 Vermerge, presenting our many body entropies and entanglement with polynomially many measurements. Thank you so much. Thank you very much. Can you hear me well in the back? Can we be higher? No. Maybe I just speak louder. First I would like to thank the organizers of course for this wonderful opportunity for me to be here because obviously the first time I attend a machine learning conference and I've learned so much. Thank you. It's much easier to learn about this in live in a conference than reading papers, especially with my background. So today I will discuss about measurement protocols for quantum processors. So it can be quantum simulator or quantum computers. And what we like doing is try to accumulate a massive data set from the quantum computer. It's a classical data set. And then we have some tricks to measure quantities that we think are interesting to measure, in particular entanglement. So I think there are opportunities to find connection with machine learning ideas and I look forward to feedbacks about it. First because we use, sorry, a massive data set, as I said, so there's a lot of data to analyze. And also because we can provide from this data physics properties and these physics properties are highly non-trivial. We measure typically entanglement on quite large systems. So maybe in turn these physical properties I could imagine could be used to represent your states in a better way. So I'll start with what I hope is a gentle introduction to techniques to measure entanglement in quantum computers and simulators. I will skip kind of a mathematical aspect behind the protocol. My aim is to show you what is state-of-the-art in terms of experimental recipe, typical numbers for statistical errors and so on, and typical limitations. And then I will move to this new result which we are quite happy about on how to extend this existing toolbox to really measure entanglement and related quantities in very large systems using very few physical assumptions. And I forgot to mention that this work was done in collaboration with a very nice team. I would like to thank in particular to mention Lorenzo Piroli who really helped me and helped us to have faith in this idea that one can eventually measure entanglement in very big systems. Marco and Maxime were super helpful for the numerics on matrix project operators and Ignacio Ciara and Peter Zoller are always useful and they provided incredible help to understand what was the impact of this idea. Don't hesitate to ask questions if something isn't clear. So to define a protocol I need to define what I want to measure. For the purpose of this talk it's sufficient for me to discuss first entanglement entropies and then I will generalize to more complicated things but for a moment you can just consider I have in mind a pure state living in AB that's my quantum computer. I want to see if A is entangled with B in this pure state picture. It is sufficient to calculate entanglement entropy of a reduced density matrix in A and in particular I'm interested in the rainy entropy which is the one which is measurable for us. The second expression, we know entanglement entropies play a fundamental role in quantum information like the number one quantity that you see in quantum information notes that quantify quantum resources for in particular communication. Maybe more important for this conference in quantum stimulation is that entanglement entropy turned to be a very useful other parameter for complex phases. You can think of critical states, the quantum phase transition, you can think of topology, you can think of dynamics. And this was used for a long time in numerical simulation quantum Monte Carlo, Diarmargie to understand a given Hamiltonian and this gave hope that if one day you can measure entanglement entropies in a lab quantum simulation will benefit from that because it's only in a synthetic quantum system that you can think about measuring such a crazy quantity. I also like this last point about quantum advantage. The entanglement entropy is by essence limited in a tensor network architecture. So if you want to develop a quantum technological device and if you can prove that in this device the entanglement entropy is below some threshold you have some claim of quantum advantage with respect to a tensor network. So now let me focus specifically on the rainy entropy. We know it quantifies entanglement. It's the log of trace rho squared. Trace rho squared is called the purity. It's one if and only if the state is pure. So if I measure a purity I measure the entanglement entropy by taking the log. It's exactly the same thing. And you will see that what I would say about the purity can be then generalized. So that's a good starting point for my explanation if you want. Now to measure a purity in an experiment there are basically two families of protocols being used at the moment in experiments. I will mostly focus on randomized measurements which you see here on the left. The advantage of this is that the circuit that you have to do for measuring is the same as in tomography. So you have a single instance of your quantum state. You perform randomized basis transformations. So this u1, u2, un are single qubit rotations. Here they are taken at random. And then this is followed by a projective measurement giving you a classical bit string which is the data that you will then analyze. So I want to emphasize this is not tomography. This is the data acquisition procedure of tomography but I will do something else than tomography with the data. In particular what has been shown with randomized measurements is that the purity can be mapped to the data with an analytical function which has a very simple form. So there is this kind of bridge between data and entanglement entropies. And it's simply an average. So I have to compute some form of average over my data. The second family are called bail basis measurement. This is an earlier protocol which was proposed earlier. This time you have two copies of your quantum state. So it's more demanding as a circuit. And then you have to couple these two copies and measure in a single basis. And this is particularly used in Riedberg-Adams, an experiment of Micheloukin for instance. But it's difficult to use it in different, in other platforms like the trap tie-ons and the superconducting qubits. Typically don't have that. So I will focus on the randomized measurement part or what comes next. But keep in mind that this other method exists and in particular the statement that I will be able to make later will apply to both ideas. So this is like the cartoon for randomized measurements. It's actually a bit simplified. It's too simplistic for what I want to say. This is now how a current randomized measurement looks like in the lab. This is a picture from a paper we did with IBM where the aim was to measure purity and also other quantities. And it looks like this. So you have two blocks. You have the blocks of data acquisition here. That's the experimental task. And then you have the block of post-processing which happens on a classical computer typically by the theories. So what happens in data acquisition is independent of what you want to measure. It's very important. It's a single experimental recipe. If you have it in the lab, you can do many things. So it starts with a calibration procedure where you apply the unit theories that you wanted to apply. But there is some noise in the experiment and it's very important to be aware of that and accept this fact. And this calibration step aimed at learning this additional noise in the experiment. And one important fact which was discovered recently is that when you start with randomized measurement in the first place, if you have additional noise on top, there is a sort of effective averaging of the noise due to the fact that you are using random unit theories. It's called twirling in quantum information. This noise in the end is summarized by what we call calibration parameters. So there is just some list of numbers that you have to learn when you are ready for the next step which is the measurement. In the second step, I want to perform the actual measurement on the quantum state of interest. And we construct classical shadows. So it's a form of randomized measurement that I will explain. And the experimental step of data acquisition is over. You can store these results and go to a post-processing stage. In the post-processing stage, there are many ways. I will focus on one which will be explained later. It's just you load the data on the classical computer and then you estimate entanglement entropies or other things with the formula. It's really simple. So how does it look, this post-processing? There are a lot of work recently about these classical shadows which was briefly mentioned in the other talk. Classical shadows are a representation of your data which is information complete in the sense that if you have infinitely many measurements, you actually have the quantum state represented. And it's a simple function of the data in the sense that S is a measured bit-string. U is the unit area I applied. This object I know. F is the noise parameter that I measured independently. I can build this object as a tensor product on my classical computer. And the key result about classical shadows, what I need here, is that in expectation value, I know that this object is equal to the density matrix. It does not mean I will do tomography here. I just know that in expectation value, this object is the density matrix. And all the tricks of randomized measurements consist in building statistical estimators of functions of a density matrix based on this relation. So typically I will sum over the shadows and then contract and so on. And I will obtain exact relations between data represented by shadows to my case purity. You can play many tricks, very simple tricks of statistics. I will not spend so much time on this. What is important maybe to finish about this introduction is to say that typically the experiment with randomized measurements are done with order 100,000, 1 million measurements. Sometimes it was several million measurements. This is what you need typically for the kind of plots that I will show. So here's an example of a recent measurement of a purity in a superconducting quantum device. So good news is you see the x-axis. You can measure a purity up to n equal 13. And we are quite happy about this because it's a decent, small, many-body Hilbert space that we have here. And what is also nice to see is that when you are using this robust method, which came from last year, this is a purity of order 109 for this pretty large systems. I feel sometimes it's a bit unfair to call this device noisy quantum device a quantum nisk and all that. These are pretty good, pure states. It's an unbiased estimate. So typically if I would repeat this experiment many times, my estimator would be symmetrically distributed around the exact value. That's a question? Okay, thanks. So robust is this updated method. And we believe that's the true purity. The blue one, the standard one is the one without noise mitigation. It is much slower. And what happens is that while you are measuring, you are putting some noise. So the purity of the state decays, effectively, right? So we are measuring some kind of state affected by the measurement itself. So, oh no, this you cannot. So for instance, if you have a spontaneous emission in a system with a single qubit, the purity will actually increase with time as you will progressively go to the ground state. It's not trivial but how it should behave. That's a good question. Yeah, it's completely independent. That's why I'm here. I want to know. So that's what I propose. What I can say is that the experiment recipe looks simple to us and we were happy that some groups started using it. In particular, Google and IBM. Also, Innsbruck. And so that's some kind of established tools at the moment. So now I can comment on some technical details which will be important for the rest of this talk if it is a final review. So I'll start with bad news. For purity, we know that this protocol requires exponentially many measurements and it is expected because we are probing a very non-trivial quantity here summarizing entanglement, right? What I want to emphasize is that the corresponding exponential scaling has a coefficient A which is of order one. So we are exponential, okay? But it's not tomography. Tomography, the scaling is four to the n rank one, eight to the n for maximum rank. So it's exponentially better than tomography. This is why this measurement is possible for much larger, but what you have with tomography and much better precision actually. Still, you end up with, you see in these plots the error bar starts to increase and I'm not showing 14 obviously. What we like about it is that it's robust. Under very typical assumptions for noise, we know that this is the right purity that we measure and this is a quantitative estimation that the experiment at least now use to verify quantitatively how the device behaves. If the purity is low on Monday, maybe you have to work a bit on Tuesday to improve the device. These sort of things that they tell us. And the framework is so general that you can measure many other things like pureities. And over the years we played a lot, in particular with Pasquale, Calabrese to measure over entropies and entanglement quantifiers. So this was fun. And if you want more about what can you measure? What is the protocol? What is the error bar? We just put a review last year together with a Caltech group which summarizes what we've done there. So now I come to the second part of this talk about a new upgrade to this toolbox which is how to measure the entropies and purities with polynomially many measurements instead of exponential. And the starting point for us is that we spent so much time to develop this toolbox that we would like someone to keep using it if possible. So if I have in mind a very large, very large system, I know that the only thing I will be able to do with existing toolbox is that I can extract the purity of blocks. I will have here this block I1, I2, I3, I4. This is what I can do. Measure the purity there. I can also imagine that I can measure the purity in pairs of blocks like I2, I3. I know I can measure this very well. The question we were asking is can we somehow use this measurement of the local purities to find the purity of a global system? This is called in quantum information quantum marginal problem. That's been studied a lot for von Neumann entropy. For purity it was open actually. And the answer is yes, there is a formula. And this is the result I will present. I start with the end actually. The result is there is a formula. And we think it's useful because now we have with existing toolbox, which is already in the lab, we have a way to build an entropy meter. We can measure extensive entropies and all these sort of nice things. Basically today. Okay. So to show the formula, I will use tensor network notations because I like them. I found they are intuitive to describe entanglement. It's a toy model to find the formula and then I will show that this formula is actually general for typical states with short range entanglement. I will put the physics assumption a bit later. But let's just see how it goes. So tensor network, we have seen many interesting talks just consisting in representing matrices and tensor graphically. So a single cubic density matrix is a box with two legs. Now I want to make an entanglement quantum state. So the first rule I need to know is gate operation, unitary operation. We know that it is u rho u dagger. So graphically it's like that. It's really obvious for most of you. And now I have my toy model. I start with the minimal size I need for the formula is 12 qubits. I start initially with a mixed state because I want to make general claims about mixed states. And I apply two qubit gates. It's a depth two quantum circuit because you have two layers. And you see that with such a circuit I generate a little bit of entanglement but not too much. And the question I'm asking is is there a close relation between the purity of a big block of this system? I will consider the first six qubits as a function of a purity of smaller blocks. If I'm not able to find a formula there, there is no formula. It's really the simplest I can propose for a circuit. So now the purity looks like this. It is the reduced density matrix on the first six qubits. So I have to trace out the last six qubits. So I contract when I put a red dot it means there is a kind of a link contracting this thing. Pink also here for the second copy. I have to reduce density matrix over there. And then purity is trace whole square. On the A part I have to multiply the two matrices together. So that's why it's the same color here. I contract like that. And I contract the whole thing afterwards. Trace whole square. So do you have an idea how I can simplify this to get a simple expression? So here for instance, you see that I multiply a gate with its Hermitian conjugate. So I have u times u dagger, I know its identity. So this is the rule I can use to simplify my tensor network. And what I obtain is the following. The purity is a product of numbers. So here I have rho square, rho square and so on. So I have a product of numbers. And at the edge I have this strange number which is due to the fact that the qubit number six is entangled with qubit number seven. So the reduced density matrix on six qubits is mixed because of this entangling link. So this has to be there. And people from think statistical physics or many body physics are not surprised, but the purity has an extensive contribution coming from the fact that I started from an extensive state in terms of entropy. And it has an edge contribution because I created entanglement at a given edge. So we were happy to see this because we hope survives. There might be a way to simplify this further to express these numbers in terms of local purity. Quality that I can measure. And this I will skip. I will ask you to believe me. If you compute these tensor diagrams for subsystems, one, two, three, four, five, six, you will see the same kind of terms appearing. And in the end, you will see that you can combine these subsystems in this form, which I like very much. The purity of A is the purity of two overlapping blocks. So I have A1, A2, and then A2, A3. I'm able to make this product. And then since I counted A2 twice, I divide by trace for A2. So there is a simple fraction to estimate global pureties from local pureties. Solving, if you want, the marginal problem there. And this extends trivially to many blocks, any system size. You multiply the pureties with the overlapping blocks and you divide by the purity of the blocks. This gives you a formula which is exact for the toy model. Now, I'm done with the toy model. I want to make better claims, claims which are typical for quantum states. If you have questions, don't hesitate. Yeah. I think it is used in this follow-up work on one of the Gibbs states, but I would be interested in hearing your opinion. For the toy model, you thought I didn't use any theorem, right? Well, it's just a name, but it's one state, right? Ah, yes. It's because I have two layers, yeah. If I would have any number of layers, I could... It holds, yes. It holds, but I need larger block size. Here, the block size is involving two qubits. I would have depth four. I would need four qubits in the block. But yeah, I think I agree with you. It's a toy model which is instructive by itself. Still, we want it to be a bit more complete, and we took now to extend this proof, we took a matrix product density operators, so tensor network for mixed states. And then what we showed is that the same formula holds, and we know that MPDO is a good representation of Gibbs states in 1D, the same formula holds up to corrections which are exponentially small in a typical length of the system, which is the correlation length, obviously. Meaning, if you go to the lab, measure this, you will be exponentially close to the right purity, and you can actually check that by increasing the size k, k the size of this block, you can try many ks, and then at some point you will see it's guaranteed that you will converge to the right value, exponential convergence. So that could use the toy model provided the right formula, and there was a recent work very nice by a team from Spain showing that any Gibbs state, this property. And now I want to conclude on this part and explain the practical consequences of this. And if you take this formula here, we know the statistical errors in randomized measurement for estimating each of these terms. So the only thing we have to do now which is left is do the error propagation that we make when we multiply these numbers together. And our final result is the following that if you want to estimate the purity up to a given relative error, relative error is important here because we are talking about potentially exponentially small numbers. The error that you make can be bounded using a polynomial number of measurements. This is for the experiment proven by this term. This is a real cube for a system size. So summarizing the formula uses randomized measurements to measure the purity in a large system what you can do in a lab. And this concludes this part on the purity. So now I think I have still a bit of time to go on. 10 minutes, wonderful. I will perhaps not talk too much about this numerical verification of the scaling. I can explain for the experimenter that we can actually predict how many measurements you would need to measure a certain block size, a certain total system. This is mostly technical. I would like to use the last minutes to discuss about entanglement detection of mixed states to present how these ideas could also be applied in this context. So now I have I want to discuss mixed state entanglement. So remember the beginning of this talk I had A and B in a pure state. Now I want to go beyond and treat noisy systems or subsystems. I have A, B in an environment. I want to check if A is entangled with B. A long standing topic of quantum information theory of course. And in our case what we found useful is to study entanglement detection in the context of the positive partial transpose condition, PPT from Peres and Orodechi. So we are not measuring the purity anymore. We are measuring the moments of the partial transpose density matrix. It's a linear operation you can do in the density matrix. And I forgot to put the power N here. So I take partial transpose and then moments of that. And something that we found a long time ago now is that these numbers that you can measure with randomized measurements can be used to detect entanglement. So if the third moment is smaller than the square of the second moment it implies the state is entangled. It's a condition which is a weak current PPT. And this is an experimental demonstration of this entanglement detection in a trap term system. Actually I like to tell this story because it was interesting that the entanglement detection was discovered experimentally. So we were looking at these plots not at this y-axis but we are looking at P2P3 and then we felt that P3 was kind of small when the entanglement was expected to be there. So it's really looking at the experimental data which helped us to find some interesting theory. So that's why that's what I like about measuring entanglement is the experimenter stimulates your theory and when you do a little bit of math it's very simple, you find this condition. So this is what happens. And it turns out these conditions are quite powerful you can generalize them to any other PN and then slowly go to a PPT case. You can also show using PT moments to work by Matteo Voto within the audience but you can detect different types of entanglement for instance if it is generated by a Clifford circuit or not it's also there. So obviously when we found this this new upgrade to a toolbox we thought it would be nice to see if we can detect entanglement with polynomially many measurements using the same trick as for a PUD. So what we did was what we derived we proposed to measure these PT moments in a rescaled way so we take the PT moment PN and we divide it by these numbers which are also measurable this is just a trick to focus for statistical error on absolute errors I know these numbers is of order one so I just need to study absolute errors and using the same tricks as before decomposing in blocks you can show that this number PN tilde can be measured with polynomially many measurements and now the only thing I need to check to conclude is that I need to check that for typical states in experiments the PN-PPT condition is violated this is actually our problem to know if a state is the entanglement of a given state simulated by a given criterion there we did the makes so we took a 1D Gibbs states simulated by tensor networks we went up to fairly large system sizes and we studied the detection of P3-PPT and P5-PPT the next one as a function of L and beta and we found that actually the number is not so high there is a big region actually where the protocol will detect entanglement but we can make this claim and what is interesting is that you see this kind of slope if you analyze the slope what it means in terms of the purity you will find that you will stop detecting entanglement when the purity if the entanglement entropy becomes more than other one if the entanglement entropy becomes extensive P3-PPT will fail and it's not so much of a surprise that if the state is permanent in the sense that entanglement entropy is extensive or looks extensive for the system size it fails but interestingly if you move to the next next condition the beta one P5 the slope vanishes and you get something which looks like more vertical line and in this new region I mean it was a big surprise but maybe it was not for you in this region I'm able to detect entanglement even though the entanglement entropy is large or the end and the purity exponentially small so it's actually surprising to me and this entanglement detection can be done with polynomially many resources so with this I conclude I think I presented a funny result on how to measure entanglement entropies in state-of-the-art experiments I'm very happy to discuss about maybe if it can be used for machine learning applications I would love that I think I can say it can be used for typical quantum simulation tasks like classifying quantum phases and so on we are working on now the tomography so can we use approximate factorization conditions so this expression in terms of blocks for learning about a quantum state and we also working on 2D we believe actually there is a formula for 2D I can write it down but I'm not able to prove apart from a toy model that this formula will give me the exact purity after some block size I'm not able to prove so I would be very happy to collaborate on trying this numerically thank you very much more questions thank you very much I have two things so I understand that this moment of the PPT is much easy to calculate at the whole matrix and do PPT because you only have to calculate the diagonal and you have all the moment there so you know that there is PP3 PP5 but there is a hierarchy there proven that it should be like that so it's always a hierarchy that it's proven yeah it was it was a result by we proved that PN, PPT is strictly weaker than PN plus 1 PPT and then if you go to P2 to the power N it's exactly PPT I think that I wanted to ask you is that use but it's weaker than PPT it's weaker yeah but it's very nice I think that I wanted to ask you let me remember you were saying that also this purity measure it's better than for human entropy measure in the sense of if you measure rainy entropy this is better or more accurate I wouldn't know how to measure von Neumann entropy from randomized measurements but in this PPT for instance you have to do partitions with respect to everything because this is a bipartite thing so you have to calculate a lot of things a good thing that I can do is partitioning after I measured I have full data and I can complete that thank you very much for the talk I have a question I'm not sure how silly it is did you ever think about combining these results for this particular problem of entanglement detection with kind of guessing a good entanglement witness I have not seen any mention of entanglement witnessing and you're talking about entanglement detection so do you have an opinion? so what we would like to do is to avoid using entanglement witnesses because it's not clear what is the best entanglement witness that you need for a given quantum state so what we would like to have is a single quantity you have to measure and then you hope that you can make a conclusion about it I'm thinking more in this machine learning thing because if you can guess if you can learn the right entanglement witness then this would be the most efficient way to confirm any more questions? I'm actually super surprised like polynomial many measurements that this is ever possible so could it be so there was one point where I was wondering a bit practically so in the end when you want to get this exponential convergence you need smaller and smaller blocks so in the end you deal with longer and longer products of purities that you have to compute now could it be that actually I could imagine that errors can actually proliferate rather strongly in the end so that you might practically be back to an exponential problem in an actual experiment actually the error propagation is linear and I will show you this with an example it's counter infinity you take this formula here take the log on both sides yeah instead of multiplying this thing you will add the log the log of quantities which will be of order 1 over 2 to the k where k is a block size so I can estimate this with a fixed additive error which will not depend on the total system because it's just a property local property on k right given block the effort does not scale with the total system the only remaining thing I need to do is to do the error propagation of n terms you see that if these n terms are measured independently I will pay your price n squared because of the covariance effect you see what I mean yes but maybe we will discuss this later but it's super interesting okay perfect let's thank the speaker again