 In many body quantum machine vector, please 15 plus 5 Thank you. Just a thank you the organizers for organizing this conference and giving me the opportunity to present our work here So I'll be talking about optimal control in many body quantum machines, so is it find the audio Recording in progress So, okay, so I'll be talking about optimal control in many body quantum machines So this is a work that I'm we're currently doing in collaboration with my collaborators Sive engine empathy and azimka what will cut in IIT Bombay and we are kind of like hopefully it will be done in a few more weeks so What I mean by optimal control is say we have a generic system with a some generic Hamiltonian HT Which is can be written as a sum of two parts a H naught and H one which are in general set time independent and H one is something that we cannot control and H naught says the control parameter which he controls through the through this tuning parameter lambda t Now we start at some initial time team at some arbitrary initial state psi in The state psi t of the system at any time t should evolve according to the Schrodinger equation and it depends on lambda t and the final target is we want to reach at some Say some particular final state at final time tf And so the problem that we are looking at is find the optimal form of lambda t subject to certain constraints Which maximizes or minimizes some target function example energy at the final time tf So the problem is very general It is just a very like a basic optimization problem Now there are different kinds of optimization the kind that we'll be looking here is something called crab optimization in which We say that no matter what the target is what the problem is Let's say we write this lambda t tuning parameter as a Fourier series Why because in principle any arbitrary function can be written as a Fourier series So for I mean large enough number of frequencies. So this air beer omega. These are the Fourier coefficients and Fourier frequencies and we take only say finite number of frequencies given by this m and ST is some parameter which is zero at initial and final time and one in between it just makes sure that the Control pulse is a constant at initial and final times So the optimization protocol is We numerically optimize air beer omega r to generate an optimal pulse Which minimizes a target function which can be energy which can be fidelity something Subject to certain constraints for example the strength of control that is what is the maximum amplitudes of air beer omega r So we start from an initial guess pulse air beer omega and find f and then search for a new set of air beer omega r which reduces f and we repeat the process and The expectation is after many iterations. We should reach the optimal set of air beer omega r which minimizes f Now all these are very basic like optimization problem. Everything is fine But the problem comes when you want to optimize a many body system the reason is The Hilbert space dimension goes as exponentially with the system says it is 2 to the power of n So that to find the optimal pulse for a many body system It also can be expected to increase exponentially and it is even more I mean it is a generic problem in any quantum mechanical setting but even more so in optimization because in optimization it Needs many several trial and error basis So the question is how can we find optimal pulse for many body quantum systems? And this is a very important problem also for say quantum technologies when you want to make optimal quantum machines many body quantum machines So the solution we are proposing is let's say we find the optimal pulse for a few body system Which can be done say within a few minutes or a few hours and then if we can use that same optimal pulse for a scaled up system size Now this is of course seems a too good to be a solution to be applicable in all cases However in some cases for example say quantum phase transitions, which are just phase transitions that occur at absolute zero temperature due to quantum fluctuations and It should we think that it should be applicable And there are other things like collective coupling in which say many body system acts like a single Many spins act like a single large spin But today I'll be talking about only about quantum phase transitions and the application of this method Can be done in several quantum machines like say quantum batteries, quantum heat engines But now I'll be talking only about quantum batteries how we can apply this kind of method in to optimize many body quantum batteries Before I go to quantum batteries, let me let me just just I mean maybe most of you already are aware But let me just remind you let these dynamics of quantum phase transitions That depends on something called the adiabatic impulse approximation Which basically says is say we are tuning this parameter lambda t that I introduced before say there is a critical point at lambda is equal to lambda c So if we quench if we change lambda The system evolves adiabatically I have from the critical point while non adiabatic excitations result close to criticality So what it means is the energy gap between the ground state and the first excited state? That is the delta which it goes to zero at lambda is equal to lambda c So there is a because of time energy uncertainty relation. There is an inherent timescale That is one by delta which diverges close to the critical point shown here and the kibble's mechanism which is very much relevant for this kind of systems driven out of equilibrium quantum critical systems driven out of equilibrium So what we say is there is a another timescale associated with the change of Hamiltonian that is delta by delta dot And there is a competition between these two timescales so whichever wins like I from the critical point when this delta is large the system evolves adiabatically but when this second timescale takes over it There are excitations And this lambda one hat which can be or the t hat that is a time at which we get the boundary Can be solved from this equation and which basically gives the kibble zero scaling Which says how much excitations or how much defects are there? Here tau is the rate of quench say and this new d is the dimension new z at the critical exponents So this is the basic this dynamics of quantum phase transitions And we will be using these dynamics to study quantum batteries Now quantum batteries are just batteries made with quantum systems Like classical batteries. They store energy. So the quantifying performance is given by algorithmic Which is the work capacity that is the energy minus energy minimum at the same entropy Which basically means how much work we can extract from a battery And so the question that we ask is what is the optimal way of charging a many-body quantum battery That is which will maximize this argotropy Now the kind of systems that we'll be using is uh, I mean any system can be used But as examples, uh, say we can take an integrable model which can be like this x y model Which is basically there are ferromagnetic interactions along x and y direction And there can be a transverse field along the z direction and this hd is the tuning parameter which we will Control to charge the battery and there is phase transition at h is equal to plus minus one Or we can have a non integrable model, which basically means that the integral model can be written in terms of non-interacting momentum modes But in non integrable model cannot be written like that And here also we can have a ferromagnetic interaction along z and field along x and z And again hd is the tuning parameter and we have phase transition at gamma is equal to one h is equal to zero So these are just two examples of quantum battery models Now as I said, uh, we can charge the battery by changing this hd. Now, let's see how we can charge it in this left Figure this red dashed line says that the hd we are changing linearly We are increasing linearly and again decreasing. This is the green is the quantum critical point quantum phase transition So the orthography that is the energy that is excited Excitation's are shown by this red curve. This is in the integrable model. So it increases and then finally saturates to this value However, if instead we use this crab optimization and optimize this hd Then we get everybody's strange like a pulse this blue curve And then we see that the orthography increases decreases and finally saturates to a value which is much higher like four times almost higher And this is done for hundred cubits We could optimize for hundred cubits because it is an integrable model Which has uh, which basically the number of degrees of freedom just is uh, linearly proportional to the system size So we could do the optimization for hundred However, this cannot be possible for a non-integrable system in which the hillbill space dimension the number of variables increases Exponentially with the system size So we ask can we use the optimal pulse say for 10 cubits which can be done in a matter of minutes or hours To improve the performance of a hundred cubit quantum battery If we can do so it will solve a huge problem. That is how to optimize like Hilbert systems, which has a very large Hilbert space now to The problem can be complicated because the pulse that we are be that we will use is a very complicated pulse It can be the summation of different elements a or b or omega. We don't know these are something that is optimization protocol gives us However, nevertheless, we can still use the kibble's red mechanism to try to answer this question What we can say is this delta t that is the energy gap Is always a function of this uh coefficients and the time and the rate of quench the tau So this kibble's red mechanism is always valid even in this case And we can say that there is this competition between the inherent timescale and the timescale at which we are changing the parameter And from this again, we can get this t hat. Uh, that is the impulse to adiabatic transition line And the important crucial thing is this t hat is independent of system size as well As long as the system size is much larger than the correlation length Close to the criticality this new is the correlation length exponent and the lambda is the distance from the critical point So from there and we can also say that the normalized arguably that is the energy Work capacity of the battery is a function of the probability of excitations and the normalized Hamiltonian so Basically with these kind of calculations what the basic the main message is That normalized algorithmic can be expected to be independent of system size for an arbitrary pulse And the uh implication of this can be understood from these two plots This is for an integrable system the x axis is time and the y axis is the number of spins So as can and this is the argue we are plotting the ergotropy as can be seen For a single time as we keep on increasing the number of spins for a particular single pulse The ergotropic basically remains the same There it is shown by the color So even if we keep on increasing spins the the normalized ergotropy that is the ergotropy divided by the maximum possible ergotropy Uh that normalized ergotropy remains the same And even more uh like remarkably it can be seen in this uh plot for non integrable system This right hand plot which is uh for this non integrable model in which What we did was the x axis is the number of spins So we have say a five spin battery we normal we optimize it We get the optimal pulse for the five spin five spin battery which can be found in a matter of minutes Now we use the same pulse however keep on increasing the size of the battery Now what we see is these green and blue curbs are yellow and Green curbs these are from the critical point What we see is if we just use the five pulse five qubit pulse for Seven eight or ten qubits It very rapidly decreases the ergotropy So what it means is a pulse that we got optimized for five qubits is not good enough for 10 qubits or 15 qubits So scaling up of uh contra machines is a is a problem here However, the picture changes Significantly close to criticality Here this blue curve is for quantum critical battery here again We give the we optimize for a five qubit battery and then keep on increasing the battery size However, the pile still works Good even for 10 15 or up to 17 qubits. So what it shows is how we can optimize a many body quantum machine So just to I mean that to summarize this result this five spin means there are 32 variables So we optimize a problem which contains 32 variables Then we use that solution To optimize a spin chain Optimize a battery which contains 17 spins that is 131 thousand variables So if we do just brute force method this uh many body quantum battery this 17 qubit Will take a few weeks to optimize 17 qubit quantum battery However, because of this quantum critical region, we can optimize it within a matter of minutes by optimizing a five qubit battery and uh So just uh Couple of more things for this integrable case actually the problem might be a little bit easier because in integrable We have uh if for integrable a many body system of size n we have n non-interacting quasi particles So a pulse which is optimal for quantum battery of size n works for n prime much much greater than n as well Even I from quantum critical points because in integrable systems. They are kind of non-interacting modes however, uh, so The conclusion is That say we aim to optimize a quantum critical machine of size n much much greater than one here I have only shown a battery, but this uh theory should work for any arbitrary machine or any arbitrary quantum system Now the question is how to find a power So the way we do it is we find a pulse for a system Or a quantum machine of size n prime which is much much less than n Because this n for a large n the Hilbert space dimension is large. So optimization is not a practical thing However, we can always optimize for n prime much much less than n Then we use the same optimal pulse for this n body quantum machine And this should work in close to quantum critical points And the advantage is significant reduction in optimization time of quantum machines or many body quantum systems And what it shows is quantum phase transitions are highly advantageous for optimal control of many body quantum machines Thank you Thank you victor uh questions The five and uh, excuse me Hello Our liner, sorry. Yes, please Oh, okay, the the five and uh 17th qubit example that you had uh So, um How uh, have you uh, is it possible to actually do the optimization for seven 17 spins? Uh, right. So, uh, not I mean here we used a trial of technique So which uh kind of instead of looking at the whole Hilbert space We just take only a few relevant eigen vectors So it might not be possible to do uh to do optimization Using exact diagonalization, but with some techniques like trial of techniques some approximate optimization can be done But uh, it is for 17 it can be done But we can also say have a 50 qubit machine in which case maybe the those techniques will fail So in that case, we need to optimize for this five spin and then uh use it for this mini spin Oh, no, I was curious how close the The value between uh Using the the parameters from the five spin case Is to the exact value in the seven spin? Oh, yeah, so Right, we are looking at it won't be exact, right? It won't be exact. We are looking at that. Uh, but I mean Yeah, I am not claiming that it is it is exactly matching. However, uh, it is it gives a very good result. That is the thing Oh, thank you I have a basic question about your your setup. So since you we were talking about batteries I was expecting to hear the word work And you're using ergo tropy and instead which seems like Work capacity, but okay, but you're defining it as essentially as the residual energy is what I would call that quantity So what about traditional work? Do you have any any results for what you would get out of it? Let's say a Carnot cycle with this kind of a system, right? So, I mean it is same as traditional work, but in heat engines what happens is the work is like we are getting from external heat Boots, so there can be non-unitury dynamics But here or everything is we are at present. We are looking at unitary dynamics only But is is your goal to? Maximize the ergo tropy minimize it exactly maximize usually in quantum handling say we minimize the residual energy, but here it is the opposite way how to maximize it So what is your initial state? Is it the ground state of Yes, with the ground state the ground state at a h is equal to zero Okay, so what about the excited states for for initial state? Uh, your conclusion I have to think about it because usually most of this quantum critical dynamics are Uh ground state Yeah Okay, I would guess even for excited states it should work because it the main thing that it uses is the correlation length diverges close to critical heating So I don't think it should depend on the initial state But I have to think about it. Thank you other questions The question online Can you open the chat? Okay, hi, victor. Uh, how about the optimization methods? Will they be same as crab that you used here? Uh other optimization methods, uh, I would guess so because the crab is just a technique that we are using to optimize But the main theory that we are using is that the correlation length diverges close to criticality and kibble direct mechanism That should be independent of what optimization scheme we are using Okay, other questions I did not quite get how this scaling is actually working So what how can you use the same method that you're using for five qubits for 50 qubits or more? Close to criticality what happens is that this energy gap goes to zero Between the ground state in the first excited state and from there the time scale diverges because of the time energy uncertainty relation Similarly, one can show that uh, like there is a corresponding length scale which also diverges close to criticality So this is the property of this quantum critical point phase transition. Okay. Good. I think we can move. Let's thank victor again