Probing Quantum Many Body Dynamics with Tweezer Arrays ▸ Manuel Endres (Caltech)

Summary

Highlights

• Manuel Endres explains how tweezer arrays trap single atoms by using tightly focused laser beams. 🎯
• The dynamic nature of Rydberg atoms allows for high levels of interaction in quantum systems. ⚛️
• Endres' team successfully creates maximal entanglement entropy states with a high degree of precision. 🤯
• Innovative techniques measure and benchmark quantum fidelities, pushing the boundaries of current technology. 🚀
• The future of this research could lead to significant developments in quantum computing and metrology. 💻

Key Takeaways

• Tweezer arrays are revolutionizing quantum physics with their ability to trap single atoms. 🔬
• Random many-body quantum dynamics can be naturally observed within experimental settings. 🎲
• Quantum errors can be systematically benchmarked to improve fidelity in experiments. 📊
• Experimentation with alkaline earth atoms is yielding promising results in precision quantum studies. 🧪
• The lecture highlights the potential of expanding these studies into two-dimensional quantum systems. 🌌

Overview

Chapters

• 00:00 - 10:00: Introduction and Tweezer Arrays The chapter titled 'Introduction and Tweezer Arrays' starts by acknowledging the efforts of the workshop and conference organizers. The discussion progresses into the most recent findings from the group's research focused on studying Kondo many-body dynamics and benchmarking employing tweezer arrays and Rydberg atoms. The ultimate objectives of their research vary from quantum computing and simulation to metrology and high precision measurements, and potentially extending towards quantum networks in the future. The chapter will delve into some new insights obtained from these studies.
• 10:00 - 20:00: Rydberg Atoms and Interaction The chapter explores the interface of quantum information and many-body physics, highlighting experimental and theoretical findings.
• 20:00 - 30:00: Entanglement and Measurement Techniques In this chapter, the focus is on exploring systems with up to 60 atoms to generate maximum entanglement entropy states. A comparison between quantum and classical resources is also discussed. The chapter delves into experimental systems using tightly focused laser beams, known as optical tweezers, to trap single atoms captured from a cold cloud. These techniques are utilized to generate larger scale arrays, potentially with spatial light modulations.
• 30:00 - 40:00: Quantum Many-Body Dynamics and Random State Ensembles The chapter explores devices known as modulators or twist optical deflectors, which split an input beam into multiple beams in a controlled manner. This technology is scalable and can be used to create large arrays, such as a depicted example with ten thousand tweezers using light. The experimental process involves loading these tweezers from a cold cloud, capturing fluorescence images, and then resetting the system to start anew, indicating a cycle of destructive reloading.
• 40:00 - 50:00: Benchmarking Quantum Simulators The chapter discusses the concept of benchmarking quantum simulators with a focus on the challenge of creating defect-free tweezer arrays. These arrays are essential for many-body experiments and quantum computing. The stochastic loading process involves taking single-shot pictures during repeated runs, where each time results in a different tweezer configuration. To overcome defects and generate defect-free arrays, an 'atom by atom assembly scheme' is proposed. This involves loading the array stochastically, imaging it to identify full tweezers, turning off empty ones, and then rearranging the atoms for optimized configurations.
• 50:00 - 60:00: Summary and Acknowledgements In this section, the focus is on generating defect-free arrays using advanced techniques. Different methods and geometries can be utilized in every shot, allowing for innovative and varied outcomes. The discussion highlights modern widely used methods where features are generated defect-free with significant flexibility, such as arranging hundreds of atoms in 2D and utilizing some classic 3D structures. atomic distances can be adjusted freely, offering flexible geometries and significantly faster repetition rates than traditional cold atom experiments. Some limits are acknowledged, indicating there are still challenges to be addressed.

Probing Quantum Many Body Dynamics with Tweezer Arrays ▸ Manuel Endres (Caltech) Transcription

• 00:00 - 00:30 thank you first of all thanks to the organizer for putting together a wonderful workshop and conference so i'll talk to you a little bit about most recent results in our group in you know studying condo many body dynamics and benchmarking using tweezer arrays and rip back atoms more broadly the goals in our groups range from you know quantum computing and simulation to metrology and high precision measurements even and maybe longer term also quantum networks in the day i'll focus on some new insights that
• 00:30 - 01:00 we've found both experimentally and theoretically that are put at the interface of quantum information and many body physics i think somewhat appropriate for the for the conference uh brief outline i'll give a somewhat short intro to tweezer arrays and rhythmic atoms for those who don't know it and then show a few of our single and two qubit or two atom results and then move on to some observation where we find random state ensembles and portal thomas type distribution from purely hamiltonian many body dynamics and then i'll show you how we use that to benchmark
• 01:00 - 01:30 systems with up to n equals 60 atoms towards generating maximum entanglement entropy states and and there we also perform some sort of a quantum classical comparison in terms of resources that is quite interesting okay let me jump right into it with the experimental system uh so the tool of the trader tweezers so these are tightly focused laser beams that you can use to to basically trap single atoms and that you catch from some sort of a cold cloud in experiments we generate larger scale arrays either with spatial light
• 01:30 - 02:00 modulators or so-called twist optical deflectors these are basically devices that take an input beam and split it into many beams into a control in a controlled fashion you can cross these devices and make this larger scale arrays here's for example a picture of about ten thousand tweezers just a tweezer light so it's a relatively scalable scalable platform in principle um in experiment you load these tweezers from a cold cloud and then take a fluorescence images and this is shown here and then you drop basically everything and start from scratch so this experiment are some sort of a destructive cycle of reloading and
• 02:00 - 02:30 losing atoms um and this is single shot pictures of basically repeated runs in such a tweezer array and you see every time you load basically different configuration of tweezers here's a stochastic loading process if you average everything it looks nice but you want to generate defect free arrays basically as a starting point for many body experiments or quantum computing experiments and the way to do that is also called atom by atom assembly scheme so basically you load an array i stochastically take an image identify which tweezers are full you switch off the empty tweezers and you can rearrange
• 02:30 - 03:00 basically into a defect free array that's one method there's a few different ways of doing that and you can basically in every shot generate some sort of different geometries almost edited so this by now uh some sort of more widely used technique um you can you know some features you can generate defect queries nowadays with about hundreds of atoms in only 2d and somewhat classy 3d meaning multiple layers atomic distances are freely adjustable you have flexible geometries and the repetition rates are a lot faster than traditional cool atom experiments there's some limits in terms
• 03:00 - 03:30 of the number of traps you're basically limited by laser power and the success probability basically for generating a completely defect free array actually scales exponentially bad in a sense with a single atom a success probability that you have but that's some sort of the preparation fatality that you would get in putting an atom in place um at these distances since it's neutral atoms the atoms are naturally non-interacting unless you do something particular so one option is to go to so-called dripback states and i'll show you a little bit in a minute how this works
• 03:30 - 04:00 there's other options you could use photomediated interactions by sticking these atoms for example into a cavity or let them tunnel between tweezers but i'll work on a rework on basically mostly rhythmic states let me kind of briefly introduce that um so how does it work so generally the atoms they start basically in the electronic ground state and the electronic wave function is quite small and then you can excite them to highlight states with high principal quantum number and that electronic wave function basically blows up and because of this large electronic wave function you can have a high polarizability with
• 04:00 - 04:30 respect to electric fields but it also means you can have high induced interactions basically the induced dipole dipole interactions between two atoms in redback states basically from fluctuating dipole moments and these are so-called fundawals interactions and they have a distance scaling with one over r6 and they have a prefactor in there um that is uh basically this principal quantum number to the power of 11 so they basically blow up if you go to this high line electronic states and hence you can get uh very high interactions even with the typical
• 04:30 - 05:00 distances that you have in these tweezer arrays and you can basically um switch the interactions off by going to large distances or make them extremely strong by going to close by tweezers okay um how does it work in a little bit more detail so in experiments that i'll show you we use global illumination where we drive atoms from ground state to the switchback states with some laser that basically gives you some rubbing frequency and you can map all of this to a spin model and if you just look at the single atom part here um it would look
• 05:00 - 05:30 like in a rotating wave and rotating wave approximation like this as basically a sigma x and a sigma c term that you can control with laser intensity into tuning if you include the interaction term um it will only act on the excited state to to a good approximation these are the ones that are really interacting strongly with this one over six interactions where you get this projector and then there's some sort of a matrix in front of it that's cases one over six with the distance okay and then you can change basically this matrix by using different atomic spacings in there okay um so this has been used quite widely in
• 05:30 - 06:00 quantum science on one hand for quantum simulation in many body physics you know there's a lot of you know physics for quantum magnetism and quantum phase transitions you will hear i think a lot more about that tomorrow from misha and julia um there's new insights into many body dynamics there's topological physics results and a lot of open ground and proposals for that on the other hand there's you know i think work more towards quantum computing and take the state generation just to give a little bit of a flavor for two qubits you can reach certain fidelities either encode it in this
• 06:00 - 06:30 ground to rip back or mapping it back to longer lift hyperfine states or just as a benchmark you can generate ghg states with over 20 qubits in these systems if you do it in a controlled fashion um so this is really remarkable experimental progress but i think we have only seen the tip of the iceberg and it's also a pretty young platform in that sense um there are some experimental challenges and limitations let me just walk you through that briefly just assume you may want to do some sort of mini body simulation with this even with the time constant hamiltonian you have noise on the hamiltonian so for example this ruby
• 06:30 - 07:00 frequency can basically have a noise term on it it can come from a laser intensity noise you can have the tuning noise that comes from face noise positional disorder that comes from finite temperature of atoms in the tweezers you can also have you know readout error preparation error more fundamentally there's a finite lifetime to these rip-back states that reduces some sort of the time scale you have for operations before you have a spontaneous emission event and then in alkali atoms these are atoms with single valence electrons you would typically have to go by an intermediate state to this redback
• 07:00 - 07:30 state in a two-photon transition and it gives you a little bit additional decoherence times um that's the question yeah the relative strength it depends on the specific implementation i could show you a backup slide later this is actually quite complicated yeah um in the in the results i'll show you later in the talks that they're about the same multiple of ones they're kind of tuned specifically for that okay all right let me uh jump into a kind of results from celtic a little bit so we have been
• 07:30 - 08:00 working on alkaline earth atoms alkaline earth atoms a little bit special they have two valence electrons and hence a particular level structure that had using was usually exploited in optical clocks i so the most precise optical clocks are actually based on neutral alkaline earth atoms and um and this is because they have very narrow optical transitions and metastable states they can basically exploit in these clocks and we have been trying to exploit this now in twister arrays in combination with redback atoms specifically strontium 88 for the experts and i
• 08:00 - 08:30 should say the work has been expired inspired by quantum gas microscopes in japan and there's a few other groups that have been working on these tweezer traps alkaline earth items in particular adam kaufman and jeff thompson so they have a bunch of results in parallel to us i don't want to cite them on each slide just when i give a shout out here um so we have been doing this since about 2016 had the first results in 2018 for first imaging and narrow line cooling of these alkaline earth atoms you can do sideband resource cooling because of this narrow lines and so on we've showed high
• 08:30 - 09:00 fidelity imaging we then moved on to rib bike excitations and then also showed that you can use these tweezer arrays in combination with alkaline earth atoms to build atomic clocks where you have signal atom readout and control in combination with very very high resolution optical spectroscopy okay i'll walk you through a little bit this high fidelity rigbeck results as they're important as a building block for the many body results that i want to show you so how does it works you start in some sort of an absolute ground state in the atom and then move to a meta stable state in the ad and has extremely long
• 09:00 - 09:30 lifetime compared to the time scale that you need for your experiment so that's what we're doing here and then you can think of this as a new ground state essentially from which you can go with a simple single photon excitation to a convenient hood back stator that's nice for you it has a few nice features you can actually go to very high rubbing frequencies you can basically work on a timescale is a lot faster than some of the tech coherence mechanisms um you have no decoherence from this in the immediate states the atoms can be very cold because you can do the second cooling tricks uh we also have you know
• 09:30 - 10:00 developed some new detection schemes that i don't have time to show that help you to go to very high fidelity readout of these root back states and the rhythmic states are in principle trappable which has been shown by jeff thompson actually okay let me just show you a little bit of experimental results so if you take these atoms and you just try to try single atom rubber oscillation so what you do is you basically place them in an array in a large spacing whether in the action is the ecliptable uh you choose zero detuning and just drive it and see what comes out and what comes out is almost textbook ruby oscillations
• 10:00 - 10:30 between this metastable ground state and the rootback state you see that here and you can go to like multiple oscillations you see about like you know 50 oscillations and at the time this was the first root by gravity oscillations with single alkaline earth atoms you see pretty high rubbing frequencies and the pi pulse fidelity we get is about 99.5 here even without correcting for preparation and measurement errors let me move on to kind of proof of principle of entanglement here so if we place these atoms in pairs you can think of this as a basically a
• 10:30 - 11:00 four level system so you have ground ground ground rhythm equipped by groundwork by quickbook of within that pair if these atoms are far away they're basically non-interacting and you would still see this ruby oscillations however if you put them very close by you see that the most excited state gets shifted up with this one over six interaction if that shift is much larger than this ruby frequency of the sigma x term you cannot excite to this state anymore and if you look into the physics a little bit more carefully so called blockade physics you will see that you should in ideal case excite to a superposition state in this
• 11:00 - 11:30 gr plus rg manifold with the excitation frequency it is a square root of two enhanced and we see all of this in experiment again with very high visibility we see up to kind of 60 of these oscillations from ground to this superposition state and then you can look into a little bit more details and and analyze this data more precisely using some tricks from our esteemed chair actually uh to show that this bell state fidelity that you get is higher than 99 we can only get a lower bound with these measurements here
• 11:30 - 12:00 where we correct for step preparation and measurement errors so these are nice results you see this somewhat high fidelities as a building block for quantum simulation you know information and also metrology application so this is all nice so there's a single body and two body physics and then i'll switch to many body physics now unless there's more questions good so far so good all right so let me move forward so for this you know a single qubit and two qubit system we have methods to really kind of extrapolate basically fidelities
• 12:00 - 12:30 of what we're doing and i've been starting to ask how we can actually benchmark larger scale systems so what do i mean with that and what i mean with that is uh say i try to do some quantum simulation in this case it's analog random simulation and i want to target a certain state you know and i maybe i can calculate that state for smaller systems still on a computer and some pure state psi an experiment of course i try to do that and i have certain errors some that i showed you and these errors generically would produce a mixed state in the experiment and the most simplest
• 12:30 - 13:00 way you could try to write this mixed state as a superposition or classical superposition between the true state that you want to get with the probability that would be approximately the fidelity and then one minus fidelity turns some erroneous mixed state on top of that okay um what we would like to know is the overlap fidelity and we would like to know how good are we in experiment actually preparing the true target state so this would be this quantity in all generality um you would think maybe that's very difficult because uh reconstructing the
• 13:00 - 13:30 experimental you know density operator might not be possible for large system was fundamentally hard for generic states however what we do is we use new insights into many body dynamics to find an efficient estimator for this fidelity i should say before i move on there's related work from peter for example in the audience and also from john using classical shadows and so on we have a little bit of a specific twist on that that uses some sort of the inherent uh chaotic dynamics of the system and this is what i want to get to towards the end of the talk um
• 13:30 - 14:00 just to give you an idea for what we want to do here um so again we want to do this benchmarking uh experiment so we want to know what the fidelity is to generate some anybody targets that so that's really what we want to do here and um some sort of the idea comes a little bit from from this cross-entropy type type estimators in principle if you have the output probability distribution in a system that's sort of necessarily random you could look at a correlator between the ideal output probability distribution to find certain bit strings and the one in
• 14:00 - 14:30 experiment and this correlator could give you the fidelity and it's some sort of well-known that for random circuits or any unitary circuits if they're deep enough this linear cross-entropy which is really just a correlator between probability distributions uh can give you actually the fidelity as an estimator for fidelity and it's quite nice actually because it only uses single basis readout and it requires in ideal cases uh sample basically a sample complexity is actually independent of system size in principle um however there's some sort of a twist you actually need a random
• 14:30 - 15:00 unitary circuits you would think and more fundamentally i come back to this you need actually random state ensembles in the system for this to work and i'll briefly um explain how we think about these random states ensembles and how we think we actually get them in a many-body experiment it doesn't even have a random circuit at all okay let me just get there so how to think about that so the easiest way on a very rough sense so i won't go into some sort of mathematical definitions here is to think of a set of pure states that
• 15:00 - 15:30 uniformly covers a hilbert space how can you think about that so say you have an input state size zero and a programmable device where you can just dial in some unitarian that you need very basically you randomize in a specific sense you can think of this already for a single qubit so you have some initial states that points up on the blossom and you do a randomized unitary that you can label and then for each of the labels you get a different output states like j and then if you do it right it would cover the block sphere basically homogeneously and that would be a random ensemble and then you can think of k designs and
• 15:30 - 16:00 these kind of things from from there okay um so if you're a content smarter person you might think oh is this not just statistical mechanics why do i have to keep track of all of these pure states um some sort of what we're used to in statistical mechanics would be the the sort of first moment of this data ensemble is if you just average over them so take these states and basically average over all of these projectors um and then in in some sort of this this graphical sense here on the block sphere this would be the average the geometric average of all of these states and it would be a maximally mixed state at the
• 16:00 - 16:30 center of the block field um and now thinking about measurement outcomes we can do projective measurements on these systems in particular some sort of the most fundamental quantities are probabilities for certain bit string outcomes i call this pj or c and um this first moment which is the average over all states would give you the mean for these guys so it's basically say the mean of this game really high random system would be one but it would be described by some sort of the first moment of this example however if i ask about fluctuations if i
• 16:30 - 17:00 want for example something like this i want the sum over p j squared that's the probability to get a certain width string square over all mouse settings of the of my device basically i wouldn't be described by this means i have to go to a higher order quantity and this would be the you know so-called second moment so you would have to double basically a hilbert space and this double tilde space operator then would describe this fluctuation so you can write this as a simple expectation value of this second moment of the example okay so there's some sort of the idea so
• 17:00 - 17:30 basically keeping track of all of these pure states or at least of higher order moments of down sample helps you to understand fluctuations in the system more generally if you look at the second order moment here and it's really uh so called to design this would be proportional to identity plus swap operator and this specific property enables in a sense this fidelity estimation actually have to go to four designer after fluctuation slow but like to just get the fidelity estimation formulas that's that's basically what you what you need okay um
• 17:30 - 18:00 more generally what we're interested in i'll show you this in a second is so-called full counting statistics so if i look at for example a single a single qubit and i'm in this in this random state on sample so i have basically a probability to find a certain outcome say the qubit in the zero state as a function of the program setting chair and every time i put a different unitary and i would get a different probability okay so i have to repeat for a given unitary setting many times to get the probability but for each setting then the probability is different that's an object okay and if you take this full
• 18:00 - 18:30 counting statistics here and generate a histogram so how often do i fall into a certain pin now for a single qubit you get a certain histogram out you know which one it is maybe some people do it's completely flat indeed actually okay so you do that it's a completely flat histogram if it's a random ensemble actually okay this is for d equals two and we're interested in seeing this type of full counting statistics um and this is sometimes called nd concentration because it's not concentrated around a certain value now
• 18:30 - 19:00 if you have a device that has errors on top and you average over different errors okay what happens is that if i take a certain program setting that would normally give me a pure state and i run it many many times so average over different errors what happens i get actually not a pure set out but a mixed state so i get an ensemble of mixed states out and now if i look at this full counting statistics for this example of mixed state it would actually have a much more peak structure because the fluctuation would be much much smaller so basically
• 19:00 - 19:30 the projection on the z-axis they get like less fluctuations that's some sort of the logic okay and sometimes this is called concentration and it's a coherence effect now moving to larger uh dimensional hilbert spaces you can again look at proof counting statistics of pit strings here and then you would get this like somewhat well-known porta-thomas distribution again we would call this nd concentration if you have errors in there you would see again a spikey distribution we would call this concentration because you get exponential distributions out if you count these things in the right way okay
• 19:30 - 20:00 that's also the logic okay so questions on that so maybe a lot of people know this in art okay so um okay sometimes this is all called speckled pattern so let me move forward so what we have been asking then um originally motivated by this benchmarking task but also a little bit of fundamental interest into many body dynamics is the following so does many body dynamics naturally produce random
• 20:00 - 20:30 state ensembles so some sort of a question you can phrase and i'll show you two options for how to think about that one is to sample temporarily from different evolution times and the other option is what you know i rephrase now in this you know spirit of recent developments more like measurement and choose generation of random state ensembles let me show you the first option first which is a little bit easier to understand i think uh so-called temporal sampling so imagine you have an input state this is all qubits in a zero state or some product state and you evolve with a time
• 20:30 - 21:00 independent hamiltonian to get output states psi of t for different evolution times and then i go ahead and sample these states for different times to generate an ensemble of pure states okay so basically gonna imagine you somehow go through a hilbert space and i just stop at different times and i get different states and we call this the temporal ensemble okay so it's basically some states at different evolution times t of t chain okay um and now what i can do an experiment so we do this experiment the hamiltonian would be the stripper hamiltonian and i look at this output probability
• 21:00 - 21:30 distributions so i look at the probability to find a certain bit string at a given time this is p t of c and i can plot this as a function of time so this experimental results these blue data points and the red data points for short and late time and for different you know bit strings basically that's some bit string other bits and other butt string and it's a probability to find that bit string as a function of time so you see some quite strong fluctuations here and these fluctuations seem to be dying out at light time and you see this for a bunch of different bit strings there's many many of those i'm only showing three okay
• 21:30 - 22:00 and we normalize these bit strings here specifically by the some time average and has something to do with with the fact of energy conservation i'll come back to this in a second um now i can take this experimental data and analyze it so i can generate these histograms the same way i just did that so i take it over a certain time window for all of those bit strings and i generate these histograms and what i see is that this indeed very close to a exponential distribution so we see here that this product thomas distribution is generated just in this
• 22:00 - 22:30 range dynamics and it's a fully coherent effect and now if i go to very late times errors basically seem to be piling up in the system and i see something that's much more spiky so we see something where is a concentration effect from from basically incoherent coupling to the past some of the errors are showing you okay so that's that so we see basically something that looks like a product thomas distribution in global quench dynamics here and we also see the effect of the bath or coupling to a bath so basically see the coherence where we get
• 22:30 - 23:00 this more concentrated distribution you can ask now is there some sort of um first of all why do you get this global distribution i'll come back in a second and and the other question could be okay is there something some sort of universal in this just change from porta thomas to this more concentrated probability distributions so what's the quantitative effect of actually coupling to a bath or it's a little bit of a detour but i like the slide um so what's the quantitative effect here and you can try to study this in a little bit more controlled fashion so here in this experiment that i showed you before just go to a very long time
• 23:00 - 23:30 but i can also stay at the times where i see globally opportunist distribution and then try to basically trace out parts of the system so what i do is here i divide the system in a and b and i trace out b and i look at the bit strings only in a okay so i trace out part of the information basically and this path b has a certain dimension that i call db and there's a global dimension d okay so if i just look at the full system it looks kind of both autonomous like there's some very preliminary data and if i trace out more and more i see that this distribution becomes narrower
• 23:30 - 24:00 and narrower the larger this dimension here gets i can go larger and larger with the with that system in a controlled fashion you see basically some sort of a transition where this gets narrower and narrower and in the end becomes basically gaussian okay and it turns out that even at this these intermediate dimensions um this solid line is actually a universal function that's actually a so-called erlang distribution and this is experimental data overlap is just this prediction this is just this airline distribution and has only one
• 24:00 - 24:30 input parameter and it's only the dimension of the path it's the only thing that matters here there's some sort of a universal um thing going on here basically such that you have distribution functions that are fully universal not just for the global system but also for subsystems and that is some sort of almost a quantum classical type uh you know change in in this distribution function okay and one common this is not just true for many body systems at high temperature but also for a high effective temperature but also for random circuits
• 24:30 - 25:00 okay let me um come back to some sort of uh this discretion i posed earlier so we did this sampling from different emolution times when we saw some signatures of random ensembles so see this port autonomous distribution and also for local subsystems we see something that you would expect from a random unitary circuit which is this erlang distribution so um so that is really randomly distributed and the answer is actually no um so if you think about the time evolved state so you have some projection onto an energy eigenstate and then you have some phase evolution with
• 25:00 - 25:30 the energy of that state um and at best the phases could be random here but these probabilities they cannot be random so i just take one hamiltonian one input state and then you have all of these probabilities being fixed so the overlap with energy eigenstates has to be basically fixed okay so at best you can hope for some sort of sampling from a state ensemble that's not fully random but there's only randomized faces i basically write down states like this and i sample different phases and i would expect maybe certain you know
• 25:30 - 26:00 certain uh predictions from that okay um so if the system is really fully ergonic in a sense these phases would be completely random but the probabilities would be fixed and then um at best case this temporal sample would get you something that you could call a random phase ensemble okay and this you can show that's relatively simple um depending on the structure of your energy eigenstates energy eigenvalues and you can basically classify this using uh so-called no resonance conditions based on these no resonance
• 26:00 - 26:30 conditions you get moments uh that you can calculate from this so-called random phase ensemble and uh this has um several consequences one is that the full counting statistics is brought at thomas globally and then locally uh this erlang distribution for generic measurement so what do i mean with that i mean something that has not much overlap with the energy eigen basis because if you actually measure in the energy eigen basis there will be no fluctuations basically because there's just everything is constant but if you measure something that's orthogonal to
• 26:30 - 27:00 that you will see basically this port autonomous distribution and you can predict that's not just for feedback where we see it experimentally but also for all kinds of other systems including somewhat surprisingly also uh in certain interacting models actually that have this no resonance condition that's one consequence and the other consequence is that you can calculate the second moments again of these distributions now and they include again swap operators and because of that if you do your math right you can actually do benchmarking your systems
• 27:00 - 27:30 got some some questions on napper yeah uh thanks so um i'm trying to understand these porter thomas distributions uh for generic hamiltonian you well enough for many uh important many-body hamiltonians you're going to have some sort of conservation law that restricts the base of states that participate in your dynamics right
• 27:30 - 28:00 um and right so like for the using model and stuff um so these port autonomous distributions are true even when you're excluding specific basis states from your dynamics i'm not sure what you mean with excluding uh in other words so like if i had say um uh a model yeah so like the easing model right uh conserves total magnetization for
• 28:00 - 28:30 instance or something that conserves particle number yeah right so you're going to have yeah you're going to have basis states that don't participate in the dynamics there so you're you're counting up the statistics of the bit strings you observe in a restricted basis state uh basis uh in the hilbert space right yeah the conservation i mean that the biggest issue to start with is energy conservation is also conservation losses i think you might have to be careful
• 28:30 - 29:00 that you don't measure in a basis associated with that conservation law potentially that's that's one thing and you have to think up a little bit about so here here the first to start with it has energy conservation so you cannot if you measure an energy eigenbasis you will not see a port autonomous but however most basis will have very small overlap with energizing bases and then um you can calculate this moment and so this moment if this correction term wasn't there so this delta i'm not telling you what exactly that delta term is if that correction term wasn't there and you write this out for all higher moments this is always basically exactly
• 29:00 - 29:30 the moments of a higher random ensemble with this product in front of it and this product you can take care of with the normalization that i had so you get perfect border thomas if you have that thing to all orders however there's a correction term now the correction term if you look at that correction term carefully this is this comes from energy conservation at correction term that correction term has a some sort of gives you a one over d correction for generic basis basically that is the global dimension so it's actually quite small um so that's all good that more tricky part is like some sort of the time scale
• 29:30 - 30:00 for forming all of these higher moments i think that's some sort of the thing that i'm sweeping under the rug actually a little bit because it turns out you can see something that looks like porter thomas quite easily actually but you might not have the right moments up to high order so that's a little bit more thank you there's another one in the back yeah i'll have to hurry up but uh yeah how many um interaction times do you tend to get before yeah reporter
• 30:00 - 30:30 thomas is washed out by yeah we we uh that's a good question uh we go and experiment past the time scale where the half chain entanglement entropy saturates and then some time window past that about like 50 extra or so and then it looks like that it's a little tricky second i'm not sure what how you would define in the action times there's various time scales in the hamiltonian yeah there's some pi factors in there
• 30:30 - 31:00 it's like you know it's linear in the system so it's linear in the system size in 1d that's all i can say you can rescale it into like there's like five or four terms in the hamiltonian to be careful linear to the to the system size you can go back you can see it here so here it's uh now i think for 10 for 10 atom system uh it would be
• 31:00 - 31:30 a few robbie flops like five robbie flocks which would give you full entanglement entropy so i think i think of it usually in entanglement entropy so you take half chain entanglement entropy i'll show you some plots later it goes up and saturates we go at least to the point where it's saturated it's not obvious that you should see it there already because there's some sort of a higher order thing so that's not obvious so in one d we would go in a linear time scale of and would see it if you should see it there or not that's a different question or a theoretical sign okay
• 31:30 - 32:00 it's not even clear that it should kick in you know very early on right um what's the earliest you see occurring entanglement time if you look earlier it doesn't look right but it's not obvious i agree like the time scale for these higher order moments to form is not obvious also tricky to do it in numerics because if you wanna calculate these moments you have to go
• 32:00 - 32:30 to double like even the second moment you have to go double hilbert space for something that's quite large already so it's not so obvious how to do that right that's a little bit of an open question how fast it goes we know that conditions when it forms at very low long times because i can tell you mathematically but how fast it it forms these moments moment by moment that's an open question right thanks okay maybe i'll move forward a bit more all right let me briefly briefly show you something else which is this uh measurement induced story or
• 32:30 - 33:00 like projected on samples so there's a second possibility to generate random ensembles which is you take hamiltonian dynamics at a fixed time and then do projective measurements in part of the system you know measurements in b but now i don't trace out i just keep track of the measurement results and then define for each measurement result the pure state and the remainder of the system is very similar some of the things you have been hearing about and then for each of the results in b i can get a new pure state my subsystem a i could choose to be just a single uh single qubit and
• 33:00 - 33:30 then for each of the results i could point in a basically a different direction okay and then just define some sort of an example we call this the projected ensemble at a fixed time that you can follow as a function of evolution time now okay so then you can check if this forms a random example or something close to it and it turns out in this case it can be true high random some sort of some of these conservation laws they basically drop out at uh effective infinite temperature and this also what we see in experiment or close to we see basically something that's very close to a flat distribution up to some onset of the coherence on the edges
• 33:30 - 34:00 um for effective temperatures that are finite it actually forms something more complicated the so-called scrooge distribution if you have heard that before that's some sort of a finite temperature generalization of high renderman samples okay so i just want to briefly mention this so there's a more general governing principle in there um that is that exotic many body dynamics produces what we call maximum entropy pure state of samples under constraints so there's certain constraints to the dynamics it could be energy conservation or in this case the sample has to
• 34:00 - 34:30 average to the reduced density operator so if your reduced entry operator has a certain form you have to put this in as a constraint then you can apply some some type of maximum entropy principle that generates a pure state ensemble and you would see under certain conditions that this will be fulfilled in many body dynamics so it's quite similar to standard statistical mechanics but now it's really for examples of non-orthogonal pure states here and uh like in different to standard statistical mechanics you need a label for the output so that's really the difference
• 34:30 - 35:00 here i need some sort of a programmable quantum device where i can program something different times or a different circuit setting or i need projective measurements that label my output states correctly if i average over everything i'm back to basically gibson samples or something like that so i need a specific label and this has now two consequences that one i showed you which is that you get universal fluctuations when sampling over say evolution times or partial measurement outcomes and more well-known in random unitary circuits over like basically circuit settings
• 35:00 - 35:30 and this universal fluctuations you see here as it could be this flat distribution or uh for autonomous distribution erlang type distribution as another consequence though it gives you new insights into this gross entropy type fidelity estimators because this second moment for example has swap operators and so forth okay let me very briefly talk about this story the second consequence here um um so one is um so this this matters not just for hamiltonian dynamics but also random unitary circuits and i want to briefly show you first the circuit case okay so imagine you
• 35:30 - 36:00 have a random unitary circuit you have a bunch of errors and you want to do fidelity estimation in there and you run it as a function of death and you plot the fidelity here so the fidelity would be say this dashed line here okay if you do standard across entropy estimation you get this fxcp score you would see okay it starts up here and then at some deep circuit you know linear with some sort of the circuit death here in the 1d it's a bonding case it would follow the fidelity quite closely but it has to be quite deep okay so now you can do something else you can
• 36:00 - 36:30 cook up a modified cross entropy i can i can do what i want so i write this a little bit more complicated i put a factor of two here and divide out by that and you see this different quantity follows the fidelity even at early times actually somewhat surprising that you would see that so globally the system hasn't been kind of scrambling at all and this comes indeed from this projected ensembles at least that's our understanding of it indeed you can prove with projected ensembles even at early times that if you have a single error in the device this formula matches some sort of a
• 36:30 - 37:00 return probability of the error operator when it spreads out average over the basis states in which you measure okay so as you can prove at early times with this this fidelity estimator at late times this thing will match fxcp so it's the same thing because this guy goes to two and you can show these formulas you can quite easily connect it to actually error evolution in the system so there's two things here so one is this fidelity estimation actually works at short times and this comes from these projected on samples emerging as much shorter times than globally so coming back to this timescale question for the projected
• 37:00 - 37:30 things where you have only uh random ensembles in some sort of a subsystem for measurements this can happen order one basically it can be very very fast and here you see the effect of this infidelity estimation uh and it's easy to prove using this projected unsample formalism um that this you can connect basically this cross entropy scores to return probabilities of of some sort of a time evolved error operator in the measurement basis okay um let me let me move quickly to quantum
• 37:30 - 38:00 simulator so what we want to do now is apply that formula maybe in a modified fashion to analog quantum simulators and again the experiments we're doing are relatively simple we start with all atoms in the ground set and just quench with that hamiltonian it's very similar to what i showed before and we want to do fidelity estimation so i have again the output distribution of the experiment and a certain target state and then i can cook up a fidelity estimator that's almost the same as what i showed you before but there's certain weighing factors in here that have to do with energy conservation or conservation laws you have to be a little bit careful
• 38:00 - 38:30 in how to choose those but the claim is that if you do that right that this basically estimates the fidelity of the experiment in quench dynamics with ergotic hamiltonians as a crosscheck we apply the same formalism to a model of the experiment where we get both the estimator and the true finality at the same time so here are some results for n equals 20 system as a as a function of evolution time i'm plotting the entanglement entropy here that saturates after some evolution time basically to the entanglement entropy that you can have in the blockade space
• 38:30 - 39:00 so this is close to pxp condition here and then as a function of time again you see exponential decay of the of that data point so this is this fidelity estimator and as a comparison we showed a fidelity estimator for the model as well as the true fidelity which all match so that's good um so this model is actually quite complicated it includes all kinds of different error sources and so forth and that's actually quite good so we know that this fidelity estimator works for some sort of realistic model errors in the experiment and we're very careful about that um you can ask
• 39:00 - 39:30 what about larger systems and then are these experiments in the end better than state-of-the-art numerics that's something i want to very briefly mention i should say it's very preliminary and the numbers are not final but kind of in the right ballpark correctly so what we now did is we used this benchmarking scheme to benchmark systems with up to n equals 60 atoms um and you see we're close to exponential decay you know up to 60 or up to 54 i'm showing we also have data for 60. and what we do is we basically fit this with an exponential function where the decay rate depends on n so this is
• 39:30 - 40:00 some sort of a many body fidelity decay rate that i want to basically analyze as a function of system size there's a pre-factor in here this is p0 to the end is basically spam error and this three factors around 99.6 in our case but now what i want to know which is a little bit more less trivial is this many body fidelity rate and this could in principle scale not just linearly with n but quadratic with and even with higher powers it's not obvious if you have correlated errors in the system or non-markovian errors and so forth so that's not obvious and we did that so we did this estimation and
• 40:00 - 40:30 we see that up to the system sizes that we benchmarks and within error bars and you can fit this in very careful fashion we don't see any evidence of anything larger than linear scaling and that's actually very very good so that's not obvious to start with okay if you had just like localized errors it would be linear again if you have correlated errors it could be anything good let me move forward briefly you could ask how did we actually benchmark such large systems so it's not obvious i go to 60 qubits and we go to some sort of systems where we try to build up a lot of entanglement and the way we did that
• 40:30 - 41:00 is um basically using mps methods so this is all done with mps specifically tbd and they basically expect up to some maximum time before this entanglement entropy hits a certain threshold basically you have some kind of time scale in which you can do this fidelity estimation and this is kind of uh what we do an experiment here we do array of 54 atoms you can benchmark it up to a certain time scale so these are the circles here and and using this time scale you can try to fit this exponentials basically you can try to go to longer time scales
• 41:00 - 41:30 in a with a trick where you try to prevent build of entanglement you can basically split the array into two halves and then benchmark these two halves in one shot basically and it's consistent with the 54 atom system and it's again consistent with the fact that these error rates only scale linearly and that's all all good um so this is a split array so now you're in an interesting some sort of a framework where you have methods to to basically estimate the quantum fit led in experiment and just as a sort of a reference point if i now extrapolate to 60 atoms at late times at
• 41:30 - 42:00 the time where i maximize the entanglement entropy in the pxp subspace i get a fidelity about 55 to generate a maximum entanglement entropy state and it's actually quite high so this is actually some sort of almost record i think um and now you can ask the following question um when is this fidelity still higher than some sort of the accuracy of the mps so this mps works until a certain time scale before basically accuracy goes down because your bond dimension is not high enough anymore okay so you can ask so what classical
• 42:00 - 42:30 resources do you need for the mps fidelity to be higher than the experimental fidelity so something i can answer some sort of quantitatively now so again this is how how much bond dimension essentially do i need for the mps overlap with the two targets set to be higher than the experimental one so what we do is we basically vary the mps algorithms would be the dashed line if the fidelity is higher than the experiment which is the red line i say okay that's a minimal bond dimension i need actually to somewhat be better than the experiment so sort of the logic so you can do this now as a function of system size n
• 42:30 - 43:00 um if the experiment had no errors this would be quickly exploding so this is some sort of the dashed line of course i have errors now so i don't need as high one dimension to be as accurate as the experiment and so these are the the data points that we have and you see it still keeps on growing it's of course smaller but it keeps on growing this system so that's already quite good and then what wanted that if you need a bond dimension of around four thousand it's actually quite high um in run time it would be you know around weeks and then you can also scale memory and so on so that's some sort of interesting because i can now more more clearly
• 43:00 - 43:30 quantify what i would need on a classical site to be actually so better than the experiment in a specific sense and this the resources that you need are significant but probably still doable so some of this actually extrapolated and we never ran it with that type one dimension as an outlook um we're working on defining a most specific sampling task that is all on a level of just fidelity but i haven't done anything with it so some sort of kind of useless at that point so we have to define some tasks to do with it and then say okay are we better or worse than an algorithm in that task and i should say this all in 1d and we're working now in
• 43:30 - 44:00 2d and 2d experiments should be as good approximately but numeric should get quite a bit higher harder okay good i'll skip this in the interest of time you can also use it to read out hamiltonians just a brief summary i walk you through some kind of experimental basics of atoms in in tweezer arrays and dripback interactions and showed you some sort of single two qubit results for alkaline earth atoms and then mostly talked about random state ensembles and benchmarking in the systems with some of the results published this temporal sampling story
• 44:00 - 44:30 is unpublished large scale benchmarking is unpublished so outlook is we really want to do some sort of precision quantum simulation and i think we can actually maybe pretty you know significantly under some sort of a quantum mode wondered regime within a year or so i believe okay quick acknowledgements i want to acknowledge specifically adam shawn tony choi we have been working on the experimental side and then zumbon and the student daniel on the theory side with this i would like to thank you for attention [Applause]
• 44:30 - 45:00 time for a few questions [Music] yeah thanks um this mini body or large-scale benchmarking so we've studied this function gamma and random circuits and i guess the prediction is that um as you as you increase in it goes through something like a first order phase transition
• 45:00 - 45:30 and it should become independent of n so i'm wondering if can you like maybe see that kind of effect by turning up the noise rate oh i would like to see that i think if you have an error model it has only local errors yeah that should just be linear and then for all for all internality yeah we we can discuss it we can discuss well there is i mean you know there is at least some observables i mean i think it's the same but we can discuss it yeah for fidelity maybe for different observers but
• 45:30 - 46:00 it has to be linear but we yeah we know there is this phase transition in these kind of decay rates it might be interesting to look forward let's chat a bit i would be surprised if for what we're looking at if it would be anything anything than linear for standard error model if you have correlated error is a little different okay hi cool cool results i had a question about the the fitting yeah
• 46:00 - 46:30 so getting the exponent yeah yeah getting extracting the exponent over a decade in the fidelity how kind of reliable experiences yeah and more generally i was wondering whether there are like other ways to measure this camera is there a hope that you could do lash with echo or something like change this yeah that is all very good it's all very good yeah that's amazing
• 46:30 - 47:00 that's a good question so there's different ways of doing that so um i think the the most uh convincing i believe is if you fit all the data at once so you basically do time and end scaling at the same time so you do an exponential you basically take all the data for all n's and just fit that function and you keep these parameters open you keep p0 p0 i know anyway i keep gamma 0 gamma 1 and gamma 2 open if you do this all at once gamma 2 comes up to 0 that's all i can
• 47:00 - 47:30 say with an error buff that's the most the most convincing and if i do that it fits all of the curves individually for our lens nicely so something but again this ongoing work it's not obvious how to do that so that's the hard part but it's the same with google i mean google also extrapolates and also this you know supremacy claim results the fidelity that has never been measured they don't even do time extrapolation
• 47:30 - 48:00 they only do any extrapolation what we're doing is both time and end that's a bit different so it's actually a bit more careful [Music] i'm very skeptical if you don't do it super carefully because if you just try to invert the hamiltonian here is you're going to erase some of your noise because a lot of the noise is non-markovian so a lot of the noise is shot to shot so if you take shorter shots say fluctuations yeah hamilton and imagine the hamiltonian is different from shorter shot but inverted exactly shot
• 48:00 - 48:30 to shot it will always get fidelity one but the fidelity will not be won in practice so you have to be super careful that is mirror circuit ideas from robin blume cohoot at sandia for random circuits you they're actually tricks where you could do it but you have to be super careful that you don't erase noise in your reversal sometimes you get like kind of overestimate so that's a little bit tricky and here you i think you would overestimate a lot if you would just invert okay okay maybe last question
• 48:30 - 49:00 mention going to 2d is that uh an experiment in the works yeah we're working on an upgrade and we're also really uh yeah there's supply chain issues it turns out it's very hard to get your lasers these days it's quite unfortunate yeah some lasers have been delayed by more than a half more than half a year now really cool should should come at some point once m square delivers for the experimentalist
• 49:00 - 49:30 on the record let's record it okay good we wrap up okay okay let's thank manuel again [Applause]