 Please, you can share your screen. OK, yeah. See you in one second. OK, do you guys see me? Did the presentation? Yes, we see the presentation. Thank you. OK, so I want to thank the organizers for inviting me. This is a collaboration amount of many people, but I especially want to acknowledge Andrew King, who took all the data, and Jack Raymond, who did a lot of simulations, as well as Seisuzuki, Idotishi, Nishimori, and Daniel Lidar. So I want to start by commenting on the title, Coherent Quantum Aneil. And this is coherent versus what? It is coherent versus quasi-static. So you can run quantum annealers in two different regimes. One of them is quasi-static regime, when the annealing time is much larger than thermal relaxation time. And in that case, the system follows thermal equilibrium for most of the annealing. And because of that, because quantum Monte Carlo also can generate thermal equilibrium, therefore, quantum Monte Carlo can simulate some aspects of quantum annealing, but not all. The opposite regime is coherent regime, when the annealing time is smaller than both relaxation and decoherence time, or defacing time. And in that case, the system follows Schrodinger dynamics, and nothing can simulate. Up to now, there have been several papers published on the wave machine, and all of them were in the quasi-static regime. So this is the only, the first set of data I want to show that that is using coherent regime in these machines. So let me start by just giving a little bit of introduction. And of course, this audience are more expert than other audience, so I will go fast. So the problem we consider is Isink problem. So this is a cost function, Isink cost function, S, I are just binary variables that could be plus and minus 1. And H and J are just bunch of numbers that are parameters of this cost function. And if you look at the cost function versus the all possibilities of S, then you will get a complex landscape. I cannot represent, visualize a hypercube. I am visualizing it in 1D, but this is not the truth. And this function can have global minimum and local minimum. And the goal is either find the global minimum or one of the lower energy local minimum. And the easiest algorithm you can think of is greedy descent. You can randomly start somewhere and then follow the gradient. And with high probability, you end up in a local minimum. So this algorithm is not very efficient. Unless you get lucky and you start in one of the well that takes you to the global minimum, you will end up in a local minimum. And local minimum is an approximate solution. Now, you can think of another algorithm, which is thermal annealing or simulated annealing, which is a simulation of thermal annealing. In this case, you make a physical system with energy proportional to the, I think, the problem that I mentioned before. But now I'm using Pauli matrices to represent the variables. Now, again, the same energy landscape. But now we can introduce temperature. Temperature gives me a energy scale. And if this energy scale is larger than the barrier heights, the system can jump out of the local minimum. And if I reduce temperature, gradually it goes to the global minimum. Another algorithm is quantum annealing, which is the focus of this conference. And you all know we add a transfer to the Hamiltonian with a energy scale again, the transfer is there. And now this energy scale generates tunneling and superposition. And it works as temperature. So instead of thermal fluctuations, you have quantum fluctuations. Now, again, if this energy scale is bigger than the barrier heights, your wave function will be a superposition of all states. And as you anneal and reduce gamma, the wave function will localize and eventually will localize in the global minimum. This is the, I think, the best visualization of what's happening in quantum annealing. Now, the important question is which one of these algorithms or these two algorithms are faster. But in general, whether the quantum annealing algorithm is faster than other classical algorithms. One of the early papers that attempted to answer this question was Kadoaki Nishimori paper. They simulated Schrodinger dynamics and simulated annealing. And they found that Schrodinger evolution gives you better probability than simulated annealing. Another relevant paper was by Santero, again, very early in 2000. And this time they used quantum Monte Carlo to simulate annealing. And as I mentioned, quantum Monte Carlo does not simulate dynamics of quantum annealing. But because if you have a quasi-static quantum annealing, you follow equilibrium, quantum Monte Carlo also follows equilibrium. There's some similarities. But the reason I'm mentioning this work because of two things. First of all, this plot, they don't focus on best solution. They don't focus on reaching ground state, which is what I want to emphasize at the end of the start. They focus on how the two algorithms, classical annealing and quantum annealing, reduce residual energy as a function of time. And the second thing is you see that quantum annealing have better slope than classical annealing. And I want to come back to this at the end of the talk. For now, ever since 2000, there have been several papers looking at this problem from different angles, adiabatic quantum computation, diabetic quantum computation, many different ways. But what I want to focus is from one particular angle, which is critical phenomena. And again, in the 2000s last decade, it became clear that to those who worked in this field quantum annealing field that phase transitions are the bottlenecks of annealing. And there are two types of phase transitions. First-order phase transition, second-order phase transition. I'm referencing a few pioneering works in this area. But what I'm going to focus on is the second-order phase transition. And I want to, at the end, leave you with an idea that maybe critical dynamics at the second-order phase transition may lead to a quantum advantage, some sort of quantum advantage. But for now, let me introduce to you what is second-order phase transition in simple language. So I want to use the Isingis pin chain, which is the simplest system that has second-order quantum phase transition. And so there's a chain of qubits. Every circle is a qubit. I'm representing up and down arrows to represent eigenfunctions of plus and minus sigma z. Because it's easier to visualize this way. And the Hamiltonian has this particular way. The reason we write it this way, because that is what we can run on the hardware, the hardware, the coefficients gamma and j have a particular time dependence as a function of s, which is the normalized time, t divided by ta, which is annealing time. And at the beginning, gamma is very big. So you have a large transfer speed, but j is very small. And at the end, it's the opposite. So now what happens is, so if j is negative and you have a ferromagnetic coupling, you expect that at the beginning of the annealing, where j is very large, you have paramagnetic phase, which means every qubit could be up or down with equal probability. And at the end of annealing, where the gamma is small, you get a ferromagnetic order. But in the paramagnetic order, basically, the probability of being up and down is equal for every qubit. And then every solution has z to symmetry, which is up and down is equally probable. When you go to the end of annealing, you either get up or down, not both. So therefore, the solution you get does not have z to symmetry. It's broken z to symmetry. But while the Hamiltonian steel has z to symmetry. And this is called spontaneous symmetry breaking. And whenever you have a spontaneous symmetry breaking, you should expect a quantum phase transition, a phase transition. Second order phase transition. Now, but let's walk through this phase transition and see what happens in detail. So on the right, I'm showing where we are in the annealing by this green line. And at the beginning where gamma is large, you are in the paramagnetic phase. And every qubit could be, as I mentioned, up or down. But every qubit could be up and down, independent of the neighbors, their neighbors. I recall, and when you have quantum tunneling, every qubit could tunnel up and down, independent of neighbors. And I call uncorrelated quantum fluctuations. If this is thermal, this is thermal fluctuations. As I anneal further and get closer to the critical point, and critical point is where these two blue and red line cross, the coupling between qubits becomes stronger. And then qubits cannot easily tunnel independently. They basically tunnel together. They fluctuate together. And the size of the region that they fluctuate together defines the correlation length. As we move forward towards the critical point, correlation length grows. And grows further and further. And at the critical point, correlation length grows to the whole size of the system, or infinity, if you have infinite size system. But if you go after the critical point, so here you have this, you still, I'm representing both down and up. So there's no symmetry breaking yet. But as you go further, there's symmetry breaking. So up is preferred, or down, I'm choosing up. But the system is not completely ordered yet. So there are regions of the system, regions of the system that still fluctuate. And the size of the fluctuations again define correlation length. Not the size of the order, the sizes of the fluctuations. And as you go further and further, this size shrinks and eventually you get an order phase. So what you see that the size of the correlation length depends on how close you are to this critical point. And critical point is right when these two lines cross. And indeed they have a very specific dependence, polynomial dependence, with a specific power or exponent called critical exponent. And these are universal exponents, which means the exponent does not depend on the details of Hamiltonian. Many different systems would have very different Hamiltonians and have exactly the same exponents. So another important aspect is time. As the length of correlation then grows, as I mentioned, these qubits that are correlated should fluctuate together, should tunnel together. Or if the thermal fluctuation should thermally fluctuate together. And there's a time scale associated with that. And you expect that the larger the number of the qubits, there's more mass, there's more number of qubits that return together, fluctuate together. The time scale grows and becomes slower and slower. And on the right, I'm showing time, response time, sometimes called relaxation time. But I'm keeping relaxation for thermal relaxation because this is not thermal. This is critical relaxation. So now, you see relaxation time also grows as you get closer to the critical point because correlation length grows. And again, there is a polynomial dependence, parallel dependence between relaxation time, response time, and correlation length. And a specific exponent called dynamic critical exponent. Again, this is also a universal exponent. If the annealing time is finite, there comes a point that the response time becomes larger than the annealing time. And as a result, the system does not have time to relax anymore, to respond to the change anymore. And because of that, correlation length cannot grow fast, larger than some length. And this is called Kibble-Zurich correlation length. It's called Kibble-Zurich mechanism. So if I anneal further than this point, you see that nothing happens. And the system frees. And in the end, you get this domain in the opposite direction of the rest and the kinks in between this domain. And you would expect again that the length of this domain or correlation length, Kibble-Zurich correlation length depend on annealing time because the larger the annealing time, the more you allow the system to grow its correlation length. And again, that's also true. Another power loss of the critical dynamics, critical phenomena is all about this power loss. And this exponent is called Kibble-Zurich exponent. It depends on the previous two exponents that I mentioned before, Z and nu that I mentioned before. You can think of it as a correlation length or you can think of it in terms of the kinks. And the density of kinks is inverse of correlation length. The larger the correlation length, this did the smaller the number of kinks that you would get. So again, the kink density would go as annealing time to the power of one minus, the minus one over mu, which mu is the Kibble-Zurich exponent. Okay, now our goal in this experiment that I'm gonna show you was to test these things. And instead of looking at a chain, we looked at a loop, a ring of qubits. This is 2000 qubits organized in a loop in a ring. And we ran it in this figure is just the one round run at 4.8 nanoseconds and 49 nanoseconds. And the two colors represent up and down, the spin-up and spin-down. And you see that when you have annealing time of 4.8 nanoseconds, there are, they get many, many kinks. So every region that you change color, the kink. And the length of these ordered phases, ordered regions, domains are very short. And on average, if you average over all of these, you get something out of the 15 qubit domains. If you increase the annealing time 10 times to 49 nanoseconds, now you see visually that the domains grow and then you get a lot less kinks. But you see I increased the annealing time by a factor of 10, correlation length increased by a factor of three. And three is a square root of 10 or approximately a square root of 10. And this is not a coincidence. If you go back to the formula I showed you before, that kink density is one over correlation length. And correlation length goes as annealing time to the power of minus one over mu, the kiba-zurich. These exponents z and nu are known for one d-spin-chain, one-to-spin-chain. And they're both one. And if I put one into this equation, I get two and then I get a square root two. So this square root that I showed here is exactly what you would expect using this critical phenomena, knowledge of critical phenomena. But you could do it differently. You could actually, this is the one d-chain, a spin-chain is exactly a solver problem. Like you could solve the problem exactly and this is the formula, you get an analytical solution which not only gives you the exponent in agreement that you would expect from critical phenomena, it also gives you a pre-factor. The pre-factor you would never get from critical phenomena. They only give you an exponent. But it also gives you exact pre-factor. And the pre-factor is a complex function of the schedule. So you remember gamma and j was a schedule, so it's a complex function of gamma and j and their derivatives. And if I put these numbers into this equation and plot adversal experimental data, I get this with no free parameters. So this is the experimental data, the symbols are experimental data at two values of j and different temperatures, these are temperatures. And the green lines are coherent annealing solutions with no free parameter. And they're right on top of each other for both values and it's not just these two, we measured at different values of j and all of them you get the same level of agreement. But you see, they agree up to some point and this agreement is independent of temperature, so it confirms that the system does not depend on thermal environment. The system is independent of the environment which confirms it, which agrees that it is coherent. But as you increase temperature, you see that the number of kinks grow compared to the coherent annealing and the more the temperature, larger temperature, the more deviation you get. And you would expect that because temperature can also create excitations and generate kinks. And for small j, actually it changes the slope of the function. So you would expect as you increase annealing time, the number of kinks go down, you get closer to the ground state, but it goes the opposite. At some point, it goes up. This is called antichipalphotoric behavior. It's just purely due to thermal excitations. And there was actually a paper 2020, which is a Bendo et al, which they did it numerically and they predicted that. Average kink density is not the only thing you can calculate or extract from experimental data. You can go further. You can look at other moments. So this is the first moment, second moment, three moment, and it is the first cumulants, the first cumulant, second cumulant, and third cumulant. And again, theoretically, these are not independent and there's some relation between them. You can prove that. And on the right, again, we have experimental data with theoretical lines with no free parameters. So it's not only the average, which is the top line, agrees, but also the second cumulant and third cumulant also agree pretty well with the experimental data. Another thing you can look at is kink-kink. So this is up to now. It was a single point statistics. Now you can look at two point statistics, kink-kink correlation, which is defined by this correlation link. And now we see that experimental data and exact solution, there's a little bit difference. Although the qualitative are the same, there's a pitch, the experimental result has a peak smaller than exact solution. So in order to get something similar to experiment, we had to add some disorder. So this is the result of a tensor network with disorder, added disorder. And it again, almost qualitatively and quantitatively, I'll leave the experiment later. Now, you could also do something else. You could also turn the system, move the system toward adiabatic limit by reducing the size. If you reduce the size, the gap grows, the minimum gap grows and at some point, the minimum gap becomes large enough that the expected, the average kink becomes exponential, one minus the ground is the proper complex dimension. So it goes from polynomial dependence to exponential dependence. And again, you can use exact solutions and see that average kink, there's an exact solution that average kink has to have this exponent A and the dependence on the length of the chain, with again the coefficient B that we had before. And so on the right, you see that you see the two chains of J 0.95 and minus 0.95 thermokinetic and anti-thermokinetic. And now we are plotting basically N bar or one minus ground state probability versus time, but in log linear, other than instead of log log, which it was before, and having a linear dependence in this plot means exponential dependence. And you see that all of them are like very good agreement with exponential dependence as you expect from Landau's inner formula. And if you extract the exponent from this graph and plot it versus exact solution that is this formula for different values of J, again, with no free parameters, you get these results, so we get very nice agreement. So now, for one D chain, everything seems to be the in agreement, but now you can take a step further and go to larger systems. And this is now, sorry, I wanna go back to the original question, which was residual energy. And the goal of the annealing was that to reduce residual energy, to find a solution that has small energy. And you can also, for one D chain, it's easy, you can calculate the residual energy as a function at link time, because every kink has a cost of two J and you can calculate the number of kinks based on the total number of qubits times the kink density. And we know kink density goes as T to the minus one half, therefore residual energy should go as T to the minus one half, this is just easy. But you can take this, because this is only whole store for one D chain, but you can take it further and use knowledge of critical phenomena again, so those people who are experts in critical phenomena can tell us how this exponent of residual energy can be related to other exponents. And this again, I should acknowledge Anders Sandwich who collaborated with us. And the way this exponent is related is again related to the Z plus one over nu, which was the Kiebel Zurich exponent, but the numerator has a DS, which is a dimension. For classical, it is D, the spatial dimension. For quantum, it's D plus space plus time. So therefore this formula is not just for one D, it's for any system that goes with critical dynamics, second order phase transition, and also not just for classical, also for quantum, but also for classical. If I put this, these numbers from what I know in one D, which is all of them are one, I would get again one half, so this formula agrees very well with this simple intuitive calculation of number of genes. But I can take it further to two D and three D and ask whether I get a larger exponent if I use a two D system, I use a quantum system or a classical system. And if the quantum happens to have a larger X, that means quantum have larger slopes. So I want to go back to the plot that I mentioned at the beginning. I showed at the beginning which was the Santorus paper. So you see here quantum has larger exponent than classical. So if that happens, that here, if that happens also to be the case, yeah, if that happens that we see this X quantum is bigger than X classical, then you have a theoretical support for that behavior. I want to end here and invite you to listen to Andrew King's talk tomorrow who will show the experimental results that answer this question. So with that, I want to end. Hope I convinced you that we can do annealing faster than the coherence time. Can do coherent annealing and study quantum phase transitions. I showed experimental results one D chain which agree very well with exact analytical solutions with no free parameters. Had an argument, high-level argument that critical phenomena may tell you that there should be a quantum advantage in reducing residual energy. I'm emphasizing this is really the reduction of residual energy. It's not reaching the exact solution. In reducing the residual energy compared to classical algorithms, some classical algorithms. And this is therefore not reaching the exact solution is reaching approximate solution. So if your goal is finding approximate solutions, it may tell you that that quantum dynamics can help reaching approximate solution faster than classical. And I'm not claiming quantum speed of a supremacy or anything. So that is that requires significant large amount of work. And one last thing I want to mention is that if you increase coherence, you will you can increase the advantage because you can actually get more advantage of or get larger correlation links come quantum correlation that come from quantum dynamics. And for the last two conclusions, I will again invite you to listen to Andrew King's talk tomorrow. So with that, I wanna end and invite the questions. Thank you for the talk. I think we can proceed immediately with the questions. Hi, thanks for the presentation. Yeah, I want to ask about your first bullet points here. So annealing faster than decoherence rates can enable coherent annealing. So usually decoherence rate is only defined in the weak coupling limit or when your system energy scale is very large. So why do you think that we can remain in this weak coupling limit even in this problem where your system minimum gap should close polynomially with system size? Yes, you can calculate coherence time, decoherence time. The coherence time is some time. And if you go faster than decoherence time, the environment does not have time. Does not have time to respond to your system. So even if the gap is small and you go faster than decoherence time, the environment does not have time. So is this decoherence time related to the single qubit decoherence or is that of the entire... So in this case, this is a good question, actually. So this is a critical dynamics and you have a very narrow critical region. It is important that you be coherent during the critical region because after that the dynamic is frozen. So it is not single qubit coherent. It's actually the whole system should be coherent during this narrow critical region. And that's it. So because this is a small fraction of the whole annealing time, so your system should be coherent in a small fraction of the total annealing time. And so if you see the experimental results, so let me show you the one, the experimental result, you see that, yeah. You see here like up to, this is like 10 nanoseconds up to like, I don't know, 20, 30, 40 nanoseconds. There is no dependence on temperature. So back to my question. The environment, my answer, the environment does not have time to respond to the system because relaxation time is around this time. So up to this annealing time, which is like 30, 40, this means that if you anneal with this annealing time, the system passes this critical region completely. That's why everything agrees very well with no three parameters. And I guess like do you try to study like where is this crossover from coherent to incoherent happens as a function of system size? If you can do that, I would say is this, I won't use coherent or incoherent. I would say, let me see my cursor. So this is a coherent regime. This is quasi-static regime. It's not incoherent. You still have single-chubed coherence. You still have a spectrum I grazed with with what you expect quantum organically. It's not the incoherent is dynamic in coherent. So there is a transition between coherent regime and quasi-static regime. And there's a transition region. And this transition region is the anti-chievous regime. There's a transition region between coherent and quasi-static, which the system starts seeing the environment. Okay, thanks. You're welcome. Yeah, hi, Mohammed. It's Paul Warburton here. Hi, Paul. You seem to be concluding that coherence can help reduce your residual energy, which is good. But you seem to be not explicitly concluding that coherence can increase your ground state probability if you're looking for a ground state solution. Could you comment on that? Is that not the case? Is coherence not a way to ground state solutions? So what I'm talking about is second order phase solution. What I'm saying that the dynamics of second order phase transition does not take you to your ground state. Because after second order phase transition, there could be first order phase transitions. You have to pass those first order phase transitions also coherently, which could be adiabatically, which could actually, there is no reason that adiabatic actually is better than diabetics. It becomes actually a very complicated question. And what I can say right now is second order phase transition takes you to a low energy state. Faster than, I want to leave it to tomorrow actually. I want to leave it to the first tomorrow. I want to say it is possible that if you put these numbers, calculate these critical experiments and if you find that this exponent X is bigger than classical, then there is a chance that you should see tomorrow what are these numbers. Let's just try and repeat Xi's last question, which I thought was a good question, which was, there's a crossover from the coherent regime to the quasi-static regime. Have you studied the dependence of that crossover point on the length of your chain? Another question, Xi? I don't think there's a dependence on the length of the chain. So I mean, this is, Andrew King was involved. I don't think we, it depends on the, it's very small, the chain is very small. Is Andrew there? Yes, I'm here. So can you rephrase the question? So let me just say what I've inferred the question is. Can you see a size dependent crossover between the Kibble Zurich regime and the Landozina regime? And the answer is? The question is Andrew, the question was, can you see crossover to Sermon regime as a function of size, and I don't think there was. There is a size dependent crossover to the exponential scaling of the Landozina regime. Just go here. Yeah, which it has nothing to do with open system. But I don't think that they're, I mean, these two kind of, they play against each other. There's a competition between the thermal excitations and the size in that sense. So you would see, I can show you data, actually. And I'm not sure if I'll have time to show it tomorrow, but I can show it to you today. I have a question, can I ask? Sure. So in slide number 44, there you say that whatever be the size of the king, the energy cost of every king is two times J. Why is it the case? Because every king, so you want to minimize the energy and you want to satisfy every term in the Hamiltonian. But every king means that the neighboring spins are not aligned, which means the J, you get two J instead of J times S times S. Instead of being plus one becomes minus one. So you get two Js difference between whether the king is satisfied or the king, whether the coupling is satisfied or coupling is not satisfied. That way. Thanks. You're welcome. Thanks for the talk. I have a question. So you mentioned that you cannot simulate quantum and quantum Monte Carlo currently. I used to think that if I don't know how to use pattern Monte Carlo, it's due to topological obstructions that you might not always simulate quantum annealing because there were some results by Glenn and Giuseppe on the quantumism chain where they studied the density of the kings and they showed that it goes like one over the square root of the annealing time. So that looks like they are simulating current quantum annealing, but will you call that quasi-static regime or how will you differentiate those results from current quantum annealing? So yeah, actually very good question. Very good question. So for 1D, indeed some of the dynamic because square root time is also diffusion. So if you have diffusion, you'll get also linked going as a square root of time. So some of the algorithms that you, if you run and just blindly look at the scaling, it gives you a square root of time. But none of them can generate all the aspects of, so in this paper, if you go and search, we actually study quantum Monte Carlo, simulated annealing, spin vector Monte Carlo, and none of them can generate all the aspects of the coherent annealing, especially when your chain size is small and you get 10,000 in a transition, a divided quantum composition, none of them can get exactly what we get. So if there is something similar, it should simulate in all regimes. But you are right, it can give you actually a square root of time, even similar to any kind of square root of time. But even a square root of time, it doesn't give you, so here we will be not only claiming we get a square root of time, we are also claiming that the pre-factor it gives is exactly, I mean, very nice agreement pre-factor with exact solutions. And this, you cannot get with those algorithms. So I think it's enough evidence that the system is following coherent annealing and not quantum Monte Carlo dynamics. But thanks for the nice talk. I guess you kind of backed off from this question of a quantum advantage. But I mean, if you ask the question, if you design hardware, would you prefer it to be operating in the classical regime or a quantum regime? If it performs better in the quantum regime, from some respects, that would be a form of advantage. But I guess what makes you hesitate to really kind of say it, you wanna put it within a larger computational structure and rule out all classical competitors or? First of all, I'm not showing the data in more than one day, so I will refer to the Andrew King's talk tomorrow. This was just an opening for his talk. So he will show more data in the 3DSPing glass actually. So that's one thing. What I mentioned was this work and the work of Andrew King tomorrow is all about secondary phase energy. And we cannot claim further than second order phase transition. And second order phase transition does not take you to the ground state. It just takes you to a good low energy solution. That's what I mentioned. If you wanna do the go to exact result, then you have to deal with first order phase transitions as well. And that's a totally different story. Again, hopefully in the future, we have coherent enough system that we can also adjust that. Hi, Mohammed. Vivekendan here. So this is looking at the dynamics of phase transitions. I'm just thinking about physics now. And I'm thinking back, way back, and I'd have to dig in the archive of my records to find the papers on this. If when you look at, when you actually do experiments, say on real materials, think about having that, you're changing the bulk, say the transverse field and macroscopically across your sample. It's in practice, you're not uniform everywhere. And so the phase transition is not exactly the same point everywhere in your system. Do you think you can study that with the setup you have, or have you seen any evidence of that? Because then the actual dynamics you then get for how the transition proceeds is not just the speed of the sweep, but it's also the homogeneity of it. And you can push it from one side to the other if you've got a slight gradient on your magnetic field, if you like. Yes. Indeed, in our annealing system, we can actually tune transverse field per tube and by hand generate this order that you mentioned. We can generate this order in Js and we'll also generate this order in gamma in the tunneling. And but in these systems, because we wanted to simplify things as much as possible to use the knowledge of critical phenomena, Andrew actually spent a lot of time to homogenize the system, so it was exactly opposite. He tried so hard to homogenize the system to get the result that I agree with what you would expect in homogenous systems. But in principle, yes, we can actually make it inhomogeneous and see the results of inhomogeneity. Hey, are there any other question? Is there any other question? I actually have another question. Yes, please. So we have seen that we have been doing some spin coherence state path integral with this quantum annealing, and we saw that when this mini gap appears, then our system goes into a minimally localized phase. In that case, can we perform first order phase transition when our system is in the minimally localized phase by introducing disorder or whatever, maybe anti-cubular mechanism, some kind of, so some way in using that, bring our system closer to the true optimum or to the true ground state. So for me, many body localization and this is entering the spin glass phase is the same thing, but looking at it from different angles. So what I showed was that you enter from paramagnetic into a spin glass phase. And what happens in this, you can think about phase, second order phase transition, if you average or many, many realization of this order, or if you look at an individual system, you can say, oh, this is a many body localization. It's the same thing. I didn't get your question, so you're saying, how we can go through the second, first of the phase transitions after many body. So first order phase transitions are harder beast to deal with, I would say. What has been clear to us and to many researchers that adiabatic evolution is not the answer because the gap is extremely small and adiabatic, if you wanna wait, adiabatic, it becomes very, very slow and especially slow and maybe. So there are suggestions that if you do diabatic evolution through many body, first order phase transition, you may get some proper excitation before reaching the anti-crossing and solve the problem faster. There's also a paper we have which we actually show that temperature also helps because it does the same thing. It actually excites the system to the solution before the anti-crossing. But from the perspective of experiment, we haven't reached the coherence time that we actually get to these points coherently and so I cannot answer you from our experiment. We now have coherence during second order phase transition. Thank you. Okay, we can thank again Professor Aming for his talk.