 Alright, so our next speaker is Matthias Erner, localization quantum phase transitions and graph theory for adiabatic quantum computing. Can everybody hear me? Thank you very much. So I'm Matthias, thank you for the introduction. I'm a PhD student with Kilimanjaro Quantum Tech in Barcelona in Spain and I will talk a little about localization quantum phase transitions and graph theory for adiabatic quantum computing. So what are we investigating? So well we're looking at adiabatic quantum computing that I assume does not need to be explained in this conference. We have like a driver or initial Hamiltonian interpolate to the target Hamiltonian which in this ground state has like the solution to an optimization problem that we're interested in and we are limiting our analysis to a case where the target Hamiltonian is diagonal. So it has each computational basis states associated with an energy EET0 and in this basis the driver Hamiltonian can be thought of as being proportional to the adjacency matrix of a simple D regular graph. Simple is just a graph that means it doesn't have any like each node in this graph only is connected to other nodes but not to itself and D regular means that it has D different neighbors right and this graph G has certain properties and we want to investigate what are how these properties relate to quantum phase transitions. Here the HD is defined as such as the adjacency matrix times minus 1 over D which is basically just to normalize the Hamiltonian so we always get a ground state NG of minus 1 but for this class of driver Hamiltonian the the ground state is always the the plus state on all the qubits which is like the common starting ground state. Important to note that the Hamiltonian that's usually used the sigma x driver actually falls into this class so it's that Hamiltonian a couple of others that are covered by this analysis and so how does it relate to my quantum phase relation on localization. So in the beginning of the kneeling the system is in the delocalized state it's the uniform superposition of the basis states and as we anneal so we go from s equals 0 to s equals 1. Ideally the system localizes in the global minimum optimization problem which would be our solution and this is the the the green arrow here the green dynamics which is typically associated with um polynomially closing gaps and smooth solution fidelities which are like the two quantities shown here but occasionally it happens that the system first localizes in local minimum and then later has to tunnel through potential barriers to get to the global minimum and this is typically associated with exponentially closing gaps and discontinuous solution fidelities those are the the red lines in these plots and we are going to investigate or the question is a little bit so why does it localize here in the first place so how does it come to be and we we do the analysis to figure out what are the properties of the driver Hamiltonian graph and the location of the target energies e z on this graph that cause or that cause one behavior or the other of the system um so how we do this so first to clarify the the graph we're dealing with is a graph in in solution or in a configuration space so each node is a computational basis state and the edges are the the off diagonal elements of for Hamiltonian typically introduced by the driver and in this in this on this graph we are going to have local minima and this local these local minima we assume are going to be um nearly degenerate so the energies of all the nodes in in the set v which is a set of degenerate eigenstates of degenerate target eigenstates all are very very close to the value etv and to apply we're going to apply the degenerate perturbation theory which means we have to solve this eigenvalue problem on the degenerate subspace v and the like the equations are relatively well known so the the target energy is all the same but here in the driver term we have like this hd prime which is basically just the projection of hd onto the subspace and we can think of this hd prime as a driver Hamiltonian that is the adjacency matrix of the graph gv which is the so-called induced subgraph of yeah the induced subgraph of g by v let's say so in the cartoon here that would be all the nodes in the blue area and all the edges that are entirely in the blue area and what we're looking for are now the the eigenvector which would be the principal eigenvector of the subgraph gv and the energy more importantly actually which is the principal eigenvalue of gv and this is typically or this eigenvalue problem is typically not easy to solve but we can get bounds on the eigenvalue we can get a lower bound on the energy which depends on the maximum degree of v which is basically just a node in v which has like the most neighbors and we get an upper bound that depends on some quantity phi v here that we want to discuss in a minute just a side note the those are like quantities that or these bounds are relatively easy to derive from like standard graph theory knowledge like using Gauchkin's circle theorem we can verify the upper bound and using for example the variation principle we can verify the or the other way around using Gauchkin's circle theorem we can verify the lower bound and using a variation principle we can verify the upper bound and well as I said the upper bound depends on some quantity phi v which is called the conductance the conductance of v and the conductance is defined as such it's like the the size of the edge boundary which is like this this partial v symbol here which is like the set of all the edges that get cut by the dashed line and just like the magnitude of this it would be one two three four five edges divided by the magnitude of the set v which is one two three that would be the conductance of the set v and the conductance is a relatively interesting property because for all subsets v of this of the of the set of nodes in the graph there's going to be one non-trivial minimum so there's going to be one non-empty subset of nodes with the with the smallest conductance and this conductance is called the chiga constant which is like a property of the graph in our case would be the property of the of the graph g and this chiga constant comes with several or two inequalities actually so called chiga inequalities and using these inequalities we can actually get a or these inequalities give us bounds on the spectral gap of the graph in terms of the of the chiga constant and we can massage them a little bit to our purposes and then we can basically get a connection between the chiga constant and the spectral gap of the driver Hamiltonian so how does this now relate to localization well we get like we calculate the energies of the local state of the state that would be localized in the global minimum and the state that would be localized in the local minimum there were green blue and orange here in these in these cartoons and the these straight lines here are the respective energies and we calculate those energies up to first order perturbation theory that's why there are straight lines and not curved or anything and oh i'm already going too fast sorry so we can calculate the energy of the delocalized state the green one by treating the target Hamiltonian as a perturbation on the driver Hamiltonian and by treating the driver Hamiltonian as a perturbation on h on the target we can get the the energies on the state that's localized in the global and the status localized in the local minimum and so basically we do perturbation theory twice once from the beginning towards the end and once from the end towards the beginning and how to how would you read this or how it would interpret this is that in the beginning of the anneal the systems in the localized state and it would we tune as we tune as we tune as and eventually the the delocalized energy crosses with a state that would be localized and this is where the system then has to transition into the localized state and we continue annealing and eventually the state that's localized into local minimum is going to cross with the one that's localized in the global minimum which is then the the point where it would need to tunnel from the local minimum to the global minimum and obviously that's not going to happen if these the lines between local and global minimum do not never cross but even if they cross there's a scenario when we actually get the process where the system localized directly namely when the localized localized transition happens before the delocalized to global transition so it would be this scenario the system is delocalized delocalized delocalized and then transitioned directly into the global minimum this corresponds to this type of dynamics here so we might be able to say a lot about this process if we knew this localized delocalized transition these red stars here and given the the bounds that we derived on the previous section we can actually get bounds on the location of these I called it s star here where the localized delocalized transition is happening we get a lower bound that's conditioned on the depends on the conductance of the local minimum and we get an upper bound that depends on the maximum degree so sometimes I call them like the conductance bound and the degree bound okay so about these bands there's a few things to note first they actually equal if the gv is a regular graph so yeah if gv had in all the nodes and gv have the same number of edges then these bonds are equal and then we can actually determine the s star exactly and the lower bound ends up being asymptotically tied if the local minimum can be described by a large additional in e-graph and now can we make statements about whether or not we would be seeing these localized localized transitions for particular problem instances well yes the e-global and ed localized are can be calculated I earlier just mentioned that it's possible but here are the actual expressions and these two expressions are going to be equal for some s prime and s prime is basically where the delocalized to the delocalize to global minimum transition is happening and if s prime is larger than the upper bound then we are certainly in the regime where we would see the delocalized to global minimum transition directly and if s prime is smaller than the lower bound then we are certainly in the regime where the delocalized state would first transition into a global minimum and a local minimum and then it has to turn it into a global minimum and if s prime is between those bounds then it's actually hard to tell and we can't really know And first to note is that, okay, so the upper bound, the lower bound, the conductance bounds, actually tied under, in both of these scenarios, actually. So, and that will come up with the numerical results. Maybe there's some, that's kind of work in progress, but it might, maybe that there's, or if the low energy states of the system are well enough described by either of these two scenarios, then we can actually give bounds in terms of the TR constant and as a consequence, also in terms of the spectral gap of the drive Hamiltonian, when we would, for any problem instances, or for any problem instance, not expect to see a localized, localized transition. But this might be related to like symmetries of the problem, but that's kind of like a topic that we're working on right now, so stay tuned. So let's look at some numerical results we have for that. So we first look at some very simple toy problem where we just generate random regular graphs and then some point in the graph, we locate the narrow global minimum and at some other point in the graph, we generate like an arbitrarily shaped, arbitrarily deep local minimum. We just make sure that they're disjoint, so because they're not disjoint, then this is kind of a different story, but for, now we want to assume those are disjoint, the global minimum and the local minimum are disjoint and then here in the top part, I can see basically the numerical computation of the cartoons I showed before. So in blue, we have the energy of the global minimum state. In green, we have the delocalized state and in orange, because we have bounds on it, it's like this orange shaded area, that's where the energy of the local minimum state is what happens to be. And in the black line here, I hope you can see that, yeah, seems to be right. The black line, that's actually the exact diagonalization that we get from these problem instances, for example. So systems, first in the delocalized state and then transition to the local minimum and later somewhere between the, yeah, somewhere in this shaded area, which is like the bounds that we get on the S-star, this is where the system would need to tunnel from the local minimum to the global minimum. And we can actually see that the discontinuity in the ground set for that is actually quite well, like pretty much smack that in the middle of these bounds that we're given, as well as the minimum gap. So this corresponds to the dynamic for a system, first localized and transitions then. And on the left-hand side, we see the same dynamics where the conditions are given that we would not see the localized, localized transition. So the system goes directly from delocalized to localized in the global minimum, which we can also tell by the smoother ground set fidelity and actually somewhat larger gap. But this was, if we do that, well, if we do that for several random problem instances and we assume that actually the minimum spectral gap is exactly at the localized localized transition, we can basically do that several times and plot the predicted S minimum versus the observed S minimum. So the predictor is gonna be the bounds here and the bounds seem to be crossing the diagonal fairly accurately or consistently. So this seems to be a well-working prediction. The color coding is that in the blue case, we are certainly in a scenario where we see the localized localized transition. The red regime is where we certainly do not see the localized localized transition and the orange ones are like the in-between cases. And in terms of like the green dots are showing the second order perturbation theory. That is some analysis that was done before by Amit and Shoy. And that's a paper that's linked here. That's several years of array and they use non-degenerate perturbation theory. And we can see that it seems to be working up here relatively nicely, but generally, it's not as close to the diagonal as he would like it to be. And we believe there's a reason for that, namely here the straight lines, this is by the way a plot that's taken directly from the paper, so thank you. But the straight lines are the first order energy corrections according to non-degenerate perturbation theory. And these lines will never cross. So you will not observe level crossings with first order non-degenerate perturbation theory. So they need to go to second order to first observe the level crossings. While with the general perturbation theory that we are applying, we can actually already see it at first order. So we would argue that the energy corrections that lead to the level crossing actually first order effect rather than a second order effect. And in the paper, they also present like a weighted minimum independent set problem, which we also reproduced. So the plot here to the right is like a very close reproduction of one of the plots in the actual paper. So the red line shows the exact diagonalization solution of this location of the minimum gap obtained by exact diagonalization for various problem instances of this weighted minimum independent set as characterized by the depth of the local minimum WL. And the predictions made by second order non-degenerate perturbation theory are the blue lines. And we can see that the graph theoretic bounds that we derived, okay, the lower bounds, kind of similar actually to the previous plot, very close to the exact solution while the upper bound is off and kind of actually matches more the second order non-degenerate perturbation solution. But I suspect that's more coincidence than actually systematic, but we'll think about that a little more. And with that I'm through. Thank you for the attention, a quick summary. So we discussed the origin and location of the first order quantum phase transitions with first order degenerate perturbation theory. And this allows us to map it to properties from graph theory describing those local minima and we can derive bounds on when and where or on average circumstances and where we would expect to see localized localized transitions. If it happens to be that the lower bounds actually tied like the conductance bond is tied and we can also formulate some general conditions which would tell us for a certain problem class when we would not expect to see those transitions. And yeah, we'll kind of work to come as we are researching inequalities that kind of bring the degree bond in this also. So we ought to map this to like problems or how to analyze the symmetries of a problem with that and what we can tell them about that. And with that, thank you for your attention. Very interesting, inspiring talk. I wanna ask the first question actually. So the quantity phi, how hard is it to calculate it and in terms of its computational complexity? So computational complexity, I can tell you, I have to research, I've done that, like so far I've been doing it by hand and it's tedious. But in terms of complexity, I don't know yet, no. But is there a reason to assume that it's polynomial time in general? Well, so it's more like a theoreticalness so it requires that you already know what the states are that you're looking for. So it's, you would need to know what states belong to this local minimum and that's kind of like solving the problem, right? So yeah. Yeah, I have the following question. So imagine you take some typical optimization problem like Sherrington-Kerpatrik or any other. So now you're telling us that there could be two different phase transitions. Now my question is the second one, when you are tunneling between two localized states, probably the gap will be exponentially small in n, right? And the first one, do you know, it should scale like one over n in some power? Could you say something about it, yeah? I can say very little about it, actually, other than just like the delocless definition, something that's unavoidable, basically, for this problem class. So we have to hope that it's polynomial, otherwise it is, sorry? What? You have to hope for what? We have to hope that it's polynomial in size. If you could show that it's always exponential, then we have like large problems, complicated problems on our hand. Okay, but actually there are not so many studied. There is some paper by Gnis, okay, about some, what is this problem? It's analog of Sherrington-Kerpatrik, where he was telling that the first gap is polynomial in n, and then he could have sequence of next gaps, yeah? It should be exponentially small in n, which would match to your picture. Yeah. But he always had first gap, which was polynomial in n, but probably it is model specific. I would assume, yeah, but any more questions? Thank you. Is there any reason why you might suspect that the driver Hamiltonian, you could find any advantage for making it non-stochastic? So there's one paper from Little Croissant, I believe, where they do this analysis numerically and find actually that non-stochastic Hamiltonians tend not to increase the gap, rather decrease. So I would say that generally, just generally probably not, but if you design the Hamiltonian specifically to the problem that you're trying to solve, then there's a chance, yes. Okay, thank you very much. And is there a next speaker here?