 Okay, the first talk of the last sub-session of the last session of the last day of the conference is given by a Victoria Kazatkin Identifying a phase transition with and without quantum annealers. You have 20 minutes. Let's talk about phase transitions Consider this Hamiltonian. It's called frustrated letter Hamiltonian the bottom row and the vertical lines are the cubits Coupled for magnetically and the top row coupled anti-permagnetically Also, the top row is biased towards spin up and the bottom row is slightly biased towards spin down What I described was classical part of the Hamiltonian and we also have the parameter s which interpolates between the standard transfers field Hamiltonian and this classical Hamiltonian How do we find phase transitions of that Hamiltonian? Well people already know for that particular Hamiltonian that there are a couple of order parameters Which are called staggered magnetization of the top row and magnetization of the bottom row and They can compute those parameters for various values of the parameters of the Hamiltonian s, k and u and draw these phase diagrams on the horizontal axis is u, k is equal to 1 and on the vertical axis is s and the temperature is 0. So we are looking at the order parameters for the ground state and Here the chain length is equal to 10. I talked about phase transitions So you may know that phase transitions are defined in thermodynamic limit Which means that you cannot see actually phase transitions here, although you can see something which looks very closely like phase transitions the places where the color on this graph changes quickly So this is problem number one, right? How do we even define this task given that we are dealing with finite systems and throughout this talk we will be dealing with finite systems and in particular this system as an example throughout the talk The second challenge is that we would like to find phase transitions where they are not known yet and in particular we don't know the relevant order parameters How do we even define phase transitions if we don't know what order parameters we should look at? That would be the second problem which I will also address later But first let me describe the challenge which we will be trying to solve throughout this talk The challenge is between classical algorithms or classical machines and machines which have access to a quantum computer and throughout this talk this quantum computer would be the quantum annealer Moreover throughout this talk we will consider one single framework for solving this problem using quantum annealer which is we first use quantum annealer to generate a data set of bit strings and then you use machine learning to learn phase transitions from those bit strings Although in principle we could ask machine learning to provide additional requests to quantum annealers but we will not be doing this in this talk And here lies the third problem which is the machine learning algorithm will only have access to the bit strings and bit strings provide incomplete information of the quantum state in general So let me now address all of these problems and the way people address these is with a quantity called fidelity susceptibility This is the quantity which intuitively measures a square rate of change of the underlying state These are the definitions On the top row is the definition of f which is the fidelity for pure state or for mixed state And then there is fc which is classical fidelity which is what we can measure if we can only access probability distributions over bit strings And then the fidelity susceptibility is the first non-trivial term in the Taylor expansion of these fidelity given that the state changes depending on some parameters So we know that quantum phase transitions or phase transitions in general are characterized by changes in fidelity or more specifically by maxima of the fidelity susceptibility Typically for the first order of phase transition you will see fidelity diverged to infinity and for the second order of phase transition you will see sharp maxima of the fidelity susceptibility But importantly We know that classical fidelity susceptibility in many cases tracks the fidelity susceptibility closely In particular if the Hamiltonian is real valued in the computational basis and we are looking at the ground state of that Hamiltonian which is non-degenerate then almost always the fidelity susceptibility will be equal to the classical fidelity susceptibility And in general there is the observation that on average across all measurements in which you can all bases in which you could perform the measurement classical fidelity susceptibility is half of the full fidelity susceptibility if you are measuring pure state For mixed states things get more complicated There is still some hope that our quantum robot which only processes bit strings will still be able to solve the problem of identifying phase transitions Let's talk about what our classical robot could do to solve this problem And one approach which was actually used for the phase diagram I displayed above is called Lantros also known as exact diagonalization And this approach is able to find a few low energy states Unfortunately its complexity scales exponentially as you can see here with the size of the system There is two to them here So in practice that means that we could run it for 20 qubits on a laptop in an hour or in two hours but if the system becomes larger than 30 qubits it's usually impractical to run it and for 50 plus qubits it's impossible to run it Other methods are tensor network based methods for example DMRG which works reasonably well in one dimension for systems except when there is a large long range entanglement And its complexity is polynomial but first it's very challenging to apply it to the systems which are different from 1D unless you want to pay exponential price and it breaks down if there is a large long range entanglement Another method is called QMC or SQA which can be used to sample the bit strings and then as in the quantum robot we could apply some machine learning to learn the phase transition from those bit strings This method works reasonably well unless there is a sign problem or topological abstractions but when there is a sign problem or topological abstractions then it breaks down and there are some other methods like you could do perturbative expansion if your Hamiltonian is similar to something which is well known or you can extrapolate from other system sizes or even from other systems other Hamiltonians in particular you could explore some complex extrapolation schemes using machine learning but in general it's very hard to solve this problem for complex systems in more than one dimension if the system is large enough Let's now go back to our quantum robot and as you can see there are two parts So first is we sample bit strings from quantum annealer How do we do this? A naive idea is to anneal till a specific value of the parameters and then measure those bit strings Unfortunately the annealer we had access to which is a device that doesn't allow to measure in the middle of the annealer and that's what we are interested in So as you can see the diagrams we get from the annealer here and here have these vertical lines which means that not much changes as we change S In fact it looks like we are always measuring at the end of the anneal which we are, we are just doing quench as fast as possible till the end of the anneal and as fast as possible is not fast enough So unfortunately so far we are not able to sample bit strings from quantum annealer with adequate quality although Andrew had some ideas with Andrew King had some ideas which we could explore further But it looks like for this project we are looking to demonstrate that we could apply these algorithms to some future quantum annealers where those problems will be resolved Now let's look at the second step which is machine learning So we have the bit strings, we want to get the phase transitions This is not a new topic, there is some literature on that In particular there are these two papers First is machine learning phases of a matter and the second is learning phase transitions by confusion So in the first one mainly people looked at identifying order parameters or something like order parameters if phase transition, if location of phase transition is already known or identifying phase transitions if we couldn't identify it just by learning whether the state looks like zero temperature state or infinite temperature state or in our case we could apply it by looking at whether a state looks like state at s equals zero or state at s equals one Unfortunately even for that diagram above at the phase transition the state already looks far away from state at s equals zero so that is not expected to work even for that example Another method is learning phase transition by confusion and in essence it also want to work in general for our case because first it requires training of one network per potential critical value so it would be quite expensive to do and also they rely on some pre-computed features and we are learning the phase transitions from bit strings and also they are not actually learning the fidelity susceptibility or anything quantitatively close to fidelity susceptibility they suggest to identify phase transition from graphs like this where this dot in the middle indicates the phase transition which may be not robust enough so we propose the method which we call bit string kfc and it describes like on this diagram first we convert the initial data set we get of the bit strings and the corresponding parameters of our Hamiltonian to a specifically crafted binary classification problem which aims at distinguishing states at different values of the parameters then we need some machine learning model which could solve that binary classification problem and if we could do this then from that machine learning model we could extract estimates of the fidelity susceptibility that's the idea in general and I won't have time to describe it in more details in this talk so after we got estimates of the classical fidelity susceptibility we then find the maximum of that classical fidelity susceptibility and those will be our estimates of the locations of the phase transition so let's see how well it performs this is again an example from the first state letter model here k is equal to u equal to 1 l is again 10 and the black line here is the true fidelity susceptibility extracted from line first, it's pretty good and the blue line is our estimate as you can see it's not perfect but the maximum of that fidelity susceptibility is quite close to the maximum of the true fidelity susceptibility so this is a second order phase transition let's look at another example which is the same frustrated letter but with u equal to 0.2 and again you can see that we got both of the peaks of the fidelity susceptibility quite closely that's all for the talks while you think about the questions let me summarize we looked at what classical algorithms could do they could solve problems up to size 20 in most of the cases but for larger sizes if the problem is complex it's hard to extrapolate from known solutions generally classical algorithms fail and in a quantum case we were able to demonstrate a machine learning algorithm which could identify the phase transitions from these strings but right now with the current quantum manilas there is a challenge with the first step which we hope will be resolved in the future with better quantum manilas questions? maybe better to wait for the microphone otherwise I will have to repeat the question thank you it was a very interesting talk could you please again go back I think to the first part of the susceptibility that you showed first part of? the trained model predictions and the Lankos comparison towards the end the performance of the model towards that one I think it was I was a 20 qubit system okay actually my question resolved itself thank you am I right that you use the bit strings for the quantum manilas method coming from the classical computations because you couldn't get the one right so here bit strings are sampled using clenches but the idea is that we could use the same algorithm with the bit strings manila that gives you the bit strings you could use those that's the point right once we have the manila which could sample those bit strings just very philosophical question what kind of phase transitions between what kind of different phases you could predict the phase transition should be it like Landau paradigm of you have to have some order parameter and spontaneous symmetry rating where you can do more predict MBL transition or some kind of transition between different symmetry protected topological phases yeah where there is no order parameter so how general the method is well the method is quite general in the sense that it doesn't need any access to any kind of order parameter it predicts the fidelity susceptibility and if the pattern which is from the bit strings to identify on which side of the phase transitions or they is very complex which machine learning cannot learn then this method will fail but in all cases we tried so far machine learning was pretty successful in learning the task like this task here from the bit strings does that answer your question I think maybe the question was more of the sense that the susceptibility is able to capture topological phase transitions on top of typical Landau-Ginsburg the fidelity susceptibility generally measures the rate of change of the underlying state so regardless of the nature of that change fidelity susceptibility itself is expected to capture all phase transitions so there are two questions now that translate to classical fidelity susceptibility of course if you have a phase transition which doesn't reflect itself in the distribution of the bit strings then the answer would be no but that's pretty rare as demonstrated by the properties of the classical fidelity that's expected to be pretty rare and the second question is if the pattern itself in the bit strings maybe it exists maybe there is a difference between probability distributions but you cannot describe it in a short amount of space or machine learning model cannot learn it then this method will also fail but we didn't find any example where it would fail well we didn't try it maybe that's a good suggestion for me to try we can discuss it after we can definitely discuss this after the talks because I'm also interested any other question? ok if not then let's thank you again