 Thank you, Spiro. I'll share my screen. You can see. Yes. OK. Can you see also my pointer? Yes, perfect. Perfect. OK, thank you very much, Spiro. Today, I'm going to talk about a three-dimensional network approach for lattice gauge theories. This is all the stuff presented here are main based on these two papers. This is for the two decays. And as we will see later for the three decays, we have very recent results that we posted on archive some months ago. And this is a work we have tried planning collaboration with Timo Felser, a PhD student here in the group of Met University of Padua, Pietro Sivvi, Carrentini's group, and of course Simone Montangelo, the ERT University of Padua. So I don't know if all of you are familiar with the concepts of lattice gauge theory. So let me very briefly review the main idea behind the model and the Hamiltonian that we studied. In the lattice gauge theory's models, usually you have two types of degrees of freedom. Because on one side, you have the quantum matter and antimatter degrees of freedom, like the degrees of freedom related to the presence of electron and positrons, for instance, in the context of quantum electrodynamics. But on the other side, you have also the presence of quantum gauge field degrees of freedom, like the electric field in quantum electrodynamics or also in the classical electrodynamics. But what makes these lattice gauge theories completely peculiar is the presence at the Hamiltonian level and into the degrees of freedom of the systems of the local symmetries. That means that are symmetries that have to be satisfied in each point of your space, one dimension to dimension, three dimensional lattice that you consider. Just to have an idea in order to keep things as simple as possible, the Gauss law in quantum electrodynamics that is represented by this equation in the continuum theory is an example of a gauge symmetry generator. What does it mean? It means that this equation relates the charge distribution, charge or anti-charge distribution, to the electric field configuration that this distribution charge generated into the space that you consider. And in particular, if you have, for instance, an electron in one dimension, ideally in a position x of the one dimensional space, this electron will generate an electric field configuration of this type that is incoming towards the particle. And this configuration satisfies this law. But if the electron move from a point x from another point x plus 1, the electric field degrees of freedom has to be changed in order to restore this local constraint. And lattice gauge theories are almost everywhere in theoretical physics because they are emergent theories, for instance, at very low energy in condensed matter systems because you can use lattice gauge theory framework for describing, for instance, high temperature superconductors or frustrated systems or spin liquids. On the other side, that's very high energy. Gauge theories are the fundamental blocks for the description of our universe. In the end, the standard model is a very huge gauge theories that combine quantum electrodynamics, quantum chromodynamics, and also the description of the weak interaction. And in particular, this model are extremely demanding from a numerical point of view because you have to consider all different types of degrees of freedom, so the quantum matter and the quantum gauge fields and the interaction between them. And so they represent an ideal goal for developing classical algorithms that can be useful, but also for benchmarking quantum simulation, quantum computation. And on the other side, for the real realistic quantum simulation and computation on an actual and future quantum platform. So, in particular, essentially this work in certain self-individual because our idea is to develop the three tensor network algorithms to run on classical supercomputing facilities like the Cineca here in Italy but also other supercomputing in Europe. But on the other side, with our idea is also to describe the lattice gauge theories models in terms of degrees of freedom that can be implemented on a realistic quantum simulator. This is, for instance, one of the first example of quantum computation on digital quantum computer from the group of Reiner's Platini's book where they essentially encode the Schrodinger model that is the quantum electrodynamics in one plus one dimension and they study the real-time dynamics of the peer production, the peer proliferation of the elect, or couples of electron and, and positron out of the vacuum. That is a problem and it's usually very difficult to take all with the classical platform of classical numerical results. I don't have the time to enter in all the details but if you are interested into the subject you can read this very complete review that is simulating lattice gauge theories with quantum technology. So, let me also very briefly introduce the main idea behind tensor network. I will not enter again into all the details because also yesterday, Simon and Oter Nicola described a little bit the main concept behind tensor networks. Essentially, the idea of the behind the tensor network is that the wave function is described in this context by a network of interconnected tensor. And the way in which you connect this tensor represents directly the amount of the entanglement that is storing to the network. And so that is the entanglement representation of the quantum states that you want to target. And suppose that you have for instance systems in one, two or three dimension as you want composed by a number of N, capital N of degrees of freedom for instance N lattice size. And each one is a D level systems like a spin system with the D levels. You can fix a basis into the illiterate space of the systems and expand any quantum states of systems for these underground states of a generic maternal systems in terms of these bases. The problem is that this coefficient, you can think of this coefficient here as a stored in a very huge multi-dimensional a real complex number with N, with a number of legs that is equal to the number of degrees of freedom of the system. And the problem is that so this tensor contains a number of parameters that scales exponentially with the system size with the number of degrees of freedom. So from a computational point of view, this is a completely inefficient representation of state because essentially, and this is true for if you want to, for instance, reach very large size of the systems. So the main idea is to replace these very huge, very big tensors with a network of smaller and interconnected tensors where, essentially they have these physical index the original tensor in this composition, but also they are interconnected with auxiliary indices that with a fixed bond dimension chi that is usually a parameter that you can tune at the beginning of your simulation. And the main advantage is that now the number of parameters in this network is scales only polynomially with the number of components or the number of physical degrees of freedom systems and the bond dimension that you use in your simulation. And in particular, essentially we can do this replacement for the, as a consequence of a very general theorem that is the area low theorem that implies that if you are interested in the low energy sector of a sufficiently local emittonians, essentially the amount of the entanglement that is contained first in the ground state is exponentially smaller with respect to a generic states of the fully the best space of the system. So also with a low or intermediate bond dimension you can reproduce properly or the entanglement content of the systems and all the correlation between the different parts of your lattice. And in particular, in the one dimensional for one dimensional systems who are one of the most of the well-known example of the tensor network and that are represented by the matrix process states where you replace each lattice sites with a computational tensor with a physical index and two bond indices with a bond dimension chi. And a possible generalization is represented by the three tensor network concepts where you have your physical lattice but then you construct a binary tree on top of this lattice by connecting the two nearest neighbor sites to a layer of tensor then connecting this layer to another additional layers on top of that. And so on up to reach the final layer of the highest layer of the tree. And in particular, these are very powerful in order to take into account also long range correlations better than multi progress states especially when you have emiltonian that contains long range interaction between the different parts of the systems. In two dimension, while in one dimension the matrix product states but also binary tree tensor networks are one of the most used algorithms for studying one dimensional problem. In two dimension, the situation is a little bit I would say in progress because the direct generalization of the MPS for two dimensional systems is represented by project entanglement state steps that in which essentially again you have one computational tensor for each lattice sites and this tensor has a physical index and four auxiliary indices with the one dimension chi. And the main advantage is that this answer automatically reproduces the area law of the entanglement because if you consider for instance a partitions of this tensor network essentially the number of links that you are cutting considering this partition scales with the area law of the partition. This is the main content of the area law theorem. But the problem is that if you want to for instance use this answer for searching the ground states and you want for instance to optimize all these tensor by minimizing the energy so the Newtonian of the two dimensional systems this optimization has a computational complexity that scales at least as chi the one dimension that you use in your simulation with the two the power of 10 that means that it's really, really difficult to scale the one dimension and typical values are for instance one dimension of the order of two, three, maximum 10 and this limit limits a little bit the possibility of essentially scales this one dimension for reproducing properly the ground states with a lot of the entanglement in the physical entanglement that you have in the ground states of the systems. On the other side, the generalization of the binary treatment for the two dimensional case is this one as I was showing this figure where essentially you start with your two dimensional lattice and then you connect two sides along for instance the x direction to a layer of top tensor then you connect these layer of top tensor between each other along the y direction and then you can, you essentially continue in the same way to construct all the top layers of the tree. The problem is that this structure this three tensor network cancer automatically do not reproduce the area of the entanglement because suppose for instance to consider a partitions in this partition here with respect to the physical lattice number of links, top links that you are cutting is not equal to number of the physical links connected by the interaction of the Hamiltonian but the main advantage is that if you want to optimize this network tensor in the ground state searching algorithm for instance variational ground state searching algorithm the optimization scales only as sky the one dimension to power or four and these allow us to reach very large one dimension of your also five, five hundreds for instance that as we will see later allow us to study also very complicated model. So the first, the first model that we consider in context of lattice gauge theory is the quantum electrodynamics in two plus one dimension. In this model you have the two dimensional lattice but we use the staggered fermions solutions in order to incorporate on the same lattice both matter and antimatter divisor freedom so the electron positrons and in particular they are represented by physical fermionic operator but with a fixed parity. So in the end you have for instance the electrons on sides with the even parity and the positrons on side with the old parity. The Hamiltonian so is composed by these opening tabs that moves the particle but this is a sort of correlating opening because if you move a particle as I show in the first one of the first slides you have also to change the electric fields on the links in the middle and this is the action of this operator that is the parallel transporter of the continuous theory discretized on the two dimensional lattice then you have the staggered mass that assign a positive or negative energy mass depending on the parity of the sides and this is due to the use of staggered fermions and then you have the free electric energy that is proportional to the square of the electric field operator that is defined on all the links of the lattice and finally you have the magnetic contributions this Hamiltonian that is related to the product of the four parallel transporter that along the shortest with some loop that you can consider on a two dimensional lattice that is the loop of a placette. In principle in the continuum theory this electric field for instance can assume any values in its spectrum but in order to do any simulation both classical or quantum you have in the end discretized this degrees of freedom and to fix the number of a loaded configuration in particular to do so we use a well-known approach for that is also used in the context of quantum computation or simulation that is the quantum link model discretization of gauge fields and the main idea is to replace this couple of kind of operators with the sense that the electric field becomes the zeta operator of a spin algebra if you want and the parallel transporter becomes the racing or you dug the opening operator the change of what you need the states of this spin algebra defined on the links of this lattice. And in particular we consider the case with the three possible electric field level for each link plus one zero minus one this is one of the simplest but non-trivial quantum electrodynamics that you can discretize on a two dimensional lattice and we create a sort of quantum simulation model with this Hamiltonian because in order to encode efficiently these types of degrees of freedom in our three tensor network concept essentially we split each link into halves then we put additional fermionic modes on the two halves of this link and we interpret the gauge field configurations as a functional one if you want these occupation number of these additional fermionic modes so taking into account the Gauss law that this is the discretized version of the Gauss law that I showed at the beginning we can define and as our local gauge invariant recent site this unit that is composed by the physical sides that tossed the physical particles of the electron and positons and we take into account the configuration of the gauge field by considering the configuration of these additional fermionic modes on the four half links around these lattice sides and then by using the Gauss law we list all the possible states for the even and the other side so for the particle and antiparticle content and essentially recognize that if you do this operation you see that with this discretization the dimension of the local little bit space for each lattice size is 35 that is a very large number comparative for instance the one dimension that used in the contents or spin systems or fermionic or bosonic model and then we use our two dimensional binary treatments for network cancer to study the property of the ground states of the systems. First of all by neglecting the magnetic contribution to this Hamiltonian so by studying the phase diagram as a function of the mass parameter of the fermions and the gauge coupling of the QED and we see that essentially we identify two different regimes when G is very large so there is an energetic cost of creating a spontaneously charged particle and electric fields out of the vacuum. The ground state is roughly the Dirac vacuum where the total density of the particle on the two dimensional lattice is roughly zero then by starting to decrease in G or increasing M towards negative value we access a regime where the total density of the particle on the two dimensional lattice is approached maximum value in the T1 and in particular by studying the correlation function through the final size scales analysis we identify a real phase transition between these two regimes and so we can study essentially we can reconstruct the phase diagram for the ground states of the systems and we identify this phase as a function of the mass parameter and the gauge coupling as the phase where if you want is the phase of our or with all the approximation of the universe where we live where you don't see the spontaneous creation in our everyday life so particle and particle so from the electromagnetic vacuum the ground state is realized is essentially characterized by the absence of particle and antiparticles and the gauge field degrees of freedom by Cheversa when we access this regime of the parameter of the Hamiltonian we see that the ground state is dominated by the presence of these dimers of charge and anti-charge sort of mesons with this configuration of particle and antiparticle and then interacting with an electric fields in the middle then we activate the magnetic contribution to the Hamiltonian so taking into account all the placate terms that we have in the magnetic sector of the Hamiltonian and we see that the net effects of these placate terms for the in where the phase essentially where we are in the regime of the of the vacuum regime the extent of the vacuum phase so where the ground state does not contain any excitation essentially the introduction of these magnetic terms has no net effects because roughly speaking the Dirac vacuum remains Dirac vacuum by Cheversa when we are in the charge crystal regime so where essentially before we observed the dimer configuration of particle and antiparticle now these magnetic terms determine trivial reorganization of the electric fields favoring a sort of cool on configurations of charge and anti-charge with all the electric fields pointing out from positive charge and incoming phase toward negative charge then we study also the regime for the two dimensional case of the final charge density sector that is very difficult to take with other numerical techniques like Monte Carlo due to the same problem in particular we see that by connecting charge into the systems essentially they are forces to reach the boundary like in a perfect conductor for this study we work with open boundary conditional course in order to minimize the incoming or upcoming electric fields towards the boundary of the systems then we try to push the limits of these three tensor network approach for taking also the same model but in three dimension I want to stress that here we are working the Hamiltonian formalism lattice gauge theory so we discretize only the spatial dimension while the times means a continuous variables and in particular by starting so for the three dimensional lattice we construct this answer for our binary tensor network again by connecting these for instance to nearest neighbor signs along the x direction to a layer of top tensor then proceeding along the y direction and then along the zeta direction up to arrive to the last layers of these three dimensional binary tree and again one of the main advantage is that because we don't have loop in the tensor net in the structure of this tensor net so the optimization complexity remains roughly the same is a bond of the order chi to the power of four and so this is a very powerful sign problem free approach also for study realistic edge theories in three plus one dimension so we consider a sense of the Hamiltonian is very similar with respect to the previous that I showed for the two dimensional case the only difference that again here we have the three dimensional lattice with the staggered particles, electron and postrons on sites with different parity and gauge feed defined on the links between these lattice sites the Hamiltonian the opening term is essentially the same but considering also the additional dimension we have the staggered mass the electric field energy one of the main difference is that in this case this Hamiltonian term is much more complicated because you have to consider not only the placket placket terms on the plane X, Y but also on the plane X, Z and Z, Y and but each one of these placket term is composed again by the product of the parallel transport that is related to the magnetic flux generated by this the moment that essentially the motion of this charge on the lattice again we consider the discretizations of the with three possible levels for the electric field plus one zero minus one we adopt again the quantum link model discretization by replacing these degrees of freedoms in terms of spin variables and again we construct if you want this sort of quantum simulation model for this Hamiltonian by putting additional degrees additional fermionic modes on the half links around each lattice sites and by using Gauss law we list all the possible gauge invariant configuration for the event and the onsites and in this case the model is much more complicated since the local dimension for each physical sets of the lattice is 267 just for comparison this is like simulating gas systems where the models of spin is 133 so we use these onsites for the in our three tensor network arguments by optimize all these tensor for minimize the energy with respect to the Hamiltonian systems and again we study the Gauss state properties first in the regime where we neglect the magnetic interactions and again we see two physical regimes the regimes for where the mass is large and positive so when there is an high energetic cost or creating particular anti-particles out of the vacuum where the density if we study the density of the particle on the lattice its value is roughly zero but it's a versa when we access the regime where the total density of the particle is sorry where the mass is large and negative the total density is roughly one so it assumes its maximum possible value and we also consider a finite size scales analysis for studying the behavior of this order parameter between these two regimes we identify the presence of our phase transition with the universal behavior close to the transition sorry close to the transition point and we give an estimate also of the transition point between these two regimes and also for the critical exponents of these finite size scales formula we also look so at the microscopical properties of the systems in these two extremar regimes and again where we are in the regime of very large mass the ground state does not contain any expression of particle or anti-particle and gauge field degrees of freedom but it's a versa when we are in the regime of large and negative mass the ground state is again characterized by the presence of this mass also dimers of charge and anti-charge with an interaction of electric fields in the middle of course in this regime there is a night degeneracy for the ground state because you can change the direction of these diamonds along the three possible x, y, z direction without that changing the energy of this state but this is only one of the states that our algorithm choose during the variational of optimization then we consider also the effect of the of adding the magnet all the placate terms so the magnetic interaction to the previous Hamiltonian and again we identify the same phase transition between the two regime, the Dirac vacuum and the charge crystal regime but in this case also we observe this non-trivial reorganization of the electric field configuration that favors in this case for the presence of the placate terms as sort of a global entanglement configuration of charge and anti-charge with this reorganization of the electric fields on the links and we essentially we perform a finite size case analysis also in this case of this order parameter and we notice a shift an horizontal shifts of the critical point that is shifting towards positive and so if you want more physical value for the critical mass of the fermions then we try to use this approach with three tensor network also for studying a different geometries if you want for the charge configurations of the on our three dimensional lattice in particular we consider the case where through the chemical potential we enforce the presence of the maximum number of positive and negative charge that we can then count on the two if you want to plane two planes of our three dimensional lattice studying a configuration that is very similar to a sort of plane capacitor so what we did, we did the following essentially we put additional chemical potential terms for the Newtonian on localized on these lattice sides where we want to create the positive charge on this plane and the negative charge on the opposite plane here we are in the regime where we neglect the magnetic interaction so we favor this string configuration of the electric fields into the bulk of this capacitor and so essentially we start from the Dirac phase so tuning the mass in the Newtonian positive and very larger so we have this configuration this is the yellow line where by studying the mean charge density along this line of the capacitor we have the maximum charge on the first plane positive then zero charge in the middle and negative charge on the opposite plane and we have the electric field that is roughly constant into the bulk of the capacitor then we start to decrease so accessing the charge crystal regime where we favor in the end the spontaneous creation of particular anti-particle and we see that from this state essentially into the bulk of the capacitor, sorry we observe the spontaneous creation of particular anti-particle in the middle we have the breaking of this initial string of the electric field the creation of two mesons if you want and essentially that has been testified if you want by looking at this quantity because we see that by decreasing them towards negative value we favor the creation of two additional charge of opposite side in the middle of the bulk of the capacitor and we have a net screening efforts of the electric fields that is zero in the middle and in particular we see that if we plot the total density of the particle by excluding of course the density of the particle on the two planes of the capacitor that we enforce as static charge we see that with respect to the previous case without the two charged planes the density is increased for any value of them and so this means that the presence of these strong external electric fields to some extent favor the spontaneous creation of particular anti-particle at the statistical level this is very similar to what is called in a dynamical from a dynamical point of view the Schwinger effects in the context of quantum electrodynamics where you expected also from an experimental point of view you should observe this pair proliferation of electron and positrons where you apply a strong external field the problem is that the critical value on which it is expected this pair proliferation is astronomical huge so it is very, very difficult to reach this intensity for the laser for instance to produce these electric fields to observe experiment on this phenomenon here we have shown that with a lot of approximational cost at a statistical level it's possible in the end to take all these effects of the spontaneous creation of particular anti-particle in the presence of a strong external field that we apply with these two planes of the capacitor then we also analyze the- Sorry, Giuseppe. Yes. So we have a question from Saptar Shimondal who asks- Yes. So I guess not about the previous slide with the one before. How close is this critical exponent to theoretical calculation or the method? It will be nice to get some comments. Yes. To the best of my knowledge we didn't find any computation for the critical exponent for this for the three dimensional quantum electrodynamics three plus one dimensional quantum electrodynamics these critical exponents are well known for the one dimensional version of this model where you expect a free transition belonging to the two-dimensionality in the university class so beta is one over eight and nu is one but for the three plus one dimensional scenario we didn't find in literature analytical calculation, analytical computation so our estimate of the critical exponent is based only on a fit with- by applying the collapse of the order parameter for the different system sites and fitting this scaling relation with the financial scale formula that I showed in the previous slide. Okay, thank you. Okay, so then we analyze also the confinement properties of this model and why these are interesting because in the Hamiltonian, in a sense, we wrote these two coefficients as independent electrical coupling and magnetical coupling but in the end in quantum electrodynamics they are related by this formula so the square of the magnetical coupling is related to the inverse of the square of the electrical coupling and in the end, so you have only one physical gauge coupling of the theory and in particular, if you take into account this you can identify two extrema regimes the weak coupling regime so where this G is very small and so the magnetical coupling is dominant terms in the Hamiltonian but vice versa, where G, the coupling of the theory is very large, the dominant term in the Hamiltonian is the electric field that favor, as I showed at the beginning, a sort of dimer string configuration of the electric field generated by two different charge in these two extrema regime it's well known from the literature you can find for instance in some calculation in this paper in principle, they should show different behavior of the interaction potential between two probe charge suppose that you have two probe charge one positive and one negative charge and you at a fixed and distance are then you measure this interaction potential in this phase that is a confined phase like the phase of the universe where we live from the electromagnetic point of view the potential should scales as one over R plus some correction due to the quantum field effects what vice versa in this strong coupling regime the interaction potential should raise linearly with the distance and so this is a sort of confined phase similar to what is expected for instance in the context of quantum chromodynamics for the interaction potential works and this is exactly what we observe in particular, we enforce the presence of two static charge one positive and one negative charge at the distance are through the chemical potential in the miltonia then we compute in the ground states of the systems measuring this energy here then we repeat the simulations for the same parameter of the miltonian by in this case without the static charge and through this by using this difference we can give an estimate of the interaction potential energy between the two charges and we see that when we are in the regime of very small coupling for the theory essentially we obtain a behavior that is compatible at least with a fit of the type one over R vice versa when we enter it when we are in the strong coupling regime so we are in the regime where the gauge coupling of the miltonian taking into account of the previous relation between the magnetic and the electric contributions to the miltonian terms we see a linear increasing of the potential favoring these configurations of the charge and charge with these strings of the electric fields but at some point it's more convenient to break again these strings to favor the creation of two methods that becomes roughly independent sensibly from each other and so the potential remains roughly constant then also we studied the problem we take all the problem of the finite density that as I mentioned it's very very difficult to study with other numerical methods like Monte Carlo for the emergence of the strong sign problems into the computation of the observables also the ground state energy and essentially we can do this because in our algorithmic we encode the U1 global symmetry theory so we can change the number of particles into the lattice as a simulation parameter at the beginning and in particular we consider the scenario by where we inject into the system for instance 16 positive charge so on a lattice of linear size 4 and so creating a situation where we expect a total density of 1 over 4 and we see that again when we are in the regime of the Dirac vacuum for the parameters of the Hamiltonian this positive charge are forced to reach the boundary working with open boundary condition in order to minimize the upcoming electrical stores the boundary of the lattice but then by looking at surface density on cubic surface those phases are at distance L from the boundary of the systems we see that if we decrease we essentially we start with this configuration on the blue line where we have all the maximum charge localized on the boundary and then roughly zero charge into the bulk then by decreasing them so accessing the charge crystal regime where the creation of particle is favored essentially we see that we favor the there is a creation of particle and particles into the bulk that determine a sort of the screening effects of this extra static charge that we enforce into the systems so what's next the idea is of course to using these algorithms for testing validating possible experimental implementation on quantum hardware from the lattice gauge theory's point of view another additional analysis interesting analysis would be the addition of non-abelian symmetries in these models and this is not so difficult because you can map essentially also the non-abelian symmetries in terms of these additional fermionic modes that we have on the links of systems and in particular also to try to encoders the dynamic, the real-time dynamics in these algorithms for study for instance cutting properties or real-time dynamics not perturbed by the efforts like pair proliferations or sink breaking and so on that are usually extremely difficult to take all with numerical methods but also from quantum simulation so thank you very much you completed the photo thank you very much for your attention thank you Giuseppe for the nice talk are there any questions? I think we have plenty of time for questions yes Pierre? so actually I have a question on the treat-and-send network method itself I guess I could have also asked it for Simone's talk but just come to my now so when you do the tree algorithm in say two or three dimension what it feels like you're doing is some sort of coarse graining yes with a periodicity that is equal to the dimension and because you truncate every time effectively it's as if you are forcefully introducing some sort of a scaling symmetry to your system even if it may not have been there at the beginning but so as you do this can it not introduce some some bias, some artifact leaving you to a prediction that do not correspond to the to the real situation of the model or or you don't know that thing? no, I think that in principle you have not introduced in this addition I've seen this because you can think, you know to this rescaling as a isometric tensors that maintains the same properties of the symmetry that you have at the lowest level of the tree so I don't think that you are introducing additional symmetries the spurious symmetries of the with respect to the original systems that you have because if I think of doing a decimation of some sort of a spin system now normally the the terms that are introduced in the model after a step of decimation is some sort of interaction of longer range however if I understood well when you when you do the tree network you discard these interactions of longer range you effectively discard them because you don't consider longer range tensor I mean longer tensor with more with more indices yes this is true but take into account that the think a little bit to the one dimensional case okay different between the MPS the tree tensor network since the idea of putting additional tensor is exactly the I mean the it's the advantage is that you can reproduce properly and better the different long correlations between the different parts of the system comparatively to MPS where we have only a fixed number of tensor between for instance i and j that instead in this case you have all the trees to connect also the renormalization between the two parts of the system and this is exactly the same for the two or three dimensional of course this is this is an answer and this is an approximation in the sense that in principle you are not following stricking for instance the area law so in principle the the answer is yes you are cutting some degrees of freedom instead of maintaining all the tensors of the system but the main point is that for the ground state searching algorithm this is variational so you can always test your convergence by increasing the bond dimension and then truncate where you see the convergence the plot of the energy so I agree with that but what I was more concerned is when you start at 2D and also at 3D but at 2D is that the fact that you are first doing a direction then another one then the same direction then another one and so you introduce some sort of periodicity in the in the scaling in the scaling coarse-graining that you are doing and it is more specifically this symmetry that there was that is not present so this isotropy yes you introduce or periodic isotropy I'm not sure how it would be called that you introduce by truncating that that could be a superior symmetry that's the main yes yes okay this could be true yes because okay I got the point you think that in principle when you renormalize at tensor levels you are I mean you think that connecting before along the x direction then along the y direction so it's yeah it's yeah I mean introduce a symmetry into the effective scaling symmetry that you force upon your system some sort of periodicity and I'm I don't know but I was thinking that yes this is a good yes we can discuss later if you want I have to think a little bit but I I okay the point is that um at I don't know if it is related to the fact that yeah the two-dimensional Hamiltonian that we consider is uniform in the act is you have the same open terms for instance everywhere yes so I don't know I mean if you connect along x and y I would say that this operation commutes with connecting the opposite direction on y and x so I don't think you for this specific type of model when you have a uniform Hamiltonians that you are introducing its curiosity maybe this could be a problem when you have non-homogeneous Hamiltonian non-homogeneous hopping for instance terms that change with respect to the directions and so on okay okay thank you you're welcome I have a related question in one of the previous plots that you showed in the for the three-dimensional case yes but I think it's the next to last if you go to the end a bit further I think it's this one yes so if I understand correctly from the the colors of those of the sides the charges are tend to concentrate to the boundary but somehow this doesn't happen in a uniform way and it cannot happen of course because the charges are not as many as the points of the surface and I wonder if if the this somehow symmetry in the distribution of the charges on the heads has something to do with the choice of the three network or no I think that from I mean just two points you are on this plane for instance the lattice you can create in this on these sides you cannot have a positive charge because this is a side with the different parity so at at maximum you could have a negative charge but this is not the case because we enact in this example positive charge into the system then regarding the distribution I think that in principle here you have and I again I like the generesis because you can move the charge here with the same configuration at the field without that cost any energy in the middle tunnel so I think that this is only one of the configuration that they are going to choose during the optimization but there should be no difference moving for instance one side taking into account the staggering the staggering of the sides with respect to the position of these charges would it make sense to to to have a mixed type of a network where you apply this tree the brown sink into in this tree structure up to some level and then at the last level let's say you apply the standard MPS which is more uniform in space in the sense it treats all blocks of points in the same way of course this would come with a cost efficiency cost from the computational cost I would say that it could be a good idea because the point for the complexity is avoiding the loops into the network so if you have one MPS on top of all the tree of the net or the or pencil of the tree in principle you are always avoiding loops so probably the computational complexity should remain the same the pro I don't I think that this question could be related to the previous one by Pierre but I to be honest I don't think that for homogeneous Hamiltonian you are losing the if you want the translational invariance or something like relating to the pairing along x and z so yes maybe maybe it could be a good idea but I think that the uniform the translational invariance symmetry is always encoded also in this case without additional MPS on top of all and I have one more question very heavy so you have presented results for even for three dimensions which is quite challenging I guess or it's clear I mean for those who know the problem so I would dare to ask if it is possible to study also time dependence in two dimensions I know that there are some works about time dependent situations in one dimension so I wonder if given that you have reached the three dimensional we can go to the three dimensional time yes is a work in progress in the end because we are trying to encode for instance the time evolution for using trotter evolution or tbdp methods on these two dimensional three structure but we didn't try yet for this type of model we are in the testing phase with the spin systems and so on but we are under development for but yes this is a very interesting direction for the for uploading if you want for these these three tensor network algorithms if there are no more questions then actually if I may so a very very short question so as of now how many sites say in 3d are you able to to simulate like within a month let's say this one the eight times eight times is the simulation on at Chinaca for instance takes for one point roughly one month okay okay and you think it would it is possible to to get more by somehow managing to optimize the algorithm yes because yes I didn't enter into the technical details but for the moment up to now these algorithms are parallelized only with the open p paradigm so only inside the node but the the main advantage of the three tensor network is that for the application of the tensor during the minimization you can separate the application for the different parts of the tree and so applying parallel on different nodes with MPI for this and this could scale a lot the I mean improve a lot the computational time on the other side we are triangle so something but very first level um to use a GPU for tensor construction that they should be very very efficient but I mean we are test mode would say so yes but we expect to improve at least a factor two or three the computational time by using MPI okay okay but so it means that maybe you will reach l equal 9 10 11 12 yes yes but most likely not beyond we don't I am already a lot that will get me wrong no no no okay I'll take into account that as far as I know the largest simulation in the world for the lattice gauge theory is with l 140 something like that that is of course an order of money to Iger than the values that we reach but I mean we are not this is a work where we we try to up to up the application of this method for the first time to the three dimensional scenario but we are extremely confident okay I would say not maybe 100 sides for the linear side but probably in one year or less optimizing all the parts of the argument in principle we are confident to reach I would say l equal to 30 40 yeah okay okay very well thank you you're welcome