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hi good morning everybody.
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Giuseppe Carleo: Let me start sharing my screen.
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Giuseppe Carleo: Okay.
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Giuseppe Carleo: My my name is Giuseppe Galileo i'm a professor and assistant professor at be felt in in Lausanne, Switzerland and today I will I will tell you about some reason, probably that we've done.
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Giuseppe Carleo: In the context of developing quantum algorithms to to simulate quantum physical systems in the framework of vibrational methods as I will explain in in the fall.
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Giuseppe Carleo: So, since it's not many of us today and not sure about your background feel free to interrupt me at any time, if you have questions or.
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Giuseppe Carleo: If there are some potentially language barriers, we can encounter during my my presentation so first of all, as I was mentioning my.
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Giuseppe Carleo: The kind of work that are present today is concerned with vibrational methods right so vibrational levels are a very.
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Giuseppe Carleo: Broad if you want a family of methods that appear in several areas of mathematics, physics or chemistry or etc so in the in the context of physics, of what that would I mean what I mean generally interested in.
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Giuseppe Carleo: is in, as you know, in in studying in concert quantum quantum state right, so a generic quantum state side.
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Giuseppe Carleo: believes in a hyperspace or in a vector space actually if you have a system of cubits so it's a linear superposition of let's say to the end possible mistakes like for inescapable cubits so in general, this is a very complex object and.
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Giuseppe Carleo: Even you know preparing the most general if you want state in this space is is a challenging task for even if you have a quantum computer, because in general.
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Giuseppe Carleo: it's if you use if you have a limited set of resources, for example, you interest it's in general, hard to target when arbitrary constant however during is.
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Giuseppe Carleo: Important simplification in his in his in his problem and it's the fact that, in general, we're not using in describing if you want all the possible states, but all a very specific subset if you want, so this subset subset here.
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Giuseppe Carleo: Of quantum states that are physical state, so this physical states are those that, for example.
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Giuseppe Carleo: satisfied the sharing your equation, which is what i've written here the you know the statue is showing an equation, or you know the time diminishing an equation for some initial state vector.
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Giuseppe Carleo: or some you know the super diversion is the Linda buckmaster equation so In essence, we are not used to describe you know arbitrary quantum states, but only a subset of right.
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Giuseppe Carleo: So that's where essentially vibrational representations and they're the game, because he in international what we do in this in this family methods is that we, Parliament, rise.
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Giuseppe Carleo: quantum state let's say that we may depend on some parameter set of parameters data that can be very large politically and then we we have we don't have a manifold if you aren't of quantum states that's our normal that we which we try to span as much as possible, these.
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Giuseppe Carleo: set of physical state, so this is the idea that we parameter is our classical state our quantum state.
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Giuseppe Carleo: And then there are you know, historically, if you want to classical vibrational privatizations of these states, I will mostly not described us today.
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Giuseppe Carleo: Instead we focus on quantum parameter ization if you want the quantum states which is this idea of using time tracks quantum state to state that i'm sure you've seen that already doing this.
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Giuseppe Carleo: Now, one of the main interest of using if you want the parameter is quantum circuit, so this family operational States is that they are believed.
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Giuseppe Carleo: This say conjecture not something that has been proven, but they are widely believed to be more expressive if you want, then the equivalent classical recommendations that are depicted here with with in this yellow block okay.
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Giuseppe Carleo: This is again a conjecture on the representational power if you want to experiment at one stage is that that is potentially useful, for example, to address.
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Giuseppe Carleo: Some challenging problems that we cannot address efficiently with classical vibrational recommendations of quantum state, and I have to say about the disclaimer is that at this point.
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Giuseppe Carleo: or classical representations that we have a quantum states at this point in time and much more powerful than what we can practical use those parameters corner secrets that we can use all natural on the matter, but again, this is a point in time, and maybe the situation we change it.
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Giuseppe Carleo: Now, the first.
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Giuseppe Carleo: algorithm that I would like to discuss is what we call it the quantum natural gradient so it's a it's a it's described in the newspaper that that I did, in collaboration with James talks at flat iron is used in New York and.
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Giuseppe Carleo: As I can kill or wire in exile to startup company quantum computing in CAP.
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Giuseppe Carleo: So the main idea again the domain setup if you want to dissolve This approach is that we have we're dealing with some operational quantum modern so as you've seen also during this workshop, you have a unitary which is.
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Giuseppe Carleo: A set of elementary operations, you can apply on your quantum computer these will depend in general, a lot of parameters, you can figure this blob here will depend on the wall of status total is a backlog of.
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Giuseppe Carleo: operational problems, so you prefer generic one state that depends on these parameters and then you can typically do a measurement and all documents right.
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Giuseppe Carleo: So this is essentially a paradise quantum circuit and typically what we do in his device settings that we have a loss function, for example, this can be damaged your funnel system, but it could be.
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Giuseppe Carleo: For example, some function that you would like to minimize so you might use this quantum device to minimize some classical function, why not.
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Giuseppe Carleo: And then, what we do is that this step here we estimate that the loss function, we measurements on the hardware and then also it's gradient will respect you because we respect to.
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Giuseppe Carleo: The virtual pollinators and we try to minimize this was fun in some eternity minimization scheme where it step and plus one we update our parameters.
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Giuseppe Carleo: And here are written the most general or one of the most general updates, you might do where essentially the next set of parameters depends on the old set of parameters on last functional it's gradient unsolved.
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Giuseppe Carleo: But in practice, I mean one of the most widely used up bit schemes is the great incentive scheme, if you want, where the next set of parameters updated with them depleted of your loss function again last can be, and if you're interested in.
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Giuseppe Carleo: A nice way of seeing the recent update so that is depicted here also these simple.
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Giuseppe Carleo: sketch of the function is also to see these update as an iterative approach, where you minimize the cost function at each step but.
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Giuseppe Carleo: Where this comes funnel is essentially a problem between this data and mindset and times the direction you might degraded plus some extra cost function.
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Giuseppe Carleo: Which is proportional to edit which we call in Jericho and machine learning on the learning rate.
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Giuseppe Carleo: So it's pretty embarrassed and also to these, and then the the square distances between that and the times of the parameters at nw step, so you can show that these two updates are identical if you perform this minimization it's straightforward.
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Giuseppe Carleo: But this this form gives us an insight on the fact that the gradient descent is actually moving you towards directional degraded with.
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Giuseppe Carleo: An additional cost rate that you are not moving too far, if you want to norm from the previous set of parameters, otherwise this part of this was fun to be large.
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Giuseppe Carleo: And that there is another generalization if you want all these all these great incentive is very.
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Giuseppe Carleo: interesting and it's very popular in, for example in in machine learning, but also in in other branches or additional optimization, for example in the in the in the classical optimization of quantum.
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Giuseppe Carleo: Many body states that is that goes under the name of niche market in descent, so this is a generalization of the standard in the sense.
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Giuseppe Carleo: in the sense that that what you do is that you can't stir again you go in this picture, where you have this direction given by as before, by the gradient.
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Giuseppe Carleo: However, you add the penalty, that is not given by the Elf to norm, if you want them in Euclidean space between my parameters, but you can in general use another inability norm between your your parameters.
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Giuseppe Carleo: That i've.
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Giuseppe Carleo: Written here as a G she wants to G now with my an arbitrary number them using to measure distance, if you want among these parameter space.
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Giuseppe Carleo: And the main idea of this is that if you use the right norm and then you can show actually the the updates will be modified in this way, if you perform this optimization so you will have the standard, if you want to create an update times the inverse of this metric okay this g.
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Giuseppe Carleo: Now, and the main idea of why you would like to do this, is that typically if you can't just distances in terms of Euclidean distance of the other parameters.
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Giuseppe Carleo: This is not the optimal thing in the sense that you might, for example, change some all your virtual practice a lot.
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Giuseppe Carleo: But your loss function might change, not as much right, so this is telling you that, indeed, what you should look at is some other metric.
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Giuseppe Carleo: So this is somehow look at here when when we modify modify the metric so we change the variables in the sense.
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Giuseppe Carleo: That is more sensitive, if you want to your parameter ization that you're using to represent, for example, your operational for.
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Giuseppe Carleo: So just to be more concrete if we if you go to the case of quantum states so again here we are attracting quantum states with a vector parameters.
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Giuseppe Carleo: Then the natural measure of distance, if you want them longer quantum quantum states is the fidelity right, so I have say state with with parameter set of zero.
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Giuseppe Carleo: And the state with parameters that one the fidelity gives you tells us our goals are in in way financial space, if you want to state okay.
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Giuseppe Carleo: And is in us as a medic tensor if you want to dis je metrics that you see here, and you can show that these medic tensor is given by this, the former here it's the real part of what is called the quantum dramatic tension.
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Giuseppe Carleo: This can be shown and it's the action if you want all of these overlap or four.
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Giuseppe Carleo: Okay, so this is the natural metric if you want so that's why also people call this object these beings nature graded it's really the natural metric that that is in use by the distance.
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Giuseppe Carleo: Among to quantum states, so this will that will move us in directions were indeed there is a it makes sense to change parameters, because they were willing use a meaningful rotation and meaningful change or send one states, not only in the political space.
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Giuseppe Carleo: And I mean just to give you an example of why these things are nice, for example, if you.
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Giuseppe Carleo: Typically, if you do a rescaling, this is just a fake problem but imagine that I do have this loss function i'm trying to minimize it, so this is the medium or minus function, I start from here.
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Giuseppe Carleo: So if I follow the standard again and he said that we follow this broker valley was a natural gradient before these other cool.
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Giuseppe Carleo: But now imagine that, for some reason I risky of the parameters in my apartment ization.
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Giuseppe Carleo: Then you can see that, essentially, the natural good in the sense to convert in the same number of steps.
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Giuseppe Carleo: The greatest and instead we try to will zigzag a bit because it's intrinsically sensitive, for example, to rescaling in the in the parameters and it is more dramatic.
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Giuseppe Carleo: Instead, if if you were to buy or pioneers let's say by 20 where you see that immediately.
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Giuseppe Carleo: The greatness and here just stops converging, whereas the natural great and it's completely insensitive to these to these rescaling because.
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Giuseppe Carleo: Indeed, as I was mentioning we have just content right metric that is not sensitive to this kind of trivial, if you want to rescaling of the operational.
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Giuseppe Carleo: This is just to give you a quick example of why you might you might want to do this kind of things, in practice, these are these are usable and implementable in the equation to murder.
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Giuseppe Carleo: that's what we showed in this in this work, and this is, I mean as simple as small relational algorithm that quantum.
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Giuseppe Carleo: quantum city that we use to to demonstrate these in our paper so essentially we have this sequence of gates that we measure the distributed y direction and we try to minimize the discourse function.
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Giuseppe Carleo: And you can see that you need as a function of the of the optimization step typically this quantum natural goodness and converts much faster than the money left on the business.
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Giuseppe Carleo: Now there is one issue with what I presented before and it's essentially the cost of performing these are these if you want a national good in the Center updates.
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Giuseppe Carleo: That we've addressed in a in a in a more recent paper in collaboration with IBM IBM Zurich.
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Giuseppe Carleo: And essentially This brings me to this notion of fast quantum metal grading so it's a way to implement this this method also more efficient if you want on on quantum monitor.
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Giuseppe Carleo: And the domain issue with the with the process I described before is that if you want to estimate this metric tense or G, which is also known as the quantum official information metrics.
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Giuseppe Carleo: You need to evaluate essentially you need to perform on your quantum order number of measurements, which is quadratic.
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Giuseppe Carleo: In the number of variation of parameters, essentially because we have to estimate all the years of these metrics even though it's symmetric but but it's on the order of the square of the number of additional parameters.
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Giuseppe Carleo: So even if you're one of the reasons ways of efficient ways of estimating these entries, which is based on this shift up they told.
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Giuseppe Carleo: You I mean Essentially, this would cost you at least for functional evaluations per entry, so this is really an expensive thing if you have, for example, other than 100 or thousand operational parameters.
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Giuseppe Carleo: The cost of these would be extremely high in terms of measurements, you have to perform on the on.
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Giuseppe Carleo: So that's why we want an efficient operation a lot and by efficient, I mean that we would like to have a process cases that linearly with the number of operational parameters, instead of erotically as this, if you want the regional.
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Giuseppe Carleo: approach, this is, of course, something that is routinely down, for example in the machine learning applications were all methods are linear in the number operational.
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Giuseppe Carleo: And indeed, you can see that if we go back to the original problem if you factor in the cost of evaluating this functions because, again, the step of.
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Giuseppe Carleo: Standard quantum national graded is more expensive, you can see that, indeed, the the advantage of having that is.
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Giuseppe Carleo: Better convergence is washed out because the number of functions evaluation, so if you're the runtime would be essentially the same of the of the standard billing descent, at least for the special case of course this is not obviously.
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Giuseppe Carleo: So we'd like to try to try to go to to remove this limitation and somehow we used to cost us over the fundamental goods.
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Giuseppe Carleo: So the idea is to use what is what is known as soon tennis perturbation stochastic approximation as PSA is the acronym it's a technique widely used in nine be curious application in a quantum chemistry, for example in on quantum quantum.
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Giuseppe Carleo: And that sort So the idea is that, instead of, for example, if you wanted to compute the gradient of a certain fashion effort, you can you can form some.
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Giuseppe Carleo: stochastic approximations of debate in using a stochastic in the approaches, so that is that we've to financial evaluations here you have a delta, so this is a displacement with.
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Giuseppe Carleo: A vector around number vector of plus or minus one that is that has the same size if of course of your own vector of parameters, then what you do is that, essentially, you can show that these are.
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Giuseppe Carleo: Just by computing the difference if you always find a difference, but on on it on a randomized directions, as you can see here will approach, an estimate of the will approach on average.
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Giuseppe Carleo: The gradient of your function F effects of course is moving.
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Giuseppe Carleo: Indeed, you can show that other some reasonable technical conditional on the to those fans are not a non on your paramedic function.
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Giuseppe Carleo: These SPS as an unbiased estimate oh degrading so do we convert to the radiant and also, if you use it eternally in a minimization approach.
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Giuseppe Carleo: It will have guaranteed conversions and and, of course, the interest is that to do this, you don't need to complete the actual get in but just perform to only to function which.
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Giuseppe Carleo: Very step of these of these optimization now the interesting thing is that these can be of course generalize also to higher order derivatives.
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Giuseppe Carleo: So, for example, you can generate generalize these also for to the Hessian so let's say that I want additional my function F, then.
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Giuseppe Carleo: I mean, this is just a sketch, but without going too much into the details, the idea is that you can perform twice this spss estimate so because you're doing web show twice.
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Giuseppe Carleo: And you can show that you can form essentially an estimate randomized estimate of your SEM just using to now random directions that they want us to these are still burn only if you want distributed vectors and the cost of computing possession is.
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Giuseppe Carleo: is equivalent to computing for functional evaluation, instead of two as interviews.
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Giuseppe Carleo: Now why Why am I talking about as students well simply well, I will tell you these in a second sorry but before I wanted to tell you that.
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Giuseppe Carleo: There are some tricks of the trade if you want to to improve all so these SPC estimate of the lesson.
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Giuseppe Carleo: So, for example, you might want to take more direction, instead of just one so that's something that.
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Giuseppe Carleo: is useful, so if you take more than just two directions here, so if you ever if you under estimate of the so no more.
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Giuseppe Carleo: Random vectors you have a systematic multicultural style if you're an improvement of your SEM the matrix.
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Giuseppe Carleo: Then there are other tricks that you can play, for example, you doing your optimization you can you can form a running average if you're for example of your or your SEM starting from the Bu step.
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Giuseppe Carleo: Or you can ensure that also that this estimated, we have here is positive definite is not true, in general, if you look at these expressions.
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Giuseppe Carleo: So, for example, one way we use it to make these estimates positive definite is just essential to take this medics square it and then take the square root.
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Giuseppe Carleo: So this is something that will give you, by definition, a a positive definitely estimate top of the audience and while using this SPC.
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Giuseppe Carleo: So, but again, the reason why we want to use this is, if you want to do this version of the es una is that, indeed, the dis medical dental that introduced before it's nothing button.
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Giuseppe Carleo: Second derivative so in essence of the overlap function, so we can rewrite this as a as an axiom of my.
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Giuseppe Carleo: of my of the overlap so so that by just essential before evaluations of the overlap at each step my optimization I can.
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Giuseppe Carleo: I can obtain an estimate of these designs some of these medications so that he announced that I changed in my in my if you wanted monetization look this step here, where I insert.
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Giuseppe Carleo: The spss so first order SPF spss estimates for the gradient of those function and second order spss estimates.
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Giuseppe Carleo: Of the essence of this cost function that it can be estimated efficient model, and then I use these by essential to compute I invert this these metrics to compute.
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Giuseppe Carleo: The these updates of the parameters and again the cost of this step now is much reduced because these will be essentially six functional motivations for step, and this can be done efficiently, also in terms of quantum measurement.
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Giuseppe Carleo: Okay, so so, and if you go back to the small virtualization example that we saw before now, with this improved approach you see that, indeed, there is a tendency because.
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Giuseppe Carleo: We converge to to this.
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Giuseppe Carleo: To this cost function in, on average, you can more or less on average, as you can see here and also there is an improvement in the number of function calls and which is much cheaper than the new regional recruiting.
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Giuseppe Carleo: and other important to improvement is that we can access much larger secrets, so this is one specific example that.
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Giuseppe Carleo: Sri Lanka song choose, but when we have essentially under parameter secret so just the old if you want to change, he would take would require to perform on your quantum Margaret.
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Giuseppe Carleo: Media measurements that we have out with new approach you can do that we've only two.
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Giuseppe Carleo: And this difference, or is it, this is much cheaper, this can be done can will systematically improving or, if you have $1,000 the difference would be really huge.
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Giuseppe Carleo: So it's really a way to perform this approach much faster, and it has been realized also experimentally, for example on these IBM Q around which is one of the.
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Giuseppe Carleo: latest superconducting quantum products are available at IBM where they showed on on this quantum chemistry problem, so this is to migrate.
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Giuseppe Carleo: That with this new approach, so this is a queue and quantum natural get an essay you can really reach what first of all, lower energies that what you can get with the standard SPC and also typically.
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Giuseppe Carleo: The convergence that you can see from from from this is much faster in terms of iterations not only relations, but actually in terms of secrets.
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Giuseppe Carleo: Evaluation, so this is really a real experimental data taken on on the pond on the live event, so there is no if you want to simulate the results, this is truly.
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Giuseppe Carleo: What you would get on the on the phone or device if you ran our our So you can see the real advantage in using this method and again this is for a very small secret, these are balanced we become more significant and actually.
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Giuseppe Carleo: You can immediately think that for larger systems, we will be the only essentially method you can you can run if you want to do to automatically.
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Giuseppe Carleo: Now, in the second part of my talk, I would like to discuss another family of operational quantum arguments that that we've introduced.
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Giuseppe Carleo: That solve another problem, so in the first part you've seen the operational approaches that solve the problem of minimizing if you want some cross functional, for example, the expedition value of the of the energy or some if you want to quantum quantum machine learning problem.
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Giuseppe Carleo: In his other part of like to to address the issue of doing so called operational unit economics.
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Giuseppe Carleo: So what what is the problem, so the problem is that again we'd like to we have a hamiltonian, for example, hamiltonian over physical system age.
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Giuseppe Carleo: And we'd like to essentially started a unit and enhancing is by design, so these operators to me here that milestone in his time in Japan.
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Giuseppe Carleo: So the standard way, so the thing that you want to do one on there, if you want to narrow corrected hardware, is to just do let's say it trotter ization of these these exponential with some small things step that we begin by the inverse of this end, which is the number of total testers.
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Giuseppe Carleo: And again everything seems fine there are even improve and better approach than just doing the doctor evolution, if you have a inner collected.
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Giuseppe Carleo: arbor you can run, this is a very simple quantum are going to start the mix of generic system, and this will work just fine.
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Giuseppe Carleo: However, in addition, is that if you would like to go to very long times so to be going in physics very interested in starting to the dynamics of.
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Giuseppe Carleo: system typically rooms or four typically long times, and you have some nice alert so that's where I enter this nice key business citing jump rescue.
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Giuseppe Carleo: The issues that you cannot afford to.
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Giuseppe Carleo: I guess you've seen also another talks, you cannot afford to have very deep secrets or so, since the depth of the secret in the starter approach with scale like the number of 10 steps on the total time you went to simulate.
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Giuseppe Carleo: Then there is a strong interest nowadays in finding ways of simulating long time then i'm.
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Giuseppe Carleo: Using a fix of that on secret right so that's what we'd like to achieve so so to solve central or to try to circumvent if you want all these issues that Nice nice devices that so essentially the around the corner, so that the director that are present nowadays.
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Giuseppe Carleo: So how do we do that well again we use a traditional approach in the sense that, instead of now, if I mentioned that the time to.
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Giuseppe Carleo: My my state at time T or that would I do, that I I changed or I try to approximate this state with some answered state that will depend, again on some possibly large number of parameters.
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Giuseppe Carleo: These can be, for example, a parameter I couldn't see it, and then I will I will use again a sort of.
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Giuseppe Carleo: Standard the Q a's operational on the bug approach in which will measure some observable and we changed my boundaries to go to the next point in time, and I was telling the moment our business.
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Giuseppe Carleo: So the main idea is to use these are the time dependent rational arguments, which is based on.
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Giuseppe Carleo: Information on principle, it was bye bye bye Dr frankel and also mclaughlin.
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Giuseppe Carleo: And, which is widely used the invitational applications in chemistry in classical relational applications in chemistry and physics, for example, to.
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Giuseppe Carleo: Time dependent ratio will try to use the term independent emerging it's it's really why do you a method that it's been if you want translated that extended to the gaze upon no secrets Bailey and Benjamin in this paper in 2017.
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Giuseppe Carleo: So all does it work, so, in practice, I imagine that that each time step I want approximate my state with my my vibrational manifold.
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Giuseppe Carleo: So I so this will end user dynamics time dependent dynamics in my virtual parameters so and you can show that, in the sense of this version of principle.
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Giuseppe Carleo: To have the optimal dynamics in the operational set of parameters, you have to solve this equation emotional, for your additional parameters data.
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Giuseppe Carleo: Okay, so So you see that that each time step the time derivative of your virtual up around must satisfy this equation as busy you will around if you want the equation.
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Giuseppe Carleo: Where you again, you can show this is shown by newspapers that is medics and is given by the imaginary part now of the of the genetic tensor that I was introducing.
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Giuseppe Carleo: Also, in the first part of my doctor and this vector V is given by the real part of these old is overlap elements that contain also the hamiltonian of people.
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Giuseppe Carleo: So if you're satisfied is equations or motion for your operational parameters, then your optimal exams that each time step, you are really following you're doing your best to follow the the relational dynamics.
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Giuseppe Carleo: And tbi what it was that I teach timestamp, then you measure these these metrics and effect or your computer.
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Giuseppe Carleo: And then you form again estimates of these metrics and backdoor universe them and you find your your time dependent variable is and then you can use those to essentially integrating all the that we did you do the nomics of your quantum.
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Giuseppe Carleo: Now the issue as before, is that this is not efficient, in a sense of again that it does not scale linearly with the nominal operational parameters.
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Giuseppe Carleo: Because, as before, we have to do a measurements on the fundamental did is proportional to the number of additional parameters squared again essentially for the same reason that's before, because this medics is is n by N, the number.
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Giuseppe Carleo: And also another additional obligation that's before we spoke to before, is that this is required really medics immersion that must be very.
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Giuseppe Carleo: Precise, so we cannot afford to regularize to match these metrics as before, and these in practice can be a problem if you estimate is Emma on on our door, and you have stochastic north.
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Giuseppe Carleo: As I was mentioning yeah, this is the imaginary part of this wonderful dramatic, whereas the official information is the real part of this of this of the sentence.
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Giuseppe Carleo: Now.
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Giuseppe Carleo: So So how do we solve this issue so how can we make dessert dessert Gordon efficient so that's essentially what we proposed in.
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Giuseppe Carleo: This method, this is, in cooperation with my with my postdoc every person in and Stefan its own is my PhD here and was on.
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Giuseppe Carleo: But, essentially, the idea is to transform the dynamics into an explicit optimization problem, because you see that here the optimization is left in place it so it was done ready, if you want.
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Giuseppe Carleo: When you pause in the vice principal but there's no explicit optimization or we are doing right.
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Giuseppe Carleo: So we'd like to perform an explicit optimization now and we must have of course some loss function, so our loss function that would be this overlap, so the idea is that.
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Giuseppe Carleo: If i'm I see that some time Okay, where I assume that my state is described by my vibrational state that has some parameters data, then at the.
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Giuseppe Carleo: If you want to time people's epsilon, we know that the state will become we should the exact state should be the application, the exponential of mines it absolutely so again.
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Giuseppe Carleo: infinitesimal evolution times my initial state Okay, so what I can do is that I can try to find another set of parameters, so in this data Pas de de de de de de de some updated i've to my initial set of parameters.
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Giuseppe Carleo: You know, in such a way that these exact status, you can see here under it matches as much as possible, my volitional maybe for my emotional state the devotional manual.
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Giuseppe Carleo: Okay, so you can see now that my loss function is not lost foundational in these basic bananas, but the loss function in the updates that i'm doing in device parameters and.
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Giuseppe Carleo: Because data is fixed and what I want to know is, essentially, how should I changed my rational parameters in such a way that the next step, my state will be as close as possible to the exact.
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Giuseppe Carleo: And what we do with that we define is discussed functionality, you can see here that, and you can show quite easily is independent on on the timestamp Pepsi on.
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Giuseppe Carleo: When you when you take the limit for example of small steps so it's a reasonable choice because we're not depending general on the on the title page.
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Giuseppe Carleo: And again, the other is relatively straightforward what we do now is that we have an operational point, no matter where our loss function is this object.
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Giuseppe Carleo: That can be again measure the efficiently on quantum after we conclude, for example, the gradient of this loss function using some standard the shift update rules.
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Giuseppe Carleo: And then we update actually very solid the learning rate to one is essentially optimal in this case, and then we update the parameters in this way, until we convert a distance.
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Giuseppe Carleo: So this data star would be my optimal set of parameters that I have at the end of this optimization.
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Giuseppe Carleo: And you can see that everything here is a linear in a number of operational parameters, there is no step, essentially, where I have a product dependence.
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Giuseppe Carleo: Of course i'm introducing now is in small optimization that but, in practice, you can show me from experiments that this is this corporate strategy fast at each timestamp.
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Giuseppe Carleo: So this is another cartoon of what we're doing as I was mentioning, so we are sitting at time zero for example here and we want to go with time absalom so and we want to approximate the next state here.
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Giuseppe Carleo: we've we've another State so and we adapt if you wanted this vibrational parameters updates, in order that we match the overlap between.
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Giuseppe Carleo: Now, and it turns out, importantly, that this way of doing things is is completely equivalent to the standard version of principle, if you want.
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Giuseppe Carleo: In the limit in which you take epsilon that goes to do so if you take your time step that goes to zero, you can expand this loss function that we have here to first or second order actually in the time step.
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Giuseppe Carleo: In both the optimal condition integration that the meaningful that we have at each time step is exactly the same as the one that you, you get with the.
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Giuseppe Carleo: With the standard version.
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Giuseppe Carleo: So, so this is important because it tells us that we are really solving the same equation, a little more subtle, but we are doing that typical faster than what you can do in the standard approach.
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Giuseppe Carleo: So, and again I mean if you do a simple evaluation of the of the functional evaluations, essentially, how many.
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Giuseppe Carleo: operations, you have to do with your computer, so you can show that these are linear in the in the in the number of parameters and.
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Giuseppe Carleo: Then there is a again, as I was mentioning a pre factor, and which is number of subset of the localization subsets but typically we see that this number of substance.
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Giuseppe Carleo: Is not large and typically with some some 10s of optimization steps you can you converge faster, because this is actually a complex cross functional so big commerce Nice.
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Giuseppe Carleo: Okay now just to give you an example application so what we we've done is.
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Giuseppe Carleo: For example, starting the dynamics of this family have been stolen, so these these are called thomasville dicing model, so you have.
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Giuseppe Carleo: a tortoise field which is proportional to the polytechnics operators and some interaction or proportional to Sigma see.
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Giuseppe Carleo: Now then, we started and our mix of these milestones in use by making by design mcs starting from a product state, for example.
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Giuseppe Carleo: So what we do is that we use a simple answers, as you can see here, it has some rx rotations summer some to compete rotations and you see that device parameters are essentially the angles of these locations.
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Giuseppe Carleo: And then I mean you, what would you get show, for example, is the you can bottom and you can compute the dd trying to see how well our facial approach performs.
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Giuseppe Carleo: Essentially you can before you can compute the infidelity so much we are moving at each step, but if you want of this inspirational emotion at some some capital T time which here I think he's taken to be one of Jay.
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Giuseppe Carleo: And you can see that in what i'm showing here is the dependence of these infidelity so again the lower infidelity, the better.
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Giuseppe Carleo: as a function of the number of samples or, if you want us as a function of the number of measurements that you have to perform on the quantum computer and again, our main goal here is to also leave it number of measurements, we have to perform.
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Giuseppe Carleo: And you can see that typical you can region essentially much lower infidelities than the standard approach, which is this DDA.
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Giuseppe Carleo: Using the same number of suffering that we tend to the same in the budget of 10 to seven samples.
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Giuseppe Carleo: You might get this kind of infidelity whereas with the standard approach you get some of an order of magnitude larger Intel before the specific.
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Giuseppe Carleo: secret and, in general, we expect that that will be favorable scaling with the number of parameters, essentially because the standard approach is the quadratic whereas our.
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Giuseppe Carleo: we've refactored but it's still a linear linear in the number of of virtual practice so you can see here so, for example, this is the it will be scaling with the number of parameters if we increase the depth.
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Giuseppe Carleo: Of the standard approach, which is essentially what radek and then we have again our course, so why to because essentially at each step we might want to.
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Giuseppe Carleo: To.
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Giuseppe Carleo: To essentially fix some some if you wanted some convergence absolutely okay here, and you can see, for example, if we if we set an era of digital mastery we need this many samples or.
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Giuseppe Carleo: If you want the higher fidelity we might need more samples, but still, you can see that typically 10 to the minus five is something that is.
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Giuseppe Carleo: Already pretty high fidelity and you see that there is a nice improvement over some that approach that again gets much larger issue more than 40 pounds, we should have thousand pounders did the gap here would be really very good.
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Giuseppe Carleo: Okay, so I think that actually my my time is almost it's almost over, maybe I will just tell you one quick word about classical relational states, since at the beginning of my talk, I mentioned that.
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Giuseppe Carleo: At this point in time it's not so clear that parameters on secrets are more effective.
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Giuseppe Carleo: In classical a parameter states, so I wanted to tell you what those about classical rational, since it, since this is.
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Giuseppe Carleo: Something we also work on a lot, so the and essentially also the reason why I would like to mention that is because.
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Giuseppe Carleo: There is a strong, powerful between the the techniques we've developed in the classical if you want stochastic warden.
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Giuseppe Carleo: And those that are now adopted the in the funnel secrets so so, for example, this version of victory approach that is used widely in the Community is nothing but a version of what is known in the classical world has been.
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Giuseppe Carleo: In both cases, you have stochastic estimates of the gradient of the energy it's really the same approach of course run on different doctor.
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Giuseppe Carleo: But then there are also many other process, so one of these was listed the va which as an equivalence in the classical world in a time dependent rational the Gala stochastic word.
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Giuseppe Carleo: Of these natural good in the sense that we discovered in the quantum world, but it is also there was a baby well existing in the in the machine learning community and known as natural gradient, so there is really a strong interplay between classical stochastic methods and quantum punishment.
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Giuseppe Carleo: And our favorite if you want the ways open i'm addressing classically the vibrational states So these are not now secrets now i'm using gpus not.
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Giuseppe Carleo: Not coupon codes are not on the processing units are new alcorn state so it's something that we introduced a few years back.
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Giuseppe Carleo: which are essentially we are prioritizing the averages of a wave function in terms of a deep neural networks Okay, so this is a compact way of writing a deep neural network.
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Giuseppe Carleo: where you have this is your essential your vector one no numbers apply a linear transformation and nolan geology and so on the cascadia formations.
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Giuseppe Carleo: And here the rational parameters are not as before, if you want to do the angles, in my younger years but are instead the weights in his in his mattresses are in these filters, for example, Nicole Charlotte.
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Giuseppe Carleo: Now, if you try to use this approach to simulate quantum computing, so now we might use the state's classical faith to simulate a quantum city good good.
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Giuseppe Carleo: So this is a useful way to understand, for example, where the limits of quantum computing are and where the limits of classical conditioning.
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Giuseppe Carleo: So you can show that showed in his paper that you can apply actually most of the of the all the gates you needed so, for example, you can apply.
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Giuseppe Carleo: RC gates and controls the gates exactly on neural networks and you get out another neural network just slightly modified as it needs to be this.
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Giuseppe Carleo: One or this one beat and only one that you cannot apply exactly which is the other market, so if you apply how they're not on your network.
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Giuseppe Carleo: You cannot get out in general, another new letter, but you have to make an approximation, so this is where.
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Giuseppe Carleo: We have to make classical approximations, but we can do that again the classically with some rational approach, so we have on this and the classical if you want.
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Giuseppe Carleo: I mean the exact state of the traditional the other one we try to approximate these with another new Omnia took.
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Giuseppe Carleo: That, as some other parameters that are prime by minimizing the infidelity i'm skipping a lot of details here, but I can assure you that this can be done efficiently using some Monte Carlo.
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Giuseppe Carleo: And the main point I want to do here is that you can compare typically the error that you make operationally so classically in approximating these these these other monitor.
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Giuseppe Carleo: With the typical errors that you have on upon to matter because, also in the quantum model, you cannot apply exactly, so to speak, the other mert or any other data, because you have errors so every time you apply gate you have some final fidelity.
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Giuseppe Carleo: On the other end in the classical case we kind of apply exactly gates, because we have some infidelity coming from the fact that we everybody from for.
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Giuseppe Carleo: And so what we would try to do here, and it was done another paper, later on, is that we can compare essentially the variation on error, with the.
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Giuseppe Carleo: noise level, you will need on the order to have the same accuracy that we have classical so it turns out that you need to pick the law arrows on the on the on the murder or the order of gentlemen sphere so.
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Giuseppe Carleo: When you drive, for example, to to simulate some.
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Giuseppe Carleo: If you want some quantum freighters forms in this case, and more recently we've simulated.
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Giuseppe Carleo: Actually pretty complex it gets like Curie, I think, maybe this has been discussed only during the seminar this workshop, so this is a family of.
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Giuseppe Carleo: quantum migrants to approximate find minimum functions, but essentially we managed to to simulate, for example, 54 cubits QA, which is in excess of what you can do.
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Giuseppe Carleo: Large and access, I would say what you can do with high fidelity on quantum Adler nowadays to some secrets that that you can see here.
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Giuseppe Carleo: These are because financially we simulate again from first principles there's no input from the mountain.
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Giuseppe Carleo: And you can see how many gates, we have here also that we can simulate with very high fidelity with this very simple you're on it.
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Giuseppe Carleo: so easy classical conditioning is very competitive steel and there are there's a lot to be done, still on the bottom side, on the other side to have this compete with the best if you want operational techniques that we have classical.
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Giuseppe Carleo: Okay Now let me go to my outlook i've shown you essentially I mean mostly see relational quantum approaches.
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Giuseppe Carleo: to learn and to minimize energies or to do on to the next efficiently again linearly the number of parameters and the advantage of using quantum simulators.
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Giuseppe Carleo: Is that they are potentially more expressive as I was mentioned in the classical methods.
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Giuseppe Carleo: You can do arbitrary unit is, but of course the drawbacks, is that you have noise and you can do on the shelf secrets or that's why we have to invent this.
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Giuseppe Carleo: Methods where we compress if you have the dynamics, on the other hand, we have some classical virtual simulations that allow us also to do highly NATO states.
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Giuseppe Carleo: But typically I mean we are limited, because we cannot apply, for example, arbitrary unitary exactly, so we cannot apply, for example, the approximate and, in general, I mean we have to rely a little bit on some.
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Giuseppe Carleo: Non general principles on how to define this neural networks that we want to use to approximate quantum system still I mean, I would say that the competition is stuff and we are learning from both sides of the and it's a very nice interplay between these two different communities.
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Giuseppe Carleo: So thank you for for your attention and.
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Giuseppe Carleo: If there are questions now, I will take them.
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Giuseppe Carleo: Otherwise, you can also still always right.
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Giuseppe Carleo: And you.
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Marius Junge: can ask a question.
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Giuseppe Carleo: Of course, go ahead.
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Marius Junge: My name is Mary sooner, so one of your potential organizers anyway, so I want to go back to this, I think this was all five where you showed us that you with this.
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Marius Junge: That you obtain an improvement.
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Marius Junge: A little bit before.
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Marius Junge: Yes, right so so this very show you in.
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Marius Junge: Your paint an improvement was this time dependent very general principle but.
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Marius Junge: Right, I, like the page where you wrote piney pendant variation.
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Marius Junge: principle that.
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Marius Junge: One wave wave my question occurred, so I just want to clarify something if you do this time racial I mean your your your process is better because your your measurements are right, your your your measurement you have fewer fewer measurements right and the more linear.
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Marius Junge: The optimization ears just in the classical parameters patients that correct.
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Marius Junge: I mean your improvement this on the parameter side is that not not the classical.
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Giuseppe Carleo: Not the domain improvement is that.
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Giuseppe Carleo: We yeah we can solve the same so Okay, so we can so.
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Giuseppe Carleo: We can get.
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Giuseppe Carleo: In at the time, which is linear right yeah but.
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Marius Junge: you're solving essentially completely in the parameters Is that correct.
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Giuseppe Carleo: Yes, yes, yes, so so we solve for for the time that Eva, these are the parameters so that's right.
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Marius Junge: So, and then you translate that into the game picture, so we are doing some kind of classical technique to find the right quantum gates Is that correct.
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Giuseppe Carleo: Oh no everything is fixed by the secret that we are using.
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Marius Junge: So, so we have a site, yes, so that.
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Marius Junge: Okay, yes, so I was, I was wondering, because if you take the fidelity.
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Marius Junge: And i'm looking at master equations for the audience and then you often have cost functions which are different from the fidelity, for instance, relative entropy right.
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Marius Junge: For you would look at some.
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Marius Junge: Some metrics which are state dependent so so I was just wondering whether you have tried, some of these you know more non competitive matrix for optimization because they could come or capture some parts of the quantum you know the quantum process differently so so.
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Marius Junge: You know, in your.
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Marius Junge: In your definition of natural gradient you use the particular metric.
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Marius Junge: Depending on the parameter and.
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Marius Junge: You know the work i'm doing probably suggest to use other metrics which are truly non communicative right so.
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Giuseppe Carleo: uh huh yeah yeah will be very happy to read about this.
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Giuseppe Carleo: yeah I yeah.
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Marius Junge: So there's work by Parliament mass on.
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You know.
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Marius Junge: Like like evolution right it says that, in order to capture the full strength of the quantum.
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Marius Junge: computation some probably one or two has to adapt some of the costs functions accordingly that's what I was you know, some are trying to join her stand or to remark right.
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Giuseppe Carleo: yeah I mean this cost five years.
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Giuseppe Carleo: is very natural because it's just your.
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Marius Junge: Yes, but for certain phrases for convergence of.
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Marius Junge: Relative entropy they might not be the natural.
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Giuseppe Carleo: Cost function yeah okay yeah this I I don't know but it's it's interesting.
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Marius Junge: there's similar so so when can.
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Marius Junge: I think this just may be interesting to look at other cross functional.
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Giuseppe Carleo: So what you're suggesting is actually changing actually the time the veneration of visible itself.
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Marius Junge: right but probably following the same you know the optimisation shrinks could be exactly the same.
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Giuseppe Carleo: yeah this might lead to a new version of beads well that's very interesting I hadn't thought about this yeah.
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Marius Junge: Okay, thank you.
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Thank you.
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Giuseppe Carleo: Okay.
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Giuseppe Carleo: Thank you, so if there are no more questions, maybe I will.
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Giuseppe Carleo: Stop sharing my screen now.