 with Jean-Lucca Pazzarelli from Naples. Jean-Lucca, are you online? Yes, hi. OK, hi. So try to start sharing. Yes. And then I will record you. So it's 15 plus five minutes question, OK? OK. Just a second. OK, so we'll talk about variational shortcuts to a debaticity for limb-bloodium dynamics. Jean-Lucca, please. OK, go. Hi, everyone. I'm Jean-Lucca. And in this talk, I'm going to tell you about some of our recent results concerning variational shortcuts to a debaticity for open-quanning systems. But before I do that, I would like to start with a quick overview of shortcuts to a debaticity, the particular county debatical driving for closed systems to then move on to open systems. So as you well know, a debatical optimization is a technique for solving optimization problems. And the idea is, of course, to encode the solution as the ground state of an easing Hamiltonian. And in order to reach this solution, you start by initializing your system in the ground state of a non-interacting transverse field Hamiltonian, which in this picture corresponds to a simple enough cost function whose minimum is easy to find. Then by changing these two parameters, A and B, you slowly deform this cost function so as to match the one that you want to optimize. And if you do this procedure slowly, then there is a high probability that the starting minimum will evolve in the correct global minimum of the target cost function. But if you move too quickly, then there is a high probability that the system will get excited, and thus the performance of your algorithm will decrease. The adiabatic criterion tells you that in order to avoid this problem, the time of evolution must be much longer than the inverse of the minimum of the gap between the ground state and the first excited state. Otherwise, the adiabatic Landau-Zener transitions will take place and degrade the performance of your algorithm. So on the one hand, you want to move slowly in order to keep these Landau-Zener transitions under control. But on the other hand, if you move too slowly, then you will leave your system more prone to errors due to the coherence. And one of the solutions to this problem has been shortcuts for diabeticity and counter-diabatic driving. The basic idea is to redefine the original Hamiltonian to include the term that can completely suppress diabetic transitions throughout the evolution, even though the adiabatic criterion is violated. So what's the idea precisely? If you write down the Schrodinger equation in your laboratory frame, then the generator of the dynamics is the Hamiltonian H0 that I wrote in the first slide. However, you can also find a different frame of reference, for example, the one corresponding to the instantaneous eigenvectors of the Hamiltonian H0. If you do that, and if you write down the Schrodinger equation here, you realize that the generator of devolution now has two terms. One that is diagonal in the energy eigenbasis and cannot induce any diabetic transitions. And another one that is non-diagonal instead and is related to the time derivative of the transformations that brings you to this moving frame. This term is responsible for diabetic transitions. So what you can do is you can do some sort of reverse engineering in which you redefine the original Hamiltonian H0, so as to include this term in blue, so that when you go to the moving frame, these two terms cancel out exactly and you are left with just the adiabatic Hamiltonian, which as I mentioned is diagonal in the energy eigenbasis and cannot induce transitions no matter how fast your evolution is. This is counter-diabatic driving. And what's nice is that of this operator, we even have an exact analytical expression that is shown here in the slide. But the problem is that this form is practically useless in all but a very selected number of cases due to the fact that this counter-diabatic operator is highly known local. It requires that you know the spectrum of the Hamiltonian if you want to compute it and is also ill-defined around quantum critical points where these denominators can go even exponentially to zero as a function of the system size. And those are the points where you would like to use this counter-diabatic driving operator the most, of course. So in order to solve these issues, in recent years there has been a lot of research towards building approximate counter-diabatic operators that do not suffer these limitations. And one step forward came from self-sampled Kovnikov who realized that the exact counter-diabatic operator satisfies this exact operator equation here and that the solution to this equation is also the minimum of this cost function here. So rather than solving this complicated operator equation, you can make an answer to some of the form of the counter-diabatic operator or rather on the form of this adiabatic gauge potential A and then you can minimize this cost function with respect to the coefficients of this expansion. In this way, you can enforce some constraints, for example, locality constraints or some other constraints related to the limitations of your experimental apparatus. And you can build approximations that prove to be very successful also for many body quantum systems. This all works in a closed system setting. So for example, in a recent paper of ours, we studied the P-spin model with P equal to three, which is described by this Hamiltonian here. It is a fully connected easing spin system where Q-bit interacts with each other via infinite range of ferromagnetic interactions, three body interactions in particular. And we tested two different answers for this counter-diabatic driving approach, one that reflects the general features of the original model that is a tree local and that's with full connectivity and another one that is far more general. It is a truncated series expansion of the true counter-diabatic operator that is known to work well for many body quantum systems. And as you can see here from the scaling of the ground state fidelity at the end of a short time evolution as a function of the system size, the inclusion of these approximate counter-diabatic operators optimized via this variational approach is able to significantly improve the scaling of the fidelity with respect to the case without any counter-diabatic correction and also the instantaneous ground state probability remains always larger compared to the case without counter-diabatic corrections. So motivated by this success of the variational approach in building unitary counter-diabatic operators, we started wondering whether was it possible to generalize this approach also to include open quantum systems instead. And the difference is, of course, that now the dynamical equation is no longer a Schrodinger equation, but it's rather a Limblad equation in this form in which you have the Limbladian super operator acting on the systems density matrix. Actually, from what follows, it will be more convenient for me if we switch to this so-called super operator representation in which density matrices are reshaped as coherence vectors by putting one column after the other and the Limbladian super operator becomes a super matrix acting on this extended Hilbert space. If we do that, we realize that the Limblad equation is very similar to your typical Schrodinger equation. The only difference is the fact that the Limbladian super matrix is not hermitian in general. And so the spectral theorem doesn't hold and you cannot always diagonalize it. However, linear algebra tells you that it is always possible to find a similarity transformation that is an invertible transformation that brings this Limbladian super matrix into the so-called Jordan canonical form, which is a block diagonal structure where each block is associated to a given complex eigenvalue of the Limbladian super matrix. So each Jordan block is shown here. On the diagonal, you have one of the eigenvalues and on the upper diagonal, you have this sequence of ones. These eigenvalues will have no positive real parts. They will decay over time and the zero eigenvalue will correspond to the instantaneous steady state for that given Limbladian. And why am I telling all of this technical stuff? Well, the reason is that adiabaticity in open quantum systems has a very precise meaning that is related to this Jordan form. In particular, we say that a dynamic equation and an anti-evolution, sorry, for an open system is adiabatic if the dynamics of Jordan blocks associated with distinct and non-crossing eigenvalues of the Limbladian super matrix are decoupled from each other. In a similar fashion to what happens in close quantum systems, where we say that dynamics are adiabatic if the different eigenspaces of the Hamiltonian evolve independently of each other. So starting from this definition of adiabaticity, we could notice some similarities between closed and open systems. For example, also for open systems, you can define a moving frame by means of this similarity transformation. And if you write down your Limbladian equation here, you realize that you have two terms. One of them is diagonally in the Jordan block index so it cannot induce any adiabatic transitions. The other one instead is responsible for the adiabatic transitions. And if you expand it onto the left and right was an eigenbasis of eigenvectors of the Limbladian super matrix, you realize that this form is very similar to the counter adiabatic operator for closed quantum systems with the same pros and the same cons. So it is a non-local super operator. You need to know the spectrum of the Limbladian if you want to compute it. It is ill-defined whenever these Liouvilleian gaps are small and so on and so forth. So due to the similarity between the equations, it is natural, or at least I hope so, that you expect that you can generalize the existing variational approach for closed quantum systems also to open quantum systems described by these time-local Limbladian master equations. Actually, there are some small technical differences in the algebra here and there. They are hidden in the derivation. Some of them are shown, some of them are not. But if you're interested in these technical details, you can find out more in our paper here. Instead, what I want to do in what remains of this talk is, first of all, I would like to stress the fact that our approach is able to generalize the existing one in the sense that whenever the Limbladian super operator is a unitary map, we find the exact same equations that were derived by cell sample comical starting from a closed system evolution. So our method is slightly more general. And I would like instead to focus on an application in the remaining part of my talk that is a single qubit in interaction with anomic environment. So we consider this kind of Hamiltonian here. It's a single qubit and we know that in a closed system setting, we would be able to compute the exact counter-diabatic operator in this case. It is proportional to sigma-wise. Actually, one of the few cases where you can actually compute the exact counter-diabatic operator in a closed form. And we decided to couple this single qubit to a collection of harmonic oscillators and to perform the standard Born-Marcoven rotating wave approximations to derive this weak coupling adiabatic master equation here. So the Limbladian we are studying is here and I would like to stress the fact that both the unitary part and the dissipative part depend on time. So we expect also a counter-diabatic super operator to depend on time via both the unitary part and the dissipative part. However, what we found out is that by just controlling the unitary part of our ansatz for the counter-diabatic super operator, we are able to significantly improve open system adiabaticity in the sense that I was mentioning before. In particular, the instantaneous steady state of this Limbladian is the thermal state at a given inverse temperature, beta, that is the equilibrium temperature of the environment for each of the Hamiltonian's H0 of T. So suppose we initialize our qubit in the thermal state at the beginning of the evolution. Are we then able to follow this Jordan block throughout the evolution thanks to this correction? Well, it turns out that if we consider just a unitary ansatz for our counter-diabatic super operator and we optimize this coefficient by using our open system variational equation, we are able to substantially improve the fidelity with the thermal state throughout the evolution compared to the case without counter-diabatic corrections. So for example, in this case, we are considering a very low temperature environment for which the relaxation time scale in the absence of corrections will be of 200 in these units. Instead, we are considering a very fast dynamics so we're not giving the system enough time to thermalize. However, if we control the unitary part of the evolution, which is of course more convenient than having to engineer the whole dissipative part of these ansatz, we are able to get fidelities that are very close to one. Of course, the agreement is not perfect because we are not considering the full expressivity of this counter-diabatic super operator, but you surely agree that this high fidelity is very nice in this case. And also if we increase the temperature, of course, we decrease the relaxation time scale and this effect is less noticeable, as you can see from the Y scale on the Y axis, but it's still here at short times. In fact, if we study the final infidelity as a function of the annealing time, we see that at long times, the addition of this counter-diabatic correction does not yield any significant improvement, but this also happens in close quantum systems where your dynamics are already adiabatic. But at short times, you can see that we can get improvements even by eight orders of magnitude compared to the uncorrected case. And also this is a very simple system, but in the paper we also study once again the P-SPIN model and we find out basically the same outcome. So we are able to control, by just controlling the unitary part of the evolution, we are able to significantly improve the overlap with the Jordan block and suppress the abatic transitions outside of this block. So with that, I have concluded this presentation. Thank you all for your attention and if you have any questions, you can either read our paper or ask them here. Thank you. Okay, thank you. Thank you, Gianluca. Very nice talk. A few questions. Okay, thank you very much for the talk. And I was just curious, if you don't take into account this coupling with the environment and you compute naively the driving for the Hamiltonian you have, how does this compare with when you take into account the coupling with the environment? I mean, how bad is it, it is to just use the standard the contralibratic driving without the environment? Well, in that case, you are doing something different because in that case, you are just optimizing the unitary part and then you are adding dissipation on top. So you are not taking into account the dissipation in your original optimization. So I would expect different performance in this case. Instead in our approach, you optimize all together. And so basically if your goal is to suppress diabetic transitions in the sense that I was mentioning before, not in the sense of a diabetic one computation. So not between energy and states, but between the different Jordan blocks of the composition of the Linglidian super matrix. I would expect that you would get very different performance if you do in these two different ways. Other questions? Hi, thanks for the very nice talk. So can you go back to the slide where you define the adiabatic hypothesis for, yeah. So does this adiabaticity hypothesis fail when the Linglidian is not diagonalizable or these are two different things? No, no, no, no, no. Even when the Linglidian is not diagonalizable, this is the same definition of adiabaticity because when your Linglidian is not diagonalizable, you simply have that these Jordan blocks are not one-dimensional. So these are actually blocks due to the fact that the geometric multiplicity of each eigenvalue is different than its algebraic multiplicity. And so basically in each block you have each eigenvalue repeated a number of times that is related to this geometric multiplicity. And then adiabaticity refers to suppressing this coupling between these non-one-dimensional Jordan blocks. But it's actually the same thing. And the equations here take into account the fact that the Linglidian super matrix cannot always be diagonalizable. Because for example, in this case, this diagonal part that vanishes in the case of a closed system evolution. In this case, it does not always vanish due to the fact that these blocks can be more than one-dimensional. Hi. So in the open system case, there is a complication which is that the Jordan blocks actually become time dependent when you properly derive the Lillenblatt equation. And did you take that into account when you showed this nice result with high fidelity when you just control the unitary part? Yes, yes. This is a general, and this is an old numerical. So all time dependencies, et cetera, are taken into account. And it turns out that for this choice of parameters, it is actually sufficient to control just the unitary part. And also when we start in the P-spin model with the same parameters, we find out basically the same results. I don't know if it's a general rule because I expect that if dissipation plays a more dominant role in the dynamics, this will no longer be the case. But if you think about this simple case or the single qubit, a thermal state is just defined by the relative balance between the populations in the final basis. And so I guess that in this case, this method works because you are just, you are able to control these relative populations by just acting on the unitary part. You just have to induce the correct diabetic transitions to remain in the correct thermal state throughout the evolution. But this is just a guess. And also in this paper, we are proposing a method. And it's just an accident that for this choice of this lean laden by controlling the unitary part, we are able to improve our fidelities. But our main result is the method actually, which is far more general. Okay. Thank you, Gianluca. I think we can thank the speaker again. And we... Okay. Move. We move to the...