information of some many body problems the first one involving a experiment with post-election condensates and the second is more recent work that we are Doing at the moment where we try to characterize transfer transport properties of Quantum spin chains Okay, so this is the kind the part regarding the methods So today I'm going to tell you about two methods that we utilize The first one is what we call the PCA entropy and as the name suggests this is something that is done within the framework of principal component analysis and Is essentially a tool that allows you to quantify the complexity of your data set In a way that I'm going to show you now So first of all a little reminder on PCA This method can be Formulated in several ways, but in the end it's about finding a set of orthogonal directions That successively maximize the variability of your data Once you project the data points onto those directions as illustrated here One can show that such an optimization problem boils down to actually linearizing square symmetric matrix this matrix sigma Over there Which you compute from your data matrix. That's the matrix X. So here X really represents for example one of these rectangular matrices there, okay, so Suppose that you have Solve the eigenvalue problem for that matrix sigma Then you get a set of eigenvectors and a set of eigenvalues the eigenvectors you Actually use in order to define the principal components in order to find these principal directions and the eigenvalues essentially they tell you a little bit of Like what's the spread of the data in the corresponding directions and here we will only utilize the the eigenvalues and We know that these eigenvalues Are all non negative and furthermore we can always normalize the set of eigenvalues by the total sum of them So in the end of the day what they define is not in both a probability distribution And finally we use that probability distribution to define the PCA entropy According to the Shannon formula that is in this square yellow square Okay, so here is how it works in pictures So suppose that your data set looks like that image over there, which is essentially white noise If you would do the PCA analysis and compute this entropy as I just told you You would get a number that is very large And in fact it's going to be a further log n where n is the number of degrees of freedom in your data set So and this is very intuitive for a for an entropy So the more random names you have the larger the entropy will be on the other hand if you would have a Extremely ordered data set and of course, this is just a kind of toy example. This doesn't correspond to anything But if you will have something like that Then essentially you just need a single principle component to explain the whole of your data And in that case the PCA entropy will be in fact zero So the more correlations you have in the data the smaller the entropy in The PCA entropy will be okay, so If yes Yes, that's actually a very good question. So We have a story the PCA entropy in a few different Kind of context But for example if you apply it to classical systems, let's say classical static systems at the equilibrium one can always define this PCA entropy with some normalization factors such that it coincides with a thermodynamic entropy in Certain limit for example at infinite temperature. So in that sense, it has a connection with a thermodynamic entropy and There is also a recent work by Carlo Bannonia and Vittorio Vitale where they analyze this kind of entropy for Quantum systems single particle quantum systems and then there show that In fact this quantity is an estimator for the participation entropy of those systems So there are certainly some some connections with physical quantities in some limits Good. So now I'll tell you about these other methods Which I think is very interesting, but perhaps is not so well known at least in the physics community And this is known as the information imbalance Which It's a method that will give you a tool to systematically find The most informative subsets of features in your data set and here by features I mean really the coordinates of your Snapshots so you can think of features snapshot as a data point in a highly dimensional space And the coordinates of that point are the features that I mentioned here So this method works by computing What is called distance ranks and it does so by So selecting two kind of Subset of features and those define two spaces which are here. I call a space a and a space b So think that you just take a subset of coordinates of your data points and you compute distances between all your data Points and you rank them according to their relative distance And then you do the same but using another subset of coordinates. So those are a space a and a space b Now, let's say that according to space a so according to these subset of features The point I and the point J are nearest neighbors So I encode that in that matrix are a by putting number one And that's the rank of these pair of points the distance rank of these pair of points But it could as well happen that if I use this other set of coordinates In the space b the same two points will be now third nearest neighbor So the corresponding distance rank in B in this case would be three and Then the information in balance is computed essentially by this conditional expectation value of the ranks in the In the space be conditioned to the points that are nearest neighbors in the space a So at this moment this quantity might look a little bit obscure But my claim is that what this quantity tells you is how well can a space a describe the structure? of a space B and There is a way to make this formulation, you know mathematically formal in probability theory and for the experts that's Essentially an application of what's known as sclar theorem but what is important to To to point out here is that these conditional expectation value really determines the full correlation structure between the two spaces Now a little bit more in plain words how it works Suppose that a space a and a space B are perfectly Correlated in the sense that the nearest neighbors according to a are the same as the nearest neighbors according to be In that case one can easily compute this conditional expectation and show that it's actually equal to 1 and Therefore this delta From a to b will vanish as 1 over nr nr here is a number of snapshots that we have a number of points On the other case on the other extreme of things if The two space are completely or perfectly uncorrelated meaning that the nearest neighbors in a have nothing to do what of With respect to what you call nearest neighbors in B Then one can also easily compute that conditional expectation value and show that it will actually be equal to nr over 2 And therefore with this definition delta is equal to 1 so this quantity goes from 0 to 1 and the closer it is to 0 and The better the space a explains the structure of the space B now You can perhaps already anticipate that this is a perfect method to do what is known as feature selection in the following sense Suppose that I call a space B the space of all the features that I measure and I take different subsets of features here I'm just showing you showing this for two subsets a one and a two and then I compute The information in balance in the way that I show you before and I get something like in this plot there then I could say that the subset of features a two are actually quite informative about the full set of observations and Here's a concrete concrete example of that suppose that You draw points from a three-dimensional Gaussian where the standard deviation in X is only one tenth Than of the standard deviation in Y and set So your data cloud is essentially in the Y set plane and then you call Space a one only the first coordinate X and you measure distance in that way And the space a two will be the second and third coordinate and you measure distance in this way Then essentially what this analysis will tell you is that if you determine distances using the Y and set component You can determine the full structure of your data because X in this case is just like some noise Okay, so now I'm going to show you some applications of this method, but maybe I stop here in case there is questions Yeah, that's also a good question. And actually at some point we were exploring precisely that question whether we could at least estimate the entanglement entropy from the SPCA It turns out that there is not a direct relation between these two quantities But it's more like I said before you can relate the PCA entropy to the participation entropy, which is you know the the Shannon entropy of the Probability density given by the wave function in that case But as far as as we know is we cannot relate it to the entanglement entropy Yes, I'm going to show you a concrete example just now And in fact it's going to be a little bit different from the beginning because I started by considering Snapshots in a lattice and now I'm going to go to the bulk So this is a kind of quantum simulator that it's of lattice And it was a collaboration with a group of Marcus Oberthaler in Heidelberg So what we did here was to analyze Data sets that had already been used in a different work that you can check out in that really nice paper Over there. So here. Let me just mention a few technicalities of the experiment So in this experiment they prepare a spin or both the Einstein condensate using rubidium atoms and These are put in a trap that has this Enlogated shape. So it's almost one-dimensional system And then in the experiment what they basically perform is a quench dynamics. So they Initially prepared all the atoms in The hyperfine manifold with f equal to one and substrate with mf equal to zero So you can think of this as a spin one system and then by tuning some lasers in the experiment they basically Perform this kind of quench dynamics So essentially what in pictures what it happens is you start by putting all the atoms in this Substate and by doing this kind of quench to bring this level into resonance with the other two levels and that You know drives some kind of spin changing collisions in the condensate and drives it out of equilibrium So and then in this experiment they can take these state selective especially resolve snapshots of atomic Populations or densities in the on in all the relevant levels and these are the kind of snapshots that We were going to on that we are going to analyze here Where you see these kind of gray? Rows over there corresponding to different levels And like I said, these are just atomic densities at different In the different relevant levels Now there is of course a lot of work on this type of systems and it's understood based on physical arguments heuristic arguments that a good quantity to describe This condensate is what is known as a transverse a spin of the of the condensate And it can be shown that this Can be computed from those measurements by taking the imbalance of different populations as showing here so the first question that we ask here is Whether such kind of dimensional reduction It's a good one in the sense that I explained at the beginning Namely whether we would obtain a compatible description if we analyze these Kind of data in a blindfolded manner if you like so without reliant on any heuristic arguments So this is just to show you how these data sets look like so essentially they measure these Densities at different times and in the different sub states and On the so the different rows again correspond to different Realizations and the different columns correspond in this case to different spatial locations where they sample these densities So this is just to say that by simple inspection, of course, you are not going to see any kind of correlation between different between different observables Good. So first we apply the PCA entropy to such data sets and we do this as a function of time so for each time we have Six matrices for the six different observables and we compute their PCA entropy And what we see is that as the system evolves There is a clear separation between two groups of observables and there is clearly four Observables that develop a smaller PCA entropy. So in that sense They are more relevant because they seem to capture more of the correlations across that time-space evolution of this of this system We went one step further by asking ourselves which Concrete combinations of these observables could be more informative so what we do here is literally concatenate the different data sets in the As schematically showing there and we compute again the PCA entropy for all the possible combinations Here I'm just showing you for combinations with two observables And what we found is that By doing this analysis, there is clearly two specific combinations that develop a smaller PCA entropy and those combinations actually Which are indicated there correspond to what it was known Based on physics arguments that would be like the relevant combinations So finally once we have identified these pairs of Relevant or these kind of relevant pairs of observables We can go one step further and try to really define new operators that would be in the most informative ones So here I'm just showing you for three particular combinations And again what we found is that the difference of these observables Has a lower PCA entropy and therefore we interpret it as more relevant and that it's in perfect agreement with This physically motivated kind of answers Good so at this point we were quite happy But you know when you use PCA you have to bear in mind that you are kind of assuming that your date is well described by a Best fit in subspace So there is always this kind of linearity assumption that you're making so we wanted to cross check Whether this analysis was also true using other kind of methods and here's where we use this information in balance So remember that you have to compute to do this information in balance You have to compute distant ranks in two spaces. So in this case our reference space be Would be the space of all the different features that are measured for the six observables So it's so you can imagine that each data point has a further 1200 coordinates in this case and Then as a space a we take only a subset of those coordinates that correspond to the features for each of the different Observables and so we ask which of those observables describe better the structure of the full space of observations So very quickly here. I'm showing you the results again doing the three steps that I show you before and The labels here. I'm sorry that I'm not showing it showing them, but they correspond to the same levels as before so again, we see that The four observables that had the lowest PCA entropy in this case. They also have the lowest Information in balance. I remember that an information in balance close to zero tells you that those coordinates Can describe quite well the full structure of your space So in this sense, they are also the relevant observables according to the information in balance and This was fully consistent with the results that we got with the PCA PCA entropy analysis So in a way was a nice Form to cross-check the results that we were obtained Okay, so in the last part of my talk I'm gonna show you some ongoing to work and here really the main Character has been the vendora Bakuni who has a poster in the poster session And I also like to advertise the poster by Cristiano who is working on similar aspects, but for a classical static systems So here We are going to chain again a little bit the setting so we go back to the lattice so we have in mind, you know quantum spin chains and The question that we are after is whether we could say something about, you know, the transport properties of such systems So the scheme that we are going to follow is the same as I presented at the beginning And here are a few results So first of all the the setting that we have in mind is One typical setting to probe transport. Namely, we are going to study the unitary dynamics generated by some Hamiltonian with some conserved quantities And we prepare the system initially in a way that it has a kind of kink in one of those conserved quantities schematically shown there And this is for example Motivated by some recent experiments like the one showing here where they study spin transport in the Heisenberg model and in particular they are able to probe Super diffusing super diffusive regime that actually falls into the KP set kind of physics So we apply the same Kind of setting about utilizing these PCA tools And only relying on these snapshots that we are taking from our many-body wave function So here all that I'm going to show is results from classical simulations so the first part corresponds to this spin transport and What I'm showing here on the first row is The growth of the PCA entropy as a function of time So at each time we have in this case an MPS representation of the many-body wave function We can sample it efficiently using for example perfect sampling And then we do this PCA analysis we compute the entropy and we plot that as a function of time And we do we do it here for three different values of the anisotropy parameter which Has been shown in previous numerical works that they correspond to ballistic transport Super diffusive transport and diffusive transport as well. So What we found is that the growth of these PCA entropy captures or is in a way Go burn by the dynamical exponent corresponding corresponding to the different types of transport and On the right panel We show these for different values of the number of realizations and it's quite remarkable that already for a few hundred of these snapshots You can Recover a signal that allows you to extract the dynamical exponent with a very good accuracy On the lower row we plot a different quantity which is essentially The difference of the largest eigenvalue when you do this PCA analysis on the left half of the system Versus on the right half of the system So this was motivated by the physical quantity that was analyzed in this experiment this polarization transfer What we see is that again this Recover or is governed by the same kind of dynamical exponents and in fact if you plot it together with a polarization transfer is basically on top of feet and The one of the nice things of PCA is that actually you can understand very well the math that you're doing So it's only linear algebra So in this case we could understand based on some singular value the composition theorems why this quantity Was essentially computing the polarization transfer So okay, you could say very well, but in this case, you know that the system in this system the magnetization in the set Direction is conserved. So you are taking the snapshots in the in that basis So in a way is not so surprising that you can capture, you know all the transport properties And based on this observation we try to do something a little bit more challenging, which is To study energy transport So here we took three systems with different kinds of energy transport and this has been a story again in previous works And in particular we took system where the energy is Transported in a diffusive way in a ballistic and a super diffusive manner And what is shown in the lowest row is a kind of energy transfer That of course you can compute from these numerical Calculations, but you cannot as far as I know you cannot measure in experiments And on the top panel is really this analysis based on the snapshots that those you can take Certainly in the experiments and where we compute again the spreading of this PCA entropy or how information is spread as measured by the PCA entropy and what we see and I think this is really kind of remarkable is that The way in which the information is spreading is again governed by the relevant Dynamical exponents and again, I emphasize that this is only taking snapshots in one single basis Okay, so with this I'm in the end of my talk and let me just Thank all my collaborators for these works, and I thank you for your attention Thanks Robert for the nice tag. Is there any question? Yes, just one question about this Actually last remark that you did that Actually, you only measure in the Z basis. So in fact, it's not the whole wave function Do you have an intuition why it is sufficient to extract this information because in general should not be the case Yes, so At least what I think here is that as long as you know, your any energy functional has some kind of overlap on the measurement basis that you are taking It seems that that's enough to For the information that is being spread in that representation to be governed by the same dynamical exponents But that's certainly a question that we are exploring at the moment to try to put on a more solid ground Yeah, so the it's the PCI entropy if you took just some let's say the first or the component of the PCI analysis of the Snapshot then it wouldn't work. I mean, I guess it's related to the fact how this PCI entropy is connected What you mentioned briefly. Yeah. Yeah, that's also a very good point In fact for the spin transport We also check when you know to do actually something very drastic and you throw away all the eigenvalues except the first one so you compute the kind of Shannon entropy for the first eigenvalue and in that case it also captures correctly the the dynamical exponent But also if you look at the PCI spectrum in this case, you see that it is Quite drastic separation between the first eigenvalue and the rest of the spectrum For the energy transport, that's not so clear and here actually I don't remember if we try Computing this quantity only with a few of these eigenvalues but in general If you have the separation between some part of your spectrum and the rest You could as well just use You know the the most important eigenvalues I would say But I mean doing this kind of analysis is very cheap So you can in principle consider all the spectrum and get the information the full information out of it without and this deviations at later times is a problem of your I don't know how you Get the Snapshots, it's not the problem of them. You mean the saturation. Yeah, it's going Yeah, yeah That would be Kind of finest size effect that I think and yeah, it's one should also note that the saturation for example on this Energy transfer occurs at later times in general with respect to this PCI entropy, but I think yeah, we don't expect to capture the full Information about the the energy transferred by only taking measurements in a single basis, so Yeah Yeah, thanks, so I have a question to the to the first part the Heidelberg data basically So in in the regime where you evaluated this I mean data was still I mean the state is still time evolving So there's also still dynamics happening Could you could you also get information from the structure of the eigenvectors? Basically from the PCI analysis that you get and not only the eigen So can you say something about the relevant structure? Maybe spatial structures or so? Yeah, that's a good question. Actually in this case, we didn't look at the eigenvectors We did Look at a different Kind of description of the system, so if you look at the physics of the system, there is a regime where it actually Exceeds itself similar dynamics And our original target was to describe that from the data science perspective as Kind of simplification of the data structures that that you are getting and that's more or less What you see when you look at the first plot there You see that in general the complexity as measured by this PCI entropy decreases as a function of time So we analyze a different quantity, which is known as the intrinsic dimension to measure precisely this and in that case we could also kind of extract the Universal regime for the dynamics that is revealed by a kind of plateau in this intrinsic dimension But in terms of PCI we didn't look at these eigenvectors So I cannot tell you more about it That thing kinks show you the eyes that can go there, please You mean in the second part? Yes I have a next Next please. Yes that can simulate the topological defects in the initial states So what we do here is We I mean the precise protocol that we follow here is we compute The ground state of the left part of the chain with a corresponding Hamiltonian And we put that left part in the ground state and then for the other half of the system We compute the highest excited state So minus the ground state of the other part and that generates a kind of energy kink when you look at the Initial energy density profile. So it would be something like this You have one half of the chain that has a larger energy compared to the other half And then we just do the unitary evolution of that initial state But that that things can be compared to topological defect in the initial state And since do we have one wall that divided two parts to phase in the system Yeah, essentially the question that we are asking in this setting is how the conserved quantity Is sort of redistributed when you start from this in homogeneous initial state? I don't know if you can relate it to the notion. I mean for sure There is the notion of a topological defect But I don't know if you can express it in terms of, you know the spraying of such topological defect or something like that Let's thanks Roberto again