e e e e e e all right um welcome everyone um so today I have the pleasure of Hosting Robert Wang um yeah welcome Robert um why let's give everyone a couple of minutes to join um while we're waiting where are you tuning from yeah right now I'm on the East Coast um visiting Princeton at the moment nice um I'm tuning from uh Washington DC area and we have a rainy day is it raining in Princeton as well uh it's a little bit cloudy oh okay all right yeah yeah thanks for joining us um so yeah uh today everyone we have uh as I said we have the pleasure of Hosting Robert Wang uh Robert is a current a senior research scientist at Google Quantum Ai and a visiting scientist at MIT in 2025 he will join Caltech as an assistant professor of theoretical physics where he also completed his PhD under the mentorship of John preskill and Tomo vidic um so Robert research like Robert has done research in in different areas of quantum information Theory and learning theory um and he has won numerous Awards I don't think we can go over all of them but just to mention a few he he won the Milton and Francis clauser doctoral prize award he also won the Google PhD Fellowship um boing Quantum Creator prize Ben PC true doctoral prize and as I mentioned many many others um yeah and today Robert is going to tell us about certifying almost all Quantum estate with few single Cubit measurements um please take it away Robert yeah thank you so much for the introduction and for inviting me here uh I'm Robert and today I'll be talking about certifying almost all Quantum states with few single Cubit measurements this is a joint work with John presal and Med Sol manifer and if you have any questions feel free to just uh just just ask and Ali Raza will be able to pass on the questions yes yes please ask your questions in the chat and I I will relate okay so let's get started so we all know that in Quantum information sense a very important thing is to create Quantum many body systems with very intricate entanglement across the entire system so by creating such a highly entangled Quantum system one can perform very powerful Quantum computation that have advantage over classical computers for sensing about the world sensing various aspects of the world um with enhance sensitivity or allow us to communicate um through a quantum network Etc and hence in Quantum information sense all the experimental effort has been really trying to understand how we can best control and and generate these Quantum systems with intricate entanglement however in an actual experimental setup we will always be subject to all kinds of errors and noise that happen in the lab so a lot of times when we have some Target Quantum system that we have in mind that we wanted to create um in reality we will not be able to do it perfectly in almost all cases and what we will end up creating is something that might be slightly different if we are good in how we're doing the control or it might be just completely different if if the if there's some failure in the in the in the system setup or or or if there's some Heating in the in the quantum system all kinds of tiny noises would just kill um or kill off the the system and cause it to be very different from the desired one and hence in the experimental setting is very important to understand if we have actually created the desired Quantum state or not and in order to do so um we need to perform certification so for those of you that have never heard of the word certification it's a very simple idea so what is certification in certification we have some desired Incubus State s so this Incubus State um you can think about it as it's we we know it in our head this is our Target State um this is something we wanted to kind of realize in the real world we can also think of Sai as being like stored on a classical computer maybe there's some representation for Sai like a tensor Network or newer Network or some other representation that stores this desired State Side um and and that's our Target State on the other hand in the experimental lab we're going to create some other Incubus State role which might be close to S if we did a good job or it might be kind of far away from s if there's too much errors or noise in the experimental Quantum platform so the certification task is this very basic task a very fundamental one that is we just wanted to test if the state role created in the lab or we think of role as the lab state is close to the Target State side or not from the measurement data so we're going to perform some experiments on row get some classical measurement like classical data that entails what the measurement tell us and then from there try to figure out if R is close to s or not so here closeness can be measured in terms of the trace uh in terms of the fality that's the almost the most common measure that people consider or the most stringent measure that people consider which is sroi is it close to one if it's exactly to one then we have prepared it perfectly but if it's close to one then we also doing a good job or if it's far away from one so like if it's 0.5 or very close to zero then we know that we're not we're not doing a good job in creating the state so that's the certification task and it's a very um old problem in in Quantum information sense and there have been a lot of different procedure that have been proposed for how to do certification how to certify that we have actually created the state we wanted to create in the lab so many techniques and many results have been proposed for doing certification however it Still Remains very challenging to certify um Quantum systems especially if we work in a mini bodied regime like we wanted to certify a Quantum system that have say hundreds of Cubit and the quantum state is highly entangled in such regime it's still experimentally very challenging to to do so and just to Showcase some of the challenges that people have been facing or some of the challenges in existing techniques let me go through a few kind of more widely known approaches for doing certification so let me start with uh approach zero this is not really an approach that people use but it's a it's a conceptually a very simple approach but it doesn't really work in practice it's based on Direct measurement so recall that um let's say we write out the target state so size the target State and now let's say the target state is generated by some Quantum circuit U apply onto the initial product State all zero then one way to measure this Fidelity is that we just given the lab state so this Quantum State here is the lab State we implement the inverse of the unitary U so apply U dagger and then and then after applying UD dagger we just measure all the incits if we see that the incits turns out to be 0 Zer then we know that this lab state or is quite close to S however if we saw that outcome is not all zero then we know there are some errors that pop up so this is a very simple approach however this approach has a very big issue which is it assumed that one get access to UD dagger so if we can assume that we can apply the inverse of U perfectly in order to run this approach then the creation circuit U should also be perfect too like they're in a way symmetric it's just running the circuit in reverse and hence in the in the world where we can really Implement UD dagger perfectly then roles should be able to be created to be equal to S perfectly and hence in this world there's no need to do any certification and hence no one really runs this approach so I call the approach zero it's very simple from a theoretical and uh and a conceptual point of view but because it requires just um a like to perform this direct measurement it requires a a resource that's as hard as just generating the state um s itself and hence this the this kind of defeats the purpose of doing certification and hence no one actually use this so now let me move on to um another approach which is based on randomized cliffer measurement so instead of implementing the specific circuit that invert the target State back to all zero State we just run a random cliffer circuit so we apply we sample a random cliffer circuit and run it and then measure in a single Cubit basis and now from this formalism known as classical shadow formalism that we um came up with several years ago one can show that the measurement data allows to efficiently predict fality which is the quantity of Interest CI and by just checking that prediction whether it's close to one or not we will be able to perform certification however um it is well known that a random cliffer circuit will be linear depths so if there's incubate the the depth of implementing random cliffer circuit on a geometrically local setting would be a polinomial Inn and hence or linear Inn and hence while it only requires depths and random cliffer circuit the depth is still pretty high it Still Remains experimentally challenging um people have actually been using this procedure um for system size n that's not too big like maybe 20 or almost 30 but but but when one wants to apply it to like hundreds of Cubit then suddenly uh a linear depths random cliffer circuit is already too too deep and it becomes experimentally challenging to actually run this procedure so can I ask a question about uh approach zero yes so if you make some assumptions about you and you dagger let's say the noise on them are similar some some symmetry assumption right you then sort of infer the errors uh by applying you dagger and ah make rigorous statements that like want certify right yeah yeah I think in so so one needs to indeed like in that sense one needs to say for example try to impose certain constraint on noise but for example if the noise is just a over rotation so like when I implement this gate I would over rotate but when I inverse implement the inverse of that gate I also over rotate then actually in those cases um we will never be able to see the over rotation because like maybe in you it would over rotate forward but then in UAG over rotate backward and cancel each other and hence um in in that regime we can also just run this procedure and see how many times we get all zero out under um some stronger assumption on noise I do think that number the number like how many times we get out zero would be have some meaning to it but then it might only be able to capture maybe some incoherent noise but not be able to capture like coherent noise and so on so so so in that regime like if one makes stronger assumption um not just assume Ro is arbitrary then then I think approach zero could still make sense and actually there are procedur that were of that form right thank you yeah thank you for the question so now let's go back to approach two um based on randomized poly measurement so so in this approach we will remove that deep circuit we now have our lap State rle and then we just perform some single CU measurement for each of the cuit we're going to measure them in a different basis to get a different um outcome and again using this classical Shadow formalism which is a general formalism for analyzing randomized measurement it allows one to also predict Fidelity however due to the lack of using any entangling um gate it's just single Cub measurement at the very end one can show that while it only requires single Cubit measurement on the lab State roll um it does have this challenge which is it requires exponential number of measurements for most Target State s and here most means uh if I just consider the entire block sphere and then I consider one one target State most of them it will require exponential number of measurement and but more precisely it means that um when for most Quantum state they're going to be highly entangled and hence for for side that's highly entangled the randomized poly measurement will require a huge number of measurements it still works for small system like n is like 10 or something but then as n becomes 100 this number of measurement becomes uh too big however I should also note that if the target State side is has low and tanglement then then in many cases this approach could also work because from the measurement outcome one can predict some local property and the local property sometimes tells us about um the Fidelity if the state is low entangled there's another related approach um called cross entry bench Mark but it's also even it's even simpler than randomized P measurement so here we have our Quantum State the lap State what we do is we just measure the whole system in the Z basis so now it will just collapse it into a bit string and we just look at the samples of b string that come out of xcp and from there one can utilize this um one can utilize the standard approach in Cross entropy Benchmark to estimate the cross entropy and establish this value called the xcb score and now by checking xcb score one can say like okay if the XB score is close to one then I say I've do a great job in creating the right state but if XB score is close to zero I say there's a lot of noise so the advantage of this is only requires single CU measurement that Z basis measurement on the lab State role but the key challenge is that actually doesn't really rigorously address the certification task because here if we only measuring Z basis effectively you're only looking at the diagonal part of the density metrix rle and we're not going to see anything about the off diagonal parts and hence if the off diagonal parts has a lot of error in a way an off diagonal part is like really the quantum part of the density metric R so if there's a lot of Errors from the quantum side um the XB cross entry Benchmark might still report a perfect xcb score if the diagonal part or the classical distribution part is is is is perfectly correct and hence even if there's a perfect xcb score the true State role might still be very far away from the target State Side however in various again like in some restrict like uh if one assume that the noise on Roll has a certain form then in some of these restricted regime or kind of more conditional regime then actually CB could be a pretty good um estimate however from a more rigorous perspective like maybe there's coherent errors Etc again xtb might fail quite badly which we will show later in the numerical experiment so just some yeah Robert one question from the audience I believe it was about approach two yes and can you give some intuition why one needs exponential in end measurements yeah that's a good question so so just to kind of say a few more words about randomized poly measurement so by performing each Cub like measuring each cubit in a random X Y or Z bases each of them complete independently so this may be x y and then this is x z and so on um we will be able to efficiently estimate any reduced deny matrices so if we have like a five body five Cubit reduced sensity matrices like this Cubit this Cubit this Cubit and some other two then we can um officially uh learn them using this approach um however the the tricky thing is that as the reduced sensy matrices start to grow um like if it's now 6 Cubit 7 Cubit 8 Cubit and so on the number of shots in order to get the error to get the estimation error down would grow exponentially in in in the subsystem size and hence um in a way once you really think of these randomized poly measurement at least when one you go through the standard classical Shadow formalism is very good at capturing local reduced sensity matrices or local observable but now if we have um this target state which is highly entangle like think of the observable that we wanted to predict is SIII cat Brasi that's a like when when sigh is highly entangled then then that observable would be highly non local and hence in order to use the standard um randomized poly measurement followed by classical Shadow uh postprocessing one would require a huge number of measurement in order to get that error down so that's some more intuition on on on on the challenge that one faces when one just applies this like randomized poly measurement is really most useful when the goal is to estimate like Honan local observables some of local observables or reduce sensity mates um entropy of subsystems but for small subsystems Etc thank you so just to summarize some of the existing challenges so as of now all existing certification protocols they either require a deep Quantum circuit before measurement and in that case the number of measurement would actually scale pretty good like in randomized cliffer measurements the the number of measurements would scale independent of system size so even if we have a thousand Cubit versus like one cubit the number of measurements to do certification to estimate fidelity is the same however if we don't allow applying deep circuit Because deep circuit are challenging then we need to um use exponentially many measurements if we wanted to perform certification on highly entangled state or um if we wanted to limit oursel to only polinomial number of measurement then in that case it only works for some specialized Target State like when s can be represented by a metrics product state or when s is is low and Tang gold and so on if or finally if uh if it somehow works for General State um Works uh yeah doesn't require deep circuit and only need a polinomial number of measurement in many cases it would now lack a rigorous guarantee that it will work in all cases it might work in some restricted cases like when the lab state is subject to White Noise um but when the lab say is subject to coherent noise or some other more um also pretty reasonable noise then they might fa pretty badly so this raises the following question which is can we rigorously certify highly entangle Quantum state by performing few single Cubit measurements so so this question is essentially left open from the previous work or if you're even more ambitious one can ask can we rigorously certify almost all Quantum state by performing only few single cubbit measurements and this is the question that we're going to set out to answer in in this talk so so the outline of the talk is as follows first I'm going to present the theorem that says the the the answer to this question is actually yes and then I'm going to talk about the protocol for establishing the theorem and finally talk about some applications one can um use these new protocol for so let me begin with the theorem statement so not that the theorem might seem a little bit magical but hopefully it will be cleared up when I talked about the protocol so the first theorem states that for almost all encb state side we will be able to certify if the state in the lab if the lab State role is close to side or not using only order n Square single Cubit measurements weall that previously if one uses like randomized poly measurement then this would be exponential in in on the other hand if we use the xcb then this theorem also doesn't hold because if we do xcb R might be um very far away from s even when one gets a perfect xcb score so this theorem says a there's actually a protocol that allows us to circumvented and work for almost all incubate Target State s so we call that s is the thing we wanted to create R is the thing we actually create in the physical world so one lives in our imagination one lives in the physical world and furthermore for the certification procedure uh applies to any role so we don't need to make any assumption on what kind of Errors is happening in the experimental lab there could be coherent error there could be white noise there could be all kinds of errors as long as there's too big up an error then the certification procedure will say oh there's too pick up an error and it will be able to catch catch those errors and if the role is actually close to S then it would also be able to say yes it is close to S and we have did good job and furthermore one um another interesting aspect is that even when the target State s has an extremely high circuit complexity um the number of measurements only grows uh as order n Square so even if there's exponential and un circuit complexity um the single C measurement is still order N squared that's another very nice aspect about this theorem so what I ask um so yeah so this theorem essentially answers the question in the in the positive it says there is if one is clever enough run the right procedure then one will be able to certify any state role with just single Cube measurement one could think about it in a different way suppose s is a highly entangled State um in that that we wanted to create intuitively it felt like single Cub measurements will only be able to Showcase what is the what is where are the local property because they're essentially very local localized measurement we're not doing any entangling gate and hence somehow it's telling us that by just looking at each Cubit individually we can somehow already see some of these high complexity entanglement which is one of the surprising aspect of this theorem because for example if you use randomized po which is also s measurement it only allow us to see local things which also matches intuitively like local measurements tell us local things but the theorem is telling us that local measurements can actually tell us about the underlying um highly complex entanglement however there's this a kind of unsatisfying part about the theorem which is that this even though which is that this only works for almost all state but not for All State and in practice what we often care about is we have a specific Target State side that we that we wanted to generate so so even though this theorem is nice it says for most of these states we will be able to certify them most of these targets say we will be able to certify them in practice we only care about one like that State Side might be very useful for sensing or for simulation Etc so then we will actually want a theorem that can be more specific given a state s how can I tell if that state s can be efficiently certified or not and in order to establish that um in order to establish that we need to define a notion called relaxation time more precisely we need to connect the task of Performing certification to somehow a kind of an abstract idea of suppose there's a there's a person randomly walking on a Boolean hyper Cube how quickly would it spread out to become close to some stationary distribution so this is a new kind of a connection that we that we discover in in this work so to be more precise suppose we have a very specific UNC Target State side that we wanted to create in the lab and now we will also choose some basis so we're going to fix a basis B where B um is a bit string so this could be the standard Z basis but it could also be some other bases like maybe some cubid is in an X basis some cubid is in a y and some Cub the Z bases choosing this bases can actually be important and could change this relaxation time um but for now we're just going to fix the bases so given side we fix the bases now when we measure side in this basis it would generate a distribution over these bit strings so we call that the measurement distribution Pi of B and we can think about Pi of b as a distribution over the Boolean hyper Cube so we write down this Boolean hyper Cube where each two different bit strings are connected if they differ by say at most one bit so now we have this Pi of B which is the a distribution on this Boolean hyper Cube we will Define a random walk on this unbid buing hyper Cube random walk is very simple suppose that we start from some point B now what we will do is we will randomly pick one of the neighbor say we pick B Prime then what the Walker would do is it will with probability proportional to Pi of B state in the B string B or with a probability proportional to Pi of B Prime jump to the B string B Prime so that's the random walk it would start from somewhere like B select a random neighbor and then either stay or jump there and then we just keep doing this step for uh some round and the notion of relaxation time um one can it's very closely connected to what's also known as mixing time they are off by a small factor but they're essentially the same essentially it's the time it takes for this random Walker to um to like after okay so initially it would start at some point where does it start it would start at a point that being sampled from the measurement distribution but once you think about it as when it start there it would be a a kind of a a point-wise distribution probability one it's there and now after one step there will be a the probability will start spreading out and then one could ask how many steps does it need to take such that it will be spread out to become kind of close to the measurement distribution it doesn't have to be exactly equal it just need to be like constant close by a small constant and that minimum time for it to take to relax to the station like starting from point start to relax to the stationary distribution is known as relaxation time or it's also closely connected to the mixing time so we let to to be the time for it to to relax to the stationary distribution time now we have the second theorem which says that for any Incubus State sign that with a chosen basis B that would induce a relaxation type to then we can certify whether the LA State Ro is close to s or not using order to single CU measurement in F way we can restrict ourself not just to General single Cube measurement but to Independent po basis measurement that is we just measure each cubit in in a random like pal like a not not fully random P bases but in some P bases then in that case we can show that we also only need order t squared single cubic measurements so so this theorem essentially allows us to focus on a specialized State pick a basis sometimes the basis would actually change the relaxation time but we can say pick the bases randomly and and and and hope for hope for the best but but yeah Pi a basis it would induce a relaxation time and now the relaxation time actually dictates how many measurement we need in order to perform certification and how one can get theorem one is essentially showing that for actually almost all state for most of the states um the relaxation time is bounded by order N squared it's only quadratic in system size and hence the number of measurements would be quadratic in system size so that's the relation between theorem 1 and Theorem two theorem two is more General and allows us to specialize to a specific state so now after talking about the theorem we can say okay so what is the what is the protocol that allows us to certify some of these highly entangled state or certify almost all state using such a small number of measurements where prior results would require an exponential number of measurements to just check if the anent is the one we want or the state is the one we want the protocol is actually very easy we just repeat the following for a few times where the few like how many times do we need it depends on relaxation time to and the protocol is like this we have our Quantum State and we're just going to do some single Cube measurement and how we how how do we do it first we will randomly pick one of the Cubit say Cubit X this okayy and then for all except that Cubit we will measure in a z basis so let's say we already fixed some bases so now the the the Z Bas is according to the fixed bases so we just measure everything in the Z basis and then for that single Cubit that we chosen the random Cubit that we chosen we will measure that random cubit in a XYZ basis so randomly pick a cubit everything else measure in a fixed basis and for that chosen Cubit measure in a random po bases and that's it for the entire procedure we just repeat that for some number of times which for Most states it will be order unsquare number of times and for a specific state with mixing time or relaxation time tow repeat that for order to square times and that finishes the experimental part so it's a very simple experiment that we do in a lab and now the question is how do we how do we postprocess these data how can these data from these simple looking experiments tell us whether the lab state that we are measuring on getting this data from is close to the Target State side or not so the post-processing looks like this and we're going to also look at one shot each time so one shot means we will will um pick one random Cubit measure everything in Z bases which generates a minus one bit and then for that one cubit we will collapse it into a random P basis so first the measurement outcome on the Z basis measurement or these pink ones there was specifi two bit string that differ exactly by one bit and the reason is because the measurement outcome give rise to a minus one bits and then there's one bit that is not yet determined which correspond to the blue Cubit here so if the blue Cubit turns out to be zero that's b0 if blue Cubit turns out to be one that's B1 so so the measurement on the Z basis actually tells us an edge it specifies a random Edge on the Boolean hyper Cube so one you you think about it as every time when we perform this measurement we will be able to see one of the random Edge on the Boolean hyper Cube B 0 and B1 and now what do we do we we we look at our ideal state so for the ideal State when we perform this measurement on the a minus one cubits after the measurement the Cubit X will be collapsed into a specific one cubit State Side b0 B1 which is proportional to this state shown here B 0 side 0 plus B1 side 1 and why could think about that as being kind of like um correspond responding the the probability amplitude for this point as well as for this point and it's like a super position of of this so in a way the the post measurement State on this Cubit essentially tells us information about how does the probability amplitude between the two vertices on this randomly chosen Edge kind of look like what is their relative probability amplitude so now what we would do is we would use this randomized P measurement on this Cubit X plus the classical Shadow formalism so what what can classical Shadow do classical Shadle um apply onto or which is a method for processing these randomized P measurement can give us a onshot on bias estimator for the Post measurement State on this Cubit the postmeasurement state for the lab State for the thing in the LA the real one not the ideal one and because it would give us an unbias estimator we can then use that unbias estimator for this single cubid state to measure the Fidelity with the ideal one cubit state so we can um essentially predict the Fidelity we we cannot predict it super accurately but we will get an again unbias estimator so what is on bias estimator just means it's a it will fluctuate but the the average of the fluctuation or the average of that random number will be the true Fidelity on that one cubit state so we call that Omega and then what we do is we just average over the entirety of experiments not that up until so far I'm just talking about one shot experiment so the procedure is pick a random Cubit measure everything else and measure that single cubit in XYZ given this one shot how do we analyze it we essentially just look at what is the post ideal postmeasurement one cubit State on Cubit X and then use the classical Shadow formalism on Cubit X to predict an unbaned estimator for the fatality with on this cbid and then we just average over all of these shots and after averaging over we will get the uh an an an estimator a pretty good estimator for the true expectation value of this one cubit Fidelity Omega and we call that one cubit Fidelity um or this kind of postmeasurement one cubit Fidelity the shadow overlap so why do we care about this quantity like what is the significance of this quantity the key Fe feature that we Pro in the work is that this Shadow overlap e Omega actually accurately tracks the true Fidelity Cyro side how does it track it essentially we can prove that if the shadow overlap is close to one it implies that the Fidelity would be close to one but there will be a slackness that corresponds to the relaxation time on the other hand if the Fidelity is close to one it also immediately imply one can show that immediately imply that the shadow overlap would be also be close to one and hence by just looking at if Shadow overlap is close to one um we will be able to check if the Fidelity is close to one they're not exactly uh the same um they like they they can move a little bit differently they can move differently which I'll show later but but as some if the goal is to do certification then that they match up very nicely like if one is close to one then the other is close to one and vice versa so now to provide more physical intuition for shadow overlap let me just write down this explicit formula so the excit formula is like this um first there will be a average over different Cubit I and then there would be a sum over different a minus one bit string corresponding to everything except for Cubit I so Sy of I is the one we choose we're going to choose that randomly and then we are going to sum over all different a minus one bit and then we're going to estimate this quantity so this B not equal to i r b notal to I once you think about it as a this is the probability that we will get this B not equal to I bit string multiply by the post measurement one cubit density metrix on the other hand the right hand side here this yellow one that's essentially the same as what is the post measurement one Cub State for the ideal Target State side and we just take the in a product between them we take the Fidelity between them and this just kind of writes out more succinctly this whole postprocessing procedure that I was talking about earlier now we can plug in some simple State let's say the lab state is all plus but actually the target set is all minus in this case um the Fidelity is zero we can also plug this in into the formula above and we will find that the shadow overlap is also all zero however things start to deviate if we say um the lap State we're actually doing quite a good job it's only the final Cubit is flipped It's A Plus instead of minus in this case the Fidelity which is a very stringent definition would still give us zero on the other hand what shadow overlap would give us if we plug into the one above is that it would actually give us n minus one/ n and more precisely if there's like K Cubit that are being flipped um then the shadow overlap would be M minus k/ n whereas the Fidelity would always be zero so Shadow overlap provides uh has this uh Hamming distance flavor um for for for some of these states so it could deviate it doesn't always track fality exactly it could deviate from Fidelity but it deviates in a very natural sense it deviates from a more like Hamming distance flavored and that's because it's kind of looking at each Cubit individually um but yet um Shadow overlap while it only relies on it only relies on performing single Cube measurement but it can tells us about um this Fidelity so so as long as Fidelity is close to one then Shadow overlap is close to one but if Fidelity is zero then Shadow overlap will be far from one but it might not be that far like it will depends on um like it has this more like Hamming distance flavor and that's corresponding to like the slackness here the slackness here essentially corresponds to um Shadow overlab having more like a Hamming distance flavor compared to Fidelity uh Robert before we move to Applications there are some questions from the audience yeah um so let me start all right so one question is do we need tow squared copies uh of this state for the protocol uh because it's the relation between to and N that needs a clarification mean is it is it mostly to to or is a question about um copies or is the question about kind of copies of this stat yes yes so in order to run this protocol I mean actually in order to run any um learning protocol for in the quantum system setting because whenever you measure the quantum state it will collapse the whole system so now you no longer have the same state anymore but what one would usually do is one would use the same physical process to prepare another sample of this state so not that a density metrix in a way is already has a notion of sample like every time we're preparing some pure state but there's a Randomness and there's noise in it so hence it becomes a density metrix so one copy of a density metrix should be thought of as like a one sample so so indeed this requires um each measurement each of the single CU measurement requires one sample of the quantum state which can be realized by just running the entire physical process for generating one state one time and that's basically a necessity for any protocol or any learning procedure if one wordss in the Quant world and can one think of this this is another question from the audience can one think of this as a directed diffusion on the space of quantum States is that oh yeah so from a in a way I guess is that more talking about like this relaxation time I think here it definitely has a lot of flavor on this diffusion one can think about the this random walk as being like I'm on a I'm on a point and then I'm kind of diffusing on this Boolean hyper Cube and the diffusion is kind of um kind of determined by the quantum state so there is a lot of flavor here but I should say this whole thing like this whole part is only in our kind of again it's in our kind of mathematical world it's in in our imagination in a way in the experimental protocol we're not doing any kind of actual diffusion of of of the of the system so H why why is the two related the the relation is like this it's like um like what the measurement protocol is essentially doing is essentially checking like there's this exponentially Big Cube and now I'm just looking at a small number of these edges like I'm randomly sample these edges based on the measure and I'm just checking kind of like I'm just checking if the if this if the if the edge behaves the right way because we're just taking this Fidelity that that essentially help us check if the H Edge in the lap State behaves the right way as the ideal State on this random Edge and now the idea is like suppose that most of these edges are behaving in the right way and what that means is when we now perform this diffusion on this Boolean hyper Cube this random walk then um the random walk will be correct and then because the random walk will now walk to become the entire State and then hence um if when we check a few edges it all look correct then it means that the whole state actually looks correct and that's how we get this um this key feature and and this is the only place where the realization time pops up and the notion of random walk pop up it's in this key feature that shadow overlap being high means Fidelity is high so so so so that's uh that's kind of the relation so that in the experimental World we're just checking edges we're making sure that when the Walker is random walking that the edges that is going to use is going to be pretty good I that's so we have many more questions but I'll let you uh proceed with applications and then if there's time uh I'll ask the rest sounds great so now that's dive into the the application after seeing the theorem and the protocol so I can ask like why do we care about certification I mean initially I did motivate it but then actually certification is really a very foundational thing that have many application if one try to be a little bit more creative in how one uses it so let me start with some more direct application which is the starting motivation for doing benchmarking that is we wanted to use Shadow overlap to check to check if the state created in the lab uh has a high fidelity to the state I wanted to generate and here I'm going to present some numerical experiments comparing to xcb for different state and for different noise model as a starting point let us look at a situation where xcb is known to work so here we have a random Target state so the target state is randomly sampled and it's fixed you just wanted to create some random state which is highly unentangled just four cubits very small and then we assume that the in the experimental setup in the lab State there will be white noise so we prepare essentially s um but subject to some Global depolarizing noise so this is a situation where it's well known that xcb works actually in a way xcb is designed so that this situation will work and now we can look at um the the plot here as the White Noise start to increase we do see that the I mean the true Fidelity would have to go down and we see that the xcb is performing pretty well because it's known to work it's designed to work in this case and it's also performing pretty well and going down on the other hand we see that the shadow overlap um is also um tracking true Fidelity very well that's kind of a sanity check to show that shadow overlap in situation where xcb Works would also work and the same holds for 20 Cubit now we can say what if it's not white noise like in the experimental setting white noise is kind of an idealized noise there could there be a lot of other noise non-unital noise coherent noise Etc so here's another situation where instead of having White Noise We have coherent noise now in this situation there's no longer a rigorous guarantee for xcb in in cerent noise xcb might felt pretty miserably um but in hard random State it's uh it is failing but it's not failing too bad so as the as the true Fidelity start to go down like as coherent noise start to increase the true Fidelity will start to go down down and actually be and also go down however we can see that it start to deviat quite bit quite quite a bit from the true Fidelity like when a true Fidelity is still around 08 it might be reporting a number that's 7 so here it's actually underestimating the true Fidelity on the other hand Shadow overlap it tracks the true Fidelity much better than XB because um in a way um Shadow overlap as we have seen from the measurement procedure even though it almost measure everything in the Z basis which are I mean xcb measure everything in Z basis Shadow overlap measures almost everything in the Z bases except for one cubit that one cubit it might Meine a random XYZ and just that tiny bit of additional change um actually help it greatly in seeing coherent noise in seeing the off diagonal elements and that's how we establish the theorem and this can also be seen from a numerical perspective for bigger system likely due to some concentration of measure type argument um XB in this case perform quite well but this is for hard random state so now we can also look at a more structured State let's say we have some product State and we just apply some phase onto it now again we look at the White Noise regime in white noise it's again xcb is known to work but in this case we saw that xcb has a much bigger fluctuation so the the standard deviation for the SCB asate is very high so here we're comparing the same number of shots xcb requires a yeah actually has a much bigger variance even though its expectation values correct it has a big fluctuation on the other hand Shadow overlap has a much smaller fluctuation and captures the true Fidelity very well for 20 Cubit um similar situation happens and now we can look at um random structure state but under coherent noise now this is really in a regime where xcb are going are are really have no guarantee at all and and when we apply it um when we look at this case when there's coherent noise in the system and this is a structure State and suddenly we see that xcb is is is going um kind of crazy so as the cerent noise start to increase somehow xcb um was saying that oh the Fidelity is actually increasing and somehow it's increasing Beyond one and and I should say this is because this is a regime where xcb is uh yeah xB is not a rigorous protocol because it only looks at diagonal it cannot look at off diagonal so coherent noise is basically affecting the off diagonal it would also affect the diagonal part but it might affect it in a in a in an unexpected way so so so there might be a lot of noise in the off diagonal part but on the diagonal part it might be less noisy or might be the opposite of noise and and in that case somehow the the the xcb is behaving in a very unexpected fashion the true fality still still going down because there's more noise shadow of up is tracking the true Fidelity well but then xcv is just going going going pretty wild in 20 Cubit similar situation still happens it's not as wild as the true fality start to go down we see that shadow of lab track it very well but xcb is is overestimating it quite substantially so so as we can see in when we have coherent noise instead of white noise xcb sometimes can under asmet fality some sometimes they can significantly over asymmetri Fidelity and and and and in a way that's just showcase that xcb has stated earlier it doesn't really have rigorous guarantee um so if you have a lab state that have some some other type of noise that are not white noise then the behavior of xcb can be can be pretty W and coherent noise is one of a point where where where this is like one of the common failure mode for using xcb so that's that's the first example for benchmarking but now we can be a bit more creative in using this certification protocol we can also use it for doing machine learning based tomography so suppose that there's a lab State uh there's a state in the lab and now here we're not trying to do benchmarking um we just wanted to learn what is that state in the lab and we wanted to essentially turn that state in the lab into a classical neuron Network representation of that state also known as neuro Quantum State I have a lot of very exciting progress in recent years so a neuron Network can represent uh various kinds of quantum State even those with volume law entanglement I one of the key Challenge and and how can we how can we use Shadow overlap in this regard essentially we can think of it this way so the lab state is fixed but then there's a there's a model for the Target State s so so think of the S as being the thing in our mind and now we are going to vary the thing the model State the Target or the side we're going to very side to maximize the shadow overlap because shadow of essentially allow us to check um the thing in the real world and the thing in our mind in our imagination so here the thing in our imagination is this neuron Network representation and then the thing in the real world is fixed now we can vary the the the thing in our imagination this neuro quum state to try to maximize Shadow overlap or maximize some other loss function based on Shadow overlap and then because Shadow overlap this has this nice property which says that if the shadow overlap is close to one It means that the lap State and the and the target state is close in a stringent fality and hence it allows us to perform a certification of the ml model so prior to this work there have been a lot of um results or or or or work that try to utilize neor Quantum state to do tomography to learn a lab state one of the key challenges that people have faced is that it's often times very hard to judge if the neuro Quantum state is hallucinating some property like we collect some training data from the experiment and now we've Ed it to train neuron Network and now the neuron Network might hallucinate things it might thought that there's some pattern in the in the quantum state but actually there's not and then if those kind of things happen which sometimes it does sometimes it doesn't then then when it does um the prediction coming out of the neuron Network can be completely wrong and hence you might thought that oh maybe my state has topological order or has some exotic property but actually there's not and and hence in those setting it could be kind of dangerous in applying neuron Network because it's a heuristic and then and then it has a it can hallucinate and so on but now if we use Shadow overlock to certify the neuron Network model to check that this is actually close to the experimental setting and now we can be certain of all the prediction that come out of the neuron Network and then we can use that neuron Network in a very rigorous fashion like now after certification the prediction from neuron Network would all be accurate because it has been certified and then we can make a build scientific statement based on these heuristic and very but very powerful new network model so that's another application so let me just showcase a little bit more in detail so let's say we have the standard neur neuro Quantum state which is just a neuron Network given some bit string it would spit out the probability amplitude that allows us to represent some wave function sign we're going to use some a variant of it because in a way in in our setup in order to estimate Shadow overlap we need to always compare between two bit strings that differ by one bit so we're going to Define this kind of relative neuro Quantum state where given two bit string that differ by one bit it will tell us what's the ratio of their probability amplitude note that this also represents s because we can just use this neur network for un times to to kind of just m multiply this ratio we fix some B string as a starting point and now just multiply this ratio and it allows us to get the the the the probability amplitude for any other Bing now we're going to train this NE network using um using Shadow overlap so we're going to consider some 120 Cubit state with extremely high circuit complexity and volume law entanglement and hence the the Purity which as like the subsistent Purity which is a measure for entanglement will go down exponentially so in in the bulk of it it will always be exponentially small and then it will come back up now when I randomly intialize this neuro Quantum State it's being initialized into some product state so the subsistent perod is uh is all one meaning that there's no in tanglement across different cuts and now what we can do is we can train a u a loss function using this kind of the same measurement data that was used to estimate Shadow overlap can we Define a lock loss that was based on the same experimental data and then after that we can use like a kind of a verification set or a test set validation set or test set to to to certify um the whether the neuron network is really doing well or not using Shadow overlap so here is just the training Dynamics so in the xaxis we have the number of training step in the neuron Network um as as it start to train originally just kind of wobbling around trying to explore the landscape and suddenly find the right structure and then it quickly goes down so the lock loss decays very quickly and then becomes very good and we can also look at the behavior but but then we don't really know if it's actually good like the lock loss just tells us it's 0.35 how do we know if 0.35 is the is the best like it actually fits the the actual experimental system or or it's it's not then what we can do is we can do at the shadow overlap the shadow overlap we see that originally zero but then after it's it's going down it's Al shoot up and it shoot up to become one and when it's when it's one we also know that the Fidelity would be one note that here even though I'm plotting the fality but in an actual experimental setup where it's it's very hard to estimate fality all we can do is uh is estimate the the shadow overlap because Shadow overlap can be estimated using single Cubit measurement so so now by looking at Shadow overlap we see that it's very good which implies that the Fidelity was would also be pretty good and together we can make sure that all the prediction from the Train new network in this case will be accurate and so now when we use the train neur Network to predict the subsistent Purity we see that it tracks the ground truth very well but furthermore we have a certification that come from it that come from shut overlock so now the the other example is we can then turn the things around we still have a Target we now fixed the target State there side but now we allow the lab state to change we try to optimize the circuit in the lab to Max maximize the shadow overlap with s and that allows us to optimize circuit to prepare our desired State here we consider um constructing some incubate matrix product state in a one-dimensional geometry using hetar CZ and T Gates we consider a setting where we do know what's like the kind of the optimal sequence for generating that MPS and that's say we just run the optimal sequence to generate it so that's what this xaxis is the the the construction step so every time we will implement the right gate to generate it and now we can compare between just look at how Fidelity and Shadow overlap Compares in this setting we see that especially for bigger system when we add in the right gate one at a time Shadow overlaps start to increase linearly towards one and at the very end it becomes one however fality has a much more like a baron Plateau structure like as we have seen before like uh even if we are putting in in the right gate it's still saying oh Fidel is zero because the last few gates are wrong only until the very end it says we can do it but not that this is the optimal sequence which in general we don't know so in practice we what we we would would see is with there what we would do is we will try to like try to train the circuit um maybe by multi Carlo or variational approach and try to make it so that it it becomes close and we can use different loss function we can say use Fidelity um or or we can use Shadow overlap so so Fidelity is bad in various regard first we don't know how to estimate it experimentally efficiently and and second it actually has this Baron Plateau feature that allows us to be very hard to train on the other hand sh the best property of Shadow overlap is that it can be estimated using single cubid um measurement which a lot of other quantity um can have a hard time doing and not only that it also has this more like a Hamming distance flavor which means when we try to train it um it it behaves much more better so here here is an example where if we train it using Shadow overl versus training using Fidelity we see that when we train it using sh Fidelity it's just kind of randomly walking and hence if we evaluat the shadow overlap it's very bad and if you evaluates the Fidelity is just zero on the other hand when we train it using shut overlap we see that if as the optimization steps start to increase the shut over left slowly steadily increases and then at the end it becomes one on the other hand fality is much more stricter so when when when we see improvement from a shadow overlap perspective we doesn't see anything in the Fidelity case only until the very end it start to shoot up and becomes one and that's because Fidelity has this more like a baron Plateau feature on the other hand Shadow overlap is better than Fidelity in two regard first one can estimate it efficiently by just performing single Cube measurement and second it doesn't have this Baron Plateau fature so that concludes some of these applications and and now just to conclude in this work we Pro that almost all Quantum States can be officially certified from few single Cube measurement but this only holds for almost all so right now it's not entirely clear if one could prove or one can show that all Quantum State can be certified efficiently using SQ measurement and to be honest it's not even clear that the following question does there exist some state that we can prove that it cannot be certified using S measurement and I think understanding that question is uh is going to help us understand if the more ambitious goal of whether one can just certify all possible Quantum State using single C measurements that's it for my talk thank you so much for listening thank you so much Robert for the very exciting talk um maybe for time we have some time for quick questions I think you already answer this question that um so there there was a question from the audience that most the St can be certified using quadratic number of samples can you give an example of a states that this method doesn't work on and I believe you just said that we just don't know if such a state exists but we can't also prove that uh they don't exist is that right yeah in a way that's what this question is about but I also add like uh the explicit like um there's actually a lot of variant kind of similar to like classical Shadow one can have different variants of it there's also a lot of variants of this procedure so here I I talk about uh like one version of it is we just pick one cubit measure everything else in a fixed basis and then measure that cubit in a random basis one can also consider variance of it where we pick two random Cubit or three and measure everything else in a fixed basis but that two cubit in a random po basis um we I do have some examples where one needs to like have some example of state where if one just use like this uh one cubit version then then then it would not be able to certify efficiently but if one uses side variance like measuring two random cubit in a random XYZ basis and suddenly it can be certified efficiently so there are that that example but but at the same time like measuring two random cubit in a XYZ bases that's still single Cube measurement and hence uh and hence the final question so so the specific one that I talked about there are counter example where one cannot certify efficiently but one can look at a few variants of it and suddenly it's not clear that there are states that um cannot be certified and the other question is um I think it's about calculating uh this or maybe the samping so they're asking if like okay classically performing the partial trace of the target state is not classically efficient in general is that a bottleneck yeah so so here the all the theorems that I talk about was mostly focusing on kind of information theoretic aspect which which means um it just talked about how many measurements do we need but then we can also ask what is the computational time like the measurement protocol is efficient but then what is the computational time that requires us to to do it so the most computational like potentially the most computation intensive part is really just this step what is the ideal postmeasurement stay or more precisely when I am given um a b string B 0 and B1 can we estimate this probability amplitude efficiently so this is also a kind of a what you think about is this is kind of a a requ like a requirement of this procedure which is also the same requirement as in like cross entry Benchmark and various other protocols so so this is um definitely a bottleneck of all of these existing protocol there have to be some way of talking about what SII is like suppose I have um I just have a SII being represented as a Quantum circuit then in general this not clear how one could estimate this but there are also a lot of representation like NE Network representation for example is explicitly designed to be able to predict what this is or tensor Network representation or if the state have certain structure that we already know of so one have to yeah so one have to have a way to access this quantity like the probability amplitude for some randomly sampled bit string if we have that then the whole procedure the whole processing would be computationally efficient as well and can be deploy in practice um but but for a very general or like very worst case situation where s is just given to us as a Quantum circuit a very deep one then then it's not clear then it's not clear so so definitely this is a limitation but I would say it's not just a limitation of our protocol but of a lot of or the wise wide range of all existing certification procedure and then like if you're given access to like this is a question from me if you're given access to a perfect quantum computer uh let's you have an ideal quantum computer you have your noisy quantum computer but you can't swap it stuff so you can't do it swap test for the certification yeah can does it allow you to calculate this efficiently uh you mean like suppose a does a perfect quantum computer allows us to calculate this yeah it would help I mean there are situations where um we we we cannot calculate it um I mean efficiently on class computer but we can calculate efficiently on quantum computer um there are situations like that that help but I think in general if I just have a very deep Quantum circuit for preparing S I just give you that circuit and now I ask you what is this probability amplitude for some randomly sample bit string I'm not entirely sure what the complexity of it is um but I suspect in some worst case regime it will it might be Beyond bqp and hence not be computable efficiently on quantum computer yes okay um yeah thank you so much for this great talk um and and I yeah we'll see see the audience in the next seminar after the holidays all right all right thank you so much bye everyone for