 And this is the title, Digistice Counter-Diabatic Quantum Algorithms, sorry. Okay, okay, thank you for your introduction and thank you the organizer to give me the opportunity to present our work about this digitalized counter-diabatic driving. I come from the University of Basque Country in Spain. And today I will talk about this counter-diabatic driving, which is one of the main methods in the shortcut to adiabaticity. In the previous talk in this conference, you have heard about this shortcut to adiabaticity to speed up the adiabatic process with the application in the quantum control, in the quantum annealing, also the adiabatic quantum computing. Here I will give some brief introduction of this background and other things, and I will focus on the digital adiabatic quantum computing, and also I apply this particularly counter-diabatic in terms to speed up the quantum algorithm to do the quantum optimization. And my talk will end up with some brief conclusion and also perspective. As you know that, okay, we are living in the next area with the decroherence in the different device. Meanwhile, in the quantum control, we need the precise and fast control of the quantum state or state of preparation in the different application with respect to the quantum information processing, the quantum simulations, and the quantum knowledge. That's why we need the shortcut to adiabaticity, because as we know that the adiabatic conditions allowed us to stay, allowed to evolve the state along the instantaneous eigenstate when the adiabatic condition is fulfilled. But as you know that in the presence of the noise or the decroherence, other things, and then maybe the adiabatic solution will be destroyed by them. That's why we need to speed up the adiabatic process, which means that we want to mimic the adiabatic process, but within the short time. That's why in the 2020, I proposed with my colleague about this concept, particularly applying these things in atomic cooling and the native people develop a lot of method, showing the same concept for instance the microberry, for example, this is the transitionist driving and Masuda developed the fast forward, you have been discussed in the conference and also the concept. Here I will focus on the concept of driving because I will use that in the adiabatic algorithm. As you know that, okay, this kind of adiabatic driving can consider as the adiabatic gauge of potential, which can be useful to cancel the internal force when we accelerate the force, you can read this picture. And based on the various transitionist driving, and we can diagnose Hamiltonian and get distance, and this is what we call the transitionist driving, and in the rice papers they call the counter diabatic driving. I also apply these things for the atomic physics in 2010. And it's good, but somehow when we diagnose the Hamiltonian, which means that we solved the problem. That's why this method could be problematic in the minimally system. And recently, the people develop this counter diabatic driving, for instance, they consider approximate gauge potential, and they minimize the actions and they estimated or calculated approximate gauge potential by using these answers. And inspired by the Fouquette engineering, somehow they can also write down the approximate gauge potential in these commutators, then they can calculate this approximate gauge transformation. And this works, we say that, okay, we can design the counter diabatic driving in minimally system without diagonally Hamiltonian. That's the advantage. Okay, here I just want to list some experiment regarding this counter diabatic driving in the atomic physics in MV center two level system and three level system. And this is for the trap time transport. This is called the item I collaborated with Chinese group and also recently they develop this local counter diabatic driving and to do the experiment implementation in the spin channel. This is the main experiment result. We can connect to the main issue about the adiabatic quantum computing, because you have already know that I just mentioned these things. In principle, we can use the adiabatic process to connect to the initial Hamiltonian to the final Hamiltonian we can include any problems in this way in this adiabatic quantum computing. And normally we need the long time for the state evolution because we want to avoid the transition. And the total learning time depends on the minimal energy gaps, as we know, and also in 2016, the Google teams also develop this idea about the digital adiabatic quantum computing. The main idea is that they write on the time evolution operators and do the totalization and we can digitalize this time evolution operators and we can do the things in the in the superconducting circuit. And so we want to combine these two things, whether we can use the counter diabatic driving in the digitalized adiabatic quantum computing to speed up it because as we know that the adiabatic shortcut can speed up the adiabatic process. So we do that. First of all, we just provide the heuristic example. Let's see, look at the spin one half single spin with time dependent on the magnetic field. And normally we can calculate the city term very easily, and then we can flip the spin from around the x direction initially and to the spin down and we can write on the Hamiltonian for this symbol problem that we just choose the net functions. And then we can write on the total Hamiltonian with the local city term. And for the two level system, this is nothing but the sigma y terms. And then we do the totalization and we do the experiment in the IBM Q stick and we saw the result and the very simple we see that okay, the C is with the city term, somehow we can achieve the high fidelity within the short time. And without the city term normally the adiabatic condition is broken down, then we will not get the high fidelity. So that's the main result. And then then we can go beyond this single simple model and go to the look at the spin. And then we can use the easy spin model by using the local city term and we write on the problem Hamiltonian and we calculate the local city term from the very method, which means that we diagonal Hamiltonian because this is simple. And we can also use the local city term calculated from the evaluation approach. And as I mentioned before, and then we can compare to the result. This is the local city term from the various method. And the first of all, we just checked the two cubits, and then we see the better performance. And then we also go to the five cubits and we see the difference. And we increase the system size, which means that we increase the cubits and the fidelity was not good because this local city term will be more approximated. So, even when we increase the interaction G, and how to deal with that, we go to use the approximated gauge potential by using the nested commutator. For the nested L equals to one, we really see the better solution when we prepare the best state, and then we also go to the GHC state, we increase the number of the cubit by increase the high order nested commutator. This is how we see that the fidelity improve a bit, but of course we can also keep the high order nested commutator to improve that, but it will increase the complicity of the circuit. But yeah, but at a certain point, and the nested commutators improve the fidelity. And this is for sure. By this result, somehow we apply to the different scenarios, for instance, the flexuralization, and also the QAOA, and also different approximation. We also apply the method learning for the QAOA, for the initial parameters. And here I just want to list some few examples about the application. For instance, adiabatic quantum flexuralization, and we can write on the problem like this, and then encode the solution of a minimal problem in the ground state of the Hamiltonian. We can write on the Hamiltonian by introducing the spring operators. Finally, the Hamiltonian can be written this way, but it could be difficult to implement because it will include the four-body interactions. But then we move to the multiplication table method, and we write on these equations after classical processing, and we can get the cost function, and we can write on the Hamiltonian. And we can always calculate the counter diabatic driving by using the definition, and we can see how speed up it. But it can be difficult because it includes the minimal interaction, as in, we can also check the different cases. For instance, we have flexuralization, the small number experiment in the IBM quantum computing, and we see the profile of the CD term, and we see the solution, which is good, and very surprising for the small numbers, we only need the two total steps by using the counter diabatic driving. And of course, when we go to the larger numbers, somehow we can choose the different operator powers, but here, as I said, we can only keep one or two spring interactions because many body interactions will be problematic. So we choose these things and the protocol efficiencies as three parameters, and we can optimize all the fix later, and then we compare the result by using different answers, the local CD term, and the two-body interactions, and then compare to the nested competitors, the first order, and we see the performance with respect to the different successful probability. And we can also check how these CD terms improve the QAOA. And here, the idea is that instead of the 2P parameters, we can introduce the parameters for the counter diabatic driving, and we use the classical optimizer to do the iteration to find the minimal integer. This is the typical here. We introduce the CD term. Of course, as I mentioned earlier, we can also use the return to neural network and to train the initial parameters and to improve the QAOA. And I want to show the result here, and we use, we consider the easing spring model, and we calculate the counter diabatic driving, just keeping the local CD term and the two-body interaction, and we find the solution for the longitudinal field easing model and the transversal easing model. We see that the counter diabatic driving is always better, and we also check how the counter diabatic driving improved the GHC, the entangled state of probability. And here, I just listed the parameters. Of course, we can also apply the listings for the classical optimization program. For instance, we calculate the three regular max-card problem, and also the SP model we really see the result. And because of time, I just take it. If you are interested, you can also check the reference here. And of course, we can also apply listings for the quantum chemistry. For instance, so finding the ground state of the hydra atoms, and we can write on the Hamiltonian, and we can calculate the CD term and to achieve the speed up. And recently, I noticed that in Chinese groupers, and they also implement this idea for finding the ground state for the hydra atoms, and it could be very interesting to extend it to the larger molecular. And here, I just want to give you one slide, how to understand these things. We can write on the spin glass, the Hamiltonian, and write on the CD term. Here we calculated the energy spectrum, and you will see that somehow this CD terms in large a bit of the energy gap, which means that we can speed up it. And in other scenes, how to understand the law of the CD term, and the CD term is not a static state, and somehow it will help the system to converge with the final ground state, as that's the law. Here, I want to conclude my talk. We introduced the counter data driving, which is the means that we can introduce arbitrary interaction, we can implement this non-statistic Hamiltonian interaction. And this result will be consistent with error corrections and can be developed in our algorithm. And here the good result is that by introducing the counter data driving, somehow we can shorten the operation time, and meanwhile we can reduce the gate counter in the quantum circuit. And here, the good advantage is that we use the approximate CD terms, which means that we don't require the knowledge of the Hamiltonian energy spectrum. We don't need to diagnose Hamiltonian first, and we use this adiabatic gauge of potential to estimate the CD term. And the counter data driving, somehow also give us the better answers or reached the answers for to adapt the ground state, which can be applied to the evaluation of EigenSolver or the QAOA. And later, we can also consider some fundamental question, also application with some perspective. For instance, we can optimize the counter data driving by using the circuit learning, by using the reinforcement machine learning. And here, I also consider how to implement this counter diabatic quantum algorithm in this case, the devices, for instance, photonic chips, or the superconductive cubes, or the trap time. For instance, for two-body interaction or even mini-body interaction, it could be difficult for the photonic chips, but somehow we can restrict to the two-body and apply the machine learning to optimize the co-efficiency and to see how we improve the implementation. And there must be some fundamental question as well. For instance, there are always interesting to look at the total errors, adiabatic errors, also the problem, the trade-off between the complicity of each total steps, and with respect to the time with other things. There are a lot of interesting problems. We should understand why this counter diabatic works better and how to improve that. Apart from this result, in our group, we also interested in some machine learning. For instance, we applied this active learning for retrieving the quantum information. We also applied the reinforced machine learning for the qubit control, and in presence of the noise. Also, we applied the recurrent neural work to improve the both side condensations control in the random environment. So if you are interested, you can also check the web page here. We have some results there. Okay, so finally, thank you for your attention. I would like to thank the teams in Biobao, also some collaboration from Shanghai University, Kusulano, and Nalaha. We have the team to develop this digital counter diabatic quantum algorithm. Thank you for your attention. Thanks. We have time for one quick question. Maybe I can ask a question, actually. I was always wondering, when you have to engineer the counter diabatic terms and things that need to be... Are there any results on how difficult that could be to find the right terms to speed up in general? That's a good question. Historically, this counter diabatic driving was proposed by Rice in the United States. They applied these things in the three-level system to speed up the stirrup. Later, Michael Berry formulated these things that he called transitionist driving. But in that time, one has to diagnose Hamiltonian. As you know, it is difficult to find the expression. Of course, you can always diagnose Hamiltonian, but for the adiabatic quantum computing, if you diagnose it, you have already solved the problem. You have solved the problem. You have to use these things to speed up. But as I said that recently, people produced in the PRL and penis papers, somehow they followed the concept from the classical mechanics. They called diabatic gauge potential. Somehow you can use this counter diabatic to compensate this internal force was when you speed up. Then with this concept, they assume this gauge potential as the answers and you don't need to find the exact CD term and you don't need to diagnose Hamiltonian. Benefits from these estimated things, somehow we can apply these things in the adiabatic quantum computing. We don't need to diagnose. Even we don't need the exact expression and we can use the answers from the diabatic gauge potential and optimize the coefficients by using different methods. As I said that you can use machine learning to do that. This is what we are doing now. Thank you. Let's thank the speaker again and let's go for the coffee break.