 Thank you, Marcello. So I would really like to thank Marcello and all the organizers for inviting me. It's great to get the opportunity to speak about my past life as an ion trapper. So the official title on the web page is Quantum Interaction to Quantum Simulators. I will try to stay true to the spirit, if not the letter of the title. But I will focus on trapped ions because of time and because of my own experience. But I hope that I can show you some more general techniques and, in general, the flavor of what it means to build a Quantum Simulator. So let me then get started. The idea of simulating physics is pretty old. You see some complicated physical system and then build a tabletop model that you can come and tune some knobs at to try to understand better your object of study. The curriculum is like 2,000 years ago, and we keep doing this with classical computers. We find some phenomenon that we cannot study analytically. Then we can just build a numerical simulation. However, the problem when we get into trying to study phenomena that are quantum mechanical in nature is that you cannot, in most cases, efficiently simulate them with a classical computer. Because, as you will know, any phenomenon that needs to be described quantum mechanically needs you to calculate usually over all possible trajectories. And then the size of your Hilbert space grows exponentially. At some point, your classical simulation methods just blow up. So then, Richard Feynman had this very neat idea over almost 40 years ago and said, well, if nature is quantum, then we need a quantum system to simulate it. And that's what we're going to be discussing. How can we use a controllable quantum system to understand another quantum system that is not directly, experimentally accessible? So for instance, two things that we are going to be looking at are magnetic systems or problems in particle physics. There are problems that are in general very complicated, where perhaps we know the basic equations that describe the physics. But we cannot really solve the equations of motion. And then we need to build some simulator to understand either, for instance, how the crown state of a magnetic system looks like, or the time evolution of some problem in particle physics. And in this talk, I will restrict myself to seeing how we can implement such problems on trap tires. And now I would like to branch a bit. And just as in classical simulations, we can make simulations in two ways, either analog or digital. For instance, we no longer build auto-race or mechanical models that describe the motion of the solar system. Now we use computers. But both approaches are equally valid. And in the exact the same way, in the quantum world, we have these two complementary ways. So let's start with the analog quantum simulation, which is perhaps the conceptually most simple one. The idea in analog quantum simulation is to design a system that mimics somehow the behavior of the other system that you want to study. And by mimic, I mean that has the same Hamiltonian or a similar enough Hamiltonian so that you can understand the physics that interest you. For example, you want to study the properties of electrons in some potential created by some crystal structure, a basic problem in condensed matter physics. You cannot go and look at your material and see your electronic orbitals. But you can, in the lab, create an optical lattice, put atoms there. And it turns out that these atoms behave very similarly to the way they would on this crystalline potential. So then you have a system where you can actually tune interaction ranges and other physical properties. And then you can study it directly. Well, the very nice thing about such an approach is that you take a system that you build in the lab. And this system already has the interactions that you are looking for. Given by, it could be, like, table to table interactions with atoms, or Coulomb interaction, or whatever it might be, they are already there. You don't have to go to great lengths to engineer them. However, this also means that you are less flexible, because you are restricted to using the interactions you already have present in your lab to simulate the physics that you want to. OK, so now for generality, I would like to mention a couple of systems that have been and are used for analog quantum simulation. I think in the past few days, you have heard much about the cold atoms. After Immanuel Bloch and Jean-Dalibard spoke, I really cannot add anything else. But this is a classical example of analog simulation, where you can, in the lab, engineer Hamiltonians that simulate interacting systems of bosons or fermions. And you not only need to think about atomic physics, if you, for instance, you can engineer micro-resonators and a solid state system, and then couple them with superconducting qubits. And in this way, what you get is a system of interacting bosons that you can arrange in lattice. And then by tuning the individual qubits that couple your resonators, you can tune the strength of the interactions or couplings between bosons. And so this might give you also much flexibility. And even with less usual systems, like photonic systems, there are experiments where you can use networks of entangled photons that mimic the correlations of chemical bonds in molecules or use interferometric setups to simulate quantum random walks. The idea that I'm trying to convey with these examples is that there is a myriad of physical systems that you have fine control on and that you can use for quantum simulation. But in this analog paradigm, you see that each physical system has their own strengths. And you need to simulate something where you already have these interactions. Now what happens if the system that you want to study is not really available in the lab? Can you still do something? The answer is yes, although it might be a bit more tricky. And the answer is digital quantum simulation. So the basic idea here is that we are trying to build a more complicated Hamiltonian out of simpler building blocks that are available in the lab. Say for instance that my system is described by this purple Hamiltonian here, which is the sum of two terms, might be two different interactions in the system, and they in general don't commute. And that's the fun part about quantum mechanics. You have these two different physical effects. And if I know how these building blocks are available in the lab, like I can implement H1 or H2, but only one at a time, then how do I do to implement the full Hamiltonian? I cannot just do H1 and then H2 because they don't commute. It's not the same as doing H2 and then H1. What Lloyd realized, based on previous work, was that if I break up these interactions in very short time steps and apply them stroboscopically one after the other, then in the limit where this time step becomes very small, I can actually approach the exact evolution as close as I want. And then I can even do higher order approximations to reduce the error. But it is this stroboscopic exchanging of the terms such that with only this very basic building blocks, you can apply them one after the other and build up a more complex Hamiltonian. And Lloyd showed that this is actually universal in the sense that you can simulate any physical Hamiltonian that has local interactions, that is interactions whose range doesn't grow with the size of the system. So this is an extremely powerful approach. It's extremely flexible because this means that most physics you can go to the lab and build once you have a sufficiently powerful quantum system. The downside is that to get a good enough approximation, you might need many steps. And also it might not be extremely straightforward to generate these single interactions in the lab. So implementing actually such a simulation might be more complicated than an analog simulation where all the physics is already there, prepacked, so to say. Yes. Can you speak up a bit, please? Yes. Yes. We are going to, actually, I hope this gets answered later. It's an excellent question. Like how do I go from any arbitrary interaction that I have in the lab to a more general interaction? The answer might not be straightforward, but so it can be done. And to make it more precisely, if the interactions that you have in the lab can be used to build a complete set of quantum gates, so for instance, a nearest neighbor interaction can usually, if you make it during a precise time, you can usually make a CNOT gate out of it. So you can turn on your nearest neighbor interaction and make it such that if this atom is on state app, then the following one will be flipped and so on. If you can find interactions in your lab that allow you to generate a universal set of gates, then in the quantum computing sense of the word, then you can apply such techniques, because you can always decompose your more complicated interactions. By the way, please stop me at any time. I think we have more to gain here by making sure that we're on the same page that we're just pitting along. All right, so now the plan of the talk for the rest of these two hours will be first. I would like to introduce quantum simulation with trapped ions. I will get a bit into the detail of the experimental implementation, not to overwhelm anyone who is not actually interested in this, but rather to give a feeling of how this actually happens in the lab. And also to show what are the building blocks, so these gates that we can do in the lab and that form this universal set of gates. And then I will show both the analog way and illustrate it with a simulation of easing models and the digital way and illustrate it with an experiment where we simulated lattice gauge theories. I try to proceed like motivated by the experiments and not the other way around. So I will show the experiment and what's necessary to understand it, and then you can stop me if anything doesn't make sense. All right, so first of all, what we need to get a trapped ion quantum simulator is ions. And how do we do this? We cannot do it with static electric fields because by Maxwell's equations, the sum of the curvatures of the electric field in all three directions have to be zero. So if you're dropping it in two directions, in the third one you are actually pushing your ion away. There's two, I'm just going to cover two ways to do it. One is a pole trap where instead of having a static field, you have a field that oscillates at radio frequency. And then you create a subtle potential that shifts sine every microseconds or so. And this confines your particle. This creates an effective confining potential that keeps your ion in the center of the trap. And then on the other direction you put a static field. This is one possibility with the time-varying fields. The other possibility is to add a magnetic field, in which case, so you have an electric quadrupole and then add a magnetic field. And your particle, again, stays confined and describes some cyclotron motion. And this also works to keep your charged particles confined in a region of space. Now, let me illustrate, once we have the ions, how we build, how we encode information in them. I will take the particular example of the calcium 40 ion, my favorite ion. But the pretty much the same idea is applied to any atomic species. So in this ion, we have a pretty simple electronic structure at least at relevant levels. So this is the crown state. This can be excited on an optical transition, at 729 nanometers, to the D state. And this transition has a natural lifetime of about one second. So this means that you, and the typical time it takes for a laser pulse to excite this transition is microseconds. So this gives you really a long time to excite this transition, encode your quantum information, and then do operations until your information is lost via a natural decay. So this kind of qubit, which is called an optical qubit, has the advantage that you can do relatively simple operations because it's all laser pulses. However, you are, in the end, limited by the natural lifetime of this transition, which is on the order of seconds, depending on the atomic species. And then your two relevant states are, we can call one or zero, or spin up and spin down, are this different electronic states of the valence electron of the ion. And you can manipulate them using laser pulses. Does this make sense so far? Yes? Yeah, exactly, exactly. OK, one other variant on this is instead of you using the optical transition as the qubit transition, you just stay here in the ground state and look closer. And it turns out that it's split with two different magnetic number states if you apply a magnetic field. And then you take these two states, call that your spin, or your qubit, or your two-level system. And you can manipulate this either directly using a ref for microwaves, depending on the species. Or by shining a two-photon transition, you shine a laser that is detuned with respect to this upper level. And then another one with a difference frequency. And then such that the sum of these two frequencies covers precisely the frequency difference between this and this level. And this allows us to manipulate this qubit. This has the big advantage that it's not lifetime limited because we're talking about two levels in the same, in the ground state. So this will, in principle, never decay, at least for relevant experimental time scales. However, interactions are now less simple because, well, first of all, you have to use two laser beams instead of one, because you need to go up and down again. Or do some microwave manipulation of your atom. But this is also a very popular alternative to encoding quantum information in ions. OK, so now to give you an idea of how these things look like, you may have, this is an in-stroke type setup, where you have a linear string of ions trapped in the pole trap. These are for electrodes where the RF electric field is applied. Then you have these end caps where you apply a DC field, come with a cooling light, and you have a beam with which you can address single ions, and a beam with which you can manipulate the whole string at the same time. And then detect the fluorescence of the ions using a CCD camera. This is another example from this time for the Monroe group. Very similar idea, only that they work with a hyperfine qubit. So they have a setup for Raman manipulation. It's the same. You can address individual ions and then have an array of PMTs to detect the fluorescence from each individual ion. Right, yes. Yes, yes. You will see later in this talk a 2D example on a penning trap. 2D crystals and even 3D crystals are possible in pole traps, but this depends very strongly on the ratio of your confining potentials. So in this case, the ions arrange themselves in a 1D case because the radial confinement is much stronger than the actual confinement. And then you're just pushing. So you have two forces at play, the mutual Coulomb repulsion between the ions, which makes them space each other, and then the external potential, the external electric fields, that will make them bunch in some direction. So if you're pushing them a lot stronger in the radial direction than in the actual direction, then they will try to spread out in the actual direction. Now, if you relax one of the radial confinement potentials, then you would get a zigzag potential, like in 2D crystal, or then a sort of pancake in 2D. And then if you relax the three directions a lot, then you would get a three-dimensional crystal, actually. OK, now you come with your laser. And what you need to do in order to do manipulations of your qubit is to find the right frequency. So if you shine now a laser at the transition, restaurant with the transition frequency of your qubit, then you will excite it to the excited state. And now remember that the ion is, and then these are the, what I have marked here with colors, are the different transitions between the different M numbers of the, so remember that each of these levels has Z-man's plate, so you have actually not one, but 12, well, 10, 12 possible transitions. And these are just the electronic transitions. Now, on top of the electronic levels of the ion, remember that it's sitting in a trap, so it's basically a harmonic oscillator. And it has motion, so a single ion will have one motion mode, and then a string of ions will have many motion modes. This, by the way, is not a video, it's just a cartoon of how this would look like. But each of these normal modes, if now I did, has a characteristic frequency, and if now I detune my laser by this frequency, and train it on the ion, it will, the ion will absorb one photon and get excited. And also, will absorb or emit or lose one quantum of, like, one photon, one quantum of vibration. And this happens if I tune my, the frequency of my laser to the side bands, so for the blue side band, my atom gains one photon and for the red side band, it loses one photon. And in this way, I can manipulate also the motion of state of the ions. And this will be very useful for actually implementing interactions and entanglement, as we'll see later. Okay, now, briefly, with the resonant beam, what we can excite our atom from the downstate to the upstate. And if I scan the length of a laser pulse, for instance, I will get a coherent population transfer or rabbit flops, so this is what happens. And this is how I do, now in terms of a block sphere picture, this is how I do X or rotations around the X or Y axis of my block sphere, just bring a population from the south pole to the north pole of the sphere. Yes, good question. This, and this is, I'm now describing this particular setup, where we have a global beam that rotates all ions at the same time, plus an addressed beam. And with this address beam in this setup, we restrict ourselves to doing addressed Z operations for technical reasons, which we can discuss if we're interested. But then, this beam is an addressed beam, so you focus down the beam to less than the inter ion distance or like a waste of two, three microns. And then talk to each ion, and you can deflect it using an electro-optical deflector and decide now I'm gonna shine my laser on ion one, ion two, et cetera. But this laser is some 10 megahertz detuned from the electronic transition, so I'm not exciting any population, I'm not doing rotations around the X or Y axis, but instead what I'm doing is I'm inducing an AC start shift, which detunes effectively my transition with respect from my laser. And now, if I put this energy difference of this detuning for a while while I'm doing my laser pulse, then what I get is a phase evolution, which is equivalent to a Z rotation around the Z axis of the block sphere. So what I can do with this technique is addressed Z operations. So addressed Z rotations, which is also one of our building blocks. So now, you might believe me if I tell you that with this global X and Y rotations and this address Z rotations, we can implement any local unitary we would like to. And what we're missing is the interesting part, which are interactions, and for that we need the motion modes. So one way of implementing interactions or making entanglement is this so-called a member servicing gate, where you couple your, so you do a two-photon transition, one will be slightly detuned from a sideband that removes one phonon, and the other one will go, like finish the frequency difference to the state where I have two spins up. So then, what I'm doing is I'm creating entanglement between the, let's say I have two ions, between the state where I have two spins down, and then the state where I flip one spin up and add a phonon, and the state where I flip the other spin up and add a phonon or I remove a phonon, and so on. And then what I get in the end, if you go through the math, is constructive interference along these four paths in such a way that the coupling strength is independent of the phonon number, of the, yes, of the phonon number. And you have this gate that creates entanglement between the spin degrees of freedom and the motion degrees of freedom. Now if you time this gate carefully in such a way that you make a full circle in phase space, let's say you time your gate to be a multiple of the trap frequency, then you will have created no entanglement with a motion degrees of freedom, and what you're left with is an effective spin-spin interaction, so you are connecting now the state down-down and up-up, so what you get actually is a sigma x, sigma x type of interaction. And this is the type of interaction that happens between all possible pairs of ions in your system. So it's an infinite range interaction which has its pros and its cons. Was there a question? So a very important thing is, so what we're doing here is a pairwise interaction between all possible ions, and now say we just want to make a nearest neighbor interaction, like to implement a synode gate between two qubits. Then what we're gonna do is we're gonna decouple some ions from this interaction, and this we can do by, for instance, this is our normal qubit transition. We can bring coherently the population from here, from this state to this state via a laser pulse, like a pipe pulse restaurant with this frequency. Now we bring the population from this state down here and then up here, and now everything we did was coherent. We now have our quantum information encoded in these two states here, and now if we come and apply our murmur servicing gate on the usual transition, this ion that we have hidden will not participate in the interaction because there's nothing here. So in this way, we can apply this decoupling scheme on particular ions in our chain and get them out from the interaction in such a way that we can decide, oh, now we're gonna make an entangling interaction only on ions, whatever. One, two, and three, out of four. And this now, together with the previous operations, is what you call a complete set of gates in that a sequence of such gates can generate any unitary evolution you would like. All you need is arbitrary local operations plus some powerful enough entangling operation, and that's it, yes. Define scalable. Yes, yes, probably not. So this scheme, so this is a very good question. Say you have whatever, 10 atoms in your register or 40 atoms in your register. This is still something that you can do. Now, such long range interactions, if you would be talking about like 200 atoms or 1,000 atoms, no, it's not. So you would want in that case to keep your register size small enough, let's say, 10 atoms or a couple tens of ions. And then you can do long range interactions here, which will provide you an advantage versus only nearest neighbor interactions. Because if you want to entangle this with this, it's quicker to just do it directly. But then at some point you have to break down and make your device more much, yeah. Okay, so decomposing actually, like you're asking how do I go from this building blocks to the more complex interactions. Say I want to implement such an interaction for a time delta t, well, I plug this into some computer code and it tells me this answer and I don't really know why such a pulse sequence reproduces the evolution under such a Hamiltonian. But I know that such a decomposition has to exist because the set of gates that I have is universal. And I can always find a brute force way to break this down and efficiently even, only that the number of gates that I get at the end might be a lot more than the optimal implementation. Well, this is an experimental question of like how much time and effort I'm willing to spend looking for implementations of my unitary operations. But in principle, it's always possible. So this is the power of digital quantum computing. Now, if I want to implement any Hamiltonian in nature, I can just break it down into pieces that don't necessarily commute. Write down these pieces as sequence of quantum gates that I can implement as laser pulses and then apply enough of them until I implement the evolution under the whole Hamiltonian. Whether this is experimentally feasible or not is a different question, but the way, so conceptually, the way is straightforward. Yes. Yes. So the time here, just take it as a free parameter. By this figure, I don't mean anything in particular more than this is possible. So my notation here is z is a rotation around the z-axis. You know, this addressed z-rotations of angle pi over 2. X is a rotation about the collective rotation around the x-axis. Can you repeat the question? Yes. Yes. Yes. And no, no. It means I need to... So this is perhaps... I'm sorry if you're not very familiar with the circuit... Sorry, who is familiar with the circuit notation of my bat? I will try to... Then I will spend a minute on this because it's quite important. Each line here represents a qubit, so one of my irons. And then each box here is a gate, an operation that I applied on this qubit. And now this, for instance, is a rotation around z of length pi over 2. Then it doesn't matter what happens in the meantime, physically speaking, I can wait or not, but ideally I'm doing nothing. And then after doing this, I do a rotation of all three qubits around x and again with an angle of pi over 2. And then I entangle them with a full entangling murmur servicing gate. And then I apply this rotation of length delta 2 over 2, where delta 2 is just some free parameter. So ideally in such a notation, between the boxes nothing happens. In such a notation, I'm not saying exactly what I'm doing in the lab. Of course in the lab I have like a pulse and then a waiting time. It's important for the experiment. In this notation I'm just saying what happens to the qubits. Could you speak up a bit? Yes. In such an architecture, we usually work, like in the digital way where we apply like signals of gates, we usually work with, I don't know, at most 14 irons, 16 perhaps. If you abandon the circuit model then you can do a couple tens of irons, let's say 30 perhaps 40. And then there's experimental constraints on how many irons you can work with. One of them being like how many, like you can only confine, the more irons you have you need to confine them more strongly and at some point they just will try to escape through the ends of your trap. I showed you a picture for one ion where I showed the electronic transitions plus the side bands and at some point this just becomes too crowded so it's very hard to avoid unintentionally exciting some additional motion modes. So all of this restricts the number of qubits that you can physically have in your trap and at some point you will need to split your architecture into smaller traps where you have some tens of irons and find a way to connect them. So this is true in general for a universal set of gates but it says nothing about the efficiency. Now in this digital quantum simulation scheme where you can decompose like two non-commuting terms into a sequence of like stroboscopic terms then the error of your approximation depends is proportional to the time step squared of course because the longer your time step the more you commit and the commutator of your terms. And you mean in the experiment or? In your life. Yes. We have some time. Right. Okay. No. There is no time evolution between the... This is a great thing about trap tie-ons. The only interactions or rotations that happen while you are doing a laser pulse. When you turn off your laser pulse of course you have these two states and they are separated by some optical frequencies like some hundreds of terahertz and then of course the actual two-level system is rotating at hundreds of terahertz but for the whole physics you stand in the rotation frame like in the rotation frame like now I stand on the natural rotation frequency of the qubit what happens now and all these manipulations happen in this frame where the only deviation you have from this natural oscillation are when you either like apply a laser pulse that rotates you or that makes some frequency difference in such a way that now you are defaced with respect to this rotating frame but in the rotating frame that all happens when you turn off your laser pulses. Yes. How do you prepare the states? By a sequence of gates. So say I want to... So the easiest state to prepare is where all the spins are down, right? That's what I can do with laser cooling and let's say I have prepared that. Now let's say I want to make it prepare an initial state where every second spin like a ferromagnetic state, right? And then what I will do is I address each individual ion that I want to rotate I will put a phase and then I will use my global rotation beam to... So one... Perhaps this is better explained by... This is not a bit involved but say I have I have prepared this and I want to flip every second spin. What I do is first apply some X rotation around pi bar 2 which will bring all my ions here, right? Now I address these two and do a Z on ion 2 of pi and a Z of pi on ion 4 of pi which will now... So it's a rotation around the Z axis, right? So now this will be like this and I flip these two on this plane and now finally I do another X pi bar 2 and now I finish rotating down so this is down, this is up, this is down and this is up. So this is how you would, for instance, prepare such a state by a carefully chosen sequence of this elementary gate. You can't prepare any... Yeah, so I mean... It is completely deterministic. So by this I mean... By this I mean something like this, right? Sorry, it's a coherent superposition of up and down. So the whole post manipulation here is deterministic and coherent. It's all unitary operations. So every laser post is a unitary gate. So to create a superposition you apply a Hadamard gate so a next rotation of pi bar 2 which will put you in a superposition of up plus down. If you change the face of your laser post to be down or up minus I down, you can change the faces and then if you apply intangling gates then you prepare entangled states. Yes, it is. Let me not get into that now but I can point you to references. It is a bit more involved. It requires a use of non-unitary lab techniques like, for instance, laser cooling or pump. Like something that involves... It has to be something that involves spontaneous decay or something non-unitary going on. These processes that I described here are all unitary. Okay. And then finally after I do all my experiments I do the measurement which is this technique is known as electron shelving. I shine a laser on my restaurant with the transition from this ground state to some other excited state, a P state and then if the atom is in the state then it will get excited and decay back and emit fluorescence and so if I look at it with a CCTV camera I see light and if it's in this dark state here since this is not connected to any other state on this frequency then nothing happens. It doesn't emit fluorescence and I see no light coming from the ion. And then if I have a string of ions I can do this procedure and get a picture of the whole string of ions and I will get something like this where I say I have eight ions and some will be in the upstate some will be on the downstate. This is a protective measurement so I do an experimental sequence make such an experiment and then see what the state of the ion is repeat a hundred times to gather statistics and this is what tells me my quantum state. And I think we already covered this like of course at some point like this breaks down for many ions and you need to break down the size of your register into smaller pieces there's a couple alternatives for this like planar traps where you have different trapping registers and then you can shuttle your ions around and make them interact or you just split your architecture into little ion traps and then try to connect them via photon interfaces. These are all fields of research under very active investigation but for the moment let us restrict ourselves to experiments that fit in one ion trap. Okay, so yes as we were discussing some minutes ago some of it has to do with the confinement of your ions so if you want to put a hundred ions there then they have Coulomb repulsion so they will try to form a longer crystal and at some point you need to they will start escaping from the ends of your trap so you need to increase the confining potential along z then they will try to escape along x and y so then you push more along x and y and at some point you just cannot confine anymore forming a crystal you can keep them confined but they will form a cloud and then you will not be able to individually address and manipulate them this is one problem another problem is that with a hundred ions you have 300 motional modes each of them has a frequency close to your electronic transition and then if your laser light is not perfectly monochromatic or if you have some other higher harmonics then you will excite this emotional frequencies unwillingly and this is bad because you will heat up your ion chain these are two problems that limit you in principle yes it's a good question I would say not 50 because when you get to 50 you all have a number of technical problems that do not allow you to I mean it's not a fundamental limitation but let's say state-of-the-art practical state-of-the-art for 10 qubits we can do the kind of stuff that I was describing yes for 50 qubits no we can do the kind of stuff that I will show you right now but yes for less than 10 qubits we have such a level of control of course you know like in these gates the next question is okay like you say you do a pi over 2 rotation but is it really a pi over 2 rotation or is it like slightly off well it's slightly off right because there's a lot of experimental errors that pile up and so the fidelity of your gates is not 100% it's something like 99.99% for single qubit gates and 99.9% for entangling gates and that's pretty close to the state-of-the-art in any quantum processing architecture we don't in the experiments that I'm going to show there is no quantum error correction going on quantum error correction has been demonstrating with such systems but at the moment for the kind of experiments that I'm going to demonstrate the infidelity of which gates is such that implementing quantum error correction which requires you to do more correction processes actually hurts you more than it helps you you first need to get your errors below some threshold so that quantum error correction starts to help you alright yes sorry can you like in case of 50 ions do the quantum states they still hold for like seconds or does it decrease to like the lifetime stays so that's a good question the lifetime of an ion chain I mean the state is preserved as long as none of your ions decays and each of them takes a second to decay and now if you consider the probability that neither of us so this is a Poisson process so like if you wait let's say one second and the probability that one ion decays is say 50% and the probability that neither of two ions decays is 25% and the probability that no three ions decays is one half to the power of three and so on so yes like the more ions you have the more you will be affected by this decay process because you want that none of your qubits decay in a given time window and the same for coherence times actually like usually for many kinds of states the coherence time goes as one over n squared so the more like actually it's like quadratically bad in the number of so perhaps we can I don't know how do you feel would you like to make a five minute break or move on raise your hands for a five minute no, less your hands for move on I see slightly less than half so let's make a five minute break and you mentioned that the lifetime decay very fast right like as I say some scales up what's the bottleneck for the scales of the so at the moment we're not limited thank you I hope Martelo isn't here I figure that alright shall we get started so easy models this is the most probably most popular model we have for ferromagnetism and you have some d-dimensional system of spins that are interacting with some sigma x-interaction so something that makes my spins rotate around the x-axis and then I have an additional magnetic field let's say along the z-direction if I only had the x-x-interaction my spins would be aligned like in a ferromagnetic state along the x-direction if I only had the magnetic field pointing in the direction all my spins would be anti-aligned with the magnetic field and the question is now what happens if I turn both terms at the same time then I get some interesting physics and that's the whole point of this experiment to understand better what's going on here now I will hear you say yeah but the easy model you can solve analytically in one dimension even in two dimensions yes sure you have to start somewhere and by the way if you go to three it's actually a very difficult problem computational speaking so it's a pretty worthwhile problem to investigate so and now in this half an hour or so I will try to cover experiments from these three papers I try to keep things even and discuss a paper from from the Monroe group from 2011 then a paper from sorry from NIST and then a paper from Rainer-Bladz group in Innsbruck alright so we need a way to implement so we have these two terms the Hamiltonian the rotation so this is the field around sigma set which in the Hamiltonian is a rotation around the z axis we know how to do this with ions right this is just a far detuned laser beam that will make an AC start shift this we saw before the break and now we need to figure out how to make these interactions but we already saw also how to make a sigma x sigma x interaction the emotional degrees of freedom of the ions and couple the spin degrees of freedom to the emotional degrees of freedom now let me illustrate but this is pretty similar to what we discussed before the break so we have this is the electronic transition that flips my spin from down to up these are all my side bands that correspond to acquiring phonons in each of these emotional different emotional modes if I have n ions and I will have 3n emotional modes and if now I cool down my ion to the ground state then all my red side bands disappear because I cannot subtract anymore phonons from the system and now I can shine two lasers at the same time we saw before we are off with respect to these emotional degrees of freedom and effectively we couple our spins from going from down down to down up and adding one phonon and then to up up and removing one phonon again so effectively we have this sigma x sigma x interaction and now the funny part is that the strength of this interaction remember is mediated by this emotional degrees of freedom so each of these lines correspond to to one particular mode they like a comm mode or some breathing mode or something like that and now the strength of interactions depends of course on the involvement of each of the individual ions in this emotional mode if one of the ions so in some modes for instance the ions will do like this and then the center ion will be static for this mode the ion is not participating at all and if I shine a laser closely tuned to this mode then this ion will not get entangled at all because it's not even participating in the motion so then if I now consider a far detuned laser beam and start counting all the different modes of my chain for instance the center of mass mode is uniform along this the string and so I have a constant positive coupling between all ions which gives me an antiferromagnetic coupling you know like positive couplings of infinite range now I have a tilt mode which where half of the ions interact with one sign and half with the other sign which gives me a coupling that is ferromagnetic of the ion and antiferromagnetic on the diagonal and these two and since now I have a laser that is of resonance then it's coupled to both of these modes so I have to sum them up and here is what I get when I sum them up now I start considering all the high order contributions and at the end what I get is an antiferromagnetic interaction whose strength decreases with distance and the strength of this interaction and then its decay with distance depends on how far I detuned my laser from the mode so if I'm very close so if I'm very far away then you can show that this interaction decreases with the distance cubed and if I'm very close for instance to the center of mass mode then since the center of mass mode is completely uniform across the whole string all the ions doing the same then the range of my interaction is infinite and then it decays as 1 over distance to the power of 0 and in between you have a range of detunings and then you will get different power laws for how the strength of this interaction decays along the chain and this is now the power that you have with such experiments that you can by just changing how far you detune this laser you can engineer the range of your easing interactions and you get very different physics depending on what's this power law that dictates how your interactions decay. Does this make sense? Because this is the building block for what we're going to see next. So now let's move on to the first experiment. So we have a pole trap very similar to the one I showed you already. We have a chain of ions now in this particular experiment we have a hyperfine qubit so the difference with respect to the gate I showed you before is that this is two Raman beams but really the conceptually it's all the same. The difference of frequency of these Raman beams gets you put it close or far detuned from these motion of sidebands and then you can engineer the range of interaction by detuning the bit note of these two beams closer or further away from your motional frequencies. So the flow of experiment is first you prepare your spins in a negative state of the magnetic field along the y-axis and then you turn on the x-interaction which means turning on these Raman beams and ramp down adiabatically the magnetic field up to some final value b. And then you start with the ground state then you turn on the interactions and ramp down the magnetic field and hopefully during the whole time if you do things slow enough you stay in the ground state so at the end you are preparing your chain in the ground state of the system that has both a field and the interactions. This is the because you did things slow enough then the system always remains in its ground state. This is so called adiabatic theorem you've probably heard about that. And then you can study was the final state very rich as a function of the final magnetic field very rich. Right, so then you can detune the laser by a different amount and see how the interactions decay with distance. This is a theoretical plot for the interaction strength. And here in the experiment since in this particular experiment they couldn't study correlations between the ions because they only measure the total light coming from the ions. You can measure for instance magnetization which is counting how many spins are up which means how many spins are bright how many ions are bright at the moment of making a protective measurement or higher orders of the magnetization like this g function which is basically just a measure of the correlations in the chain defined here as a function of the number of spins up. And then this is an order parameter for your system and you expect to have a phase transition for a large system size as you decrease the magnitude of the magnetic field you expect your spins to become more ordered in the direction of your interaction. And this is precisely what so this is the theoretical expectation you expect the actually with the magnetization you have a second order phase transition which means that the the reality of this will be discontinuous for an infinite size system and then in if you study this correlation function you expect actually a sharper transition so this is a more sensitive measurement. And then you see how the transition should become sharper as you increase the number of the sizes of your system because then you are getting closer to the thermodynamic limit. And here are the experiments to compare and you see indeed that as you reduce the value of this final magnetic field both in the magnetization and in this correlation function you see how first of all the transition becomes more ordered so the value of this order parameter increases and also that how the transition becomes sharper for larger system sizes here they are comparing two ions versus nine ions. So that's all I want to say about this experiment which I think was one of the first to study such spin models with trapped ions. Now I want to show another neat experiment where they do use a very similar using two-dimensional crystals in a panning trap. So remember here that the ions are so you have a magnetic field so the ions are continuously rotating at the frequency of some 40 kilohertz and just as before you can apply two laser fields whose frequency is detuned with respect to each other by some magnitude and you can tweak this detuning to get exactly as before to regulate the strength of the interactions to induce. So this is a theoretical this is in theory how all the normal modes of the string look like from a center of mass mode to all this some shaped eigen modes of the crystal and here is where you have oscillation frequencies on the order of hundreds of kilohertz and here is where you place where you will detune the bit note of your lasers and by changing this detuning mu r you get different power loss of the interaction strength as a function of the spin-spin separation and here you can see the geometry of these interactions for instance for the case where the detuning is very very small then your power law is almost a constant so A is almost equal to zero and then the interaction is the same between all ions and as you increase your detuning then you have approached this as a tautical inverse cubed law and the longest interaction is with the nearest neighbors and then indicates as one over distance cubed ok so this is what you expect and now you can actually go to the lab and benchmark these interactions by first preparing some spin state rotating it a bit and then turning on the interaction and letting it proceed around the Z axis where you get a mean field effect and then studying the amplitude of these oscillations and how much or how fast you are proceeding depends on the strength of interaction so now you can change the detuning of your laser which is on the X axis here and study how fast your spins are proceeding and this gives you a magnitude of the strength of interaction and see indeed how it decays according to the right power law ok so now that we have these elements in place I would like to introduce one more element which is what if now we can take pictures of our ions and actually study correlations then I could resolve individual quantum systems and not only along the Z direction but also along the X direction if I first rotate my ions by power 2 around X for instance and then measure along Z the X component of the spin so then I can measure any correlation that I want between ions in the same shot I just do a one shot measurement and I see if for instance the ions are anti-correlated or correlated and so on which I cannot do with just aggregating the brightness of all the ions coming at the same time ok so then how can I characterize my interaction well I can prepare one ion in the spin up and there on the spin down switch on my interaction which will flip spin up down to down up because this is what the sigma X sigma X interaction does and then the strength of interaction is given by the flopping speed of these two states or the rapid frequency of these oscillations and I can now plot this for every pair of ions and the coupling matrix and you see how it decays with distance like then the nearest neighbors interact the most strongly and then the further away removed you you get this power law decrease and having characterized this then you can make very neat experiments such as you prepare your chain in a state where all the ions are pointing down and then you excite one atom or once you flip one spin turn on your interactions and what this will do is ok now your spin will start flipping over with neighboring spins and this has some correlation length and you expect then to create entanglement between the spins at some rate that will depend on the strength of your interaction so the more long range your interactions are which is here then the faster your correlations spread you see here what I'm plotting here is the the excitation of these of the ions whereas for for shorter range interactions the propagation of these correlations is slower because you first have to flip the neighboring spins then the nearest neighbor the second nearest and so on so you see how this speed of propagation depends on the range of interactions which is something there's also of theoretical interest what's the speed at which the speed of sound or the speed at which information can be propagated under such interactions and then see also if you look at not just spin flipping but entanglement how entanglement propagates such that at the beginning for instance spins 3 and 5 are completely unentangled and then at some point this entanglement wave comes through them and then they get entangled and then this entangled again and so spins 2 and 6 and 1 and 7 they get entangled mutually at increasingly longer times let's say the half of this paper is precisely comparing such bounds for the details I will refer you to the paper but yes basically the idea is to relate the range to the Robinson bound well so I mean you could probably see if by localization you mean localization due to disorder you could probably see it but not in this case because this is a very ordered system so in this case I would not speak of localization I mean your initial state is localized because that's the initial state you prepare like a perfectly localized excitation and then what you have is the actually perhaps this makes a bit more sense if we look at in this picture so you can either look at it in terms of the individual sites and then you excite one ion one spin and then you see how this localized excitation propagates or you can think of it in terms of quasi-particles like normal modes in the system and then the eigenstates of your Hamiltonian are not individual excitation but delocalized spin waves so basically you have modes where for instance the middle spins are the ones most excited and then indicates towards the ends most where you have a node in the middle and so on towards a exact state where one ion is spin up and there is spin down and so on these are the actual line of state of the system and so another different way to look at it is to say okay like actually when I excite one ion say I have here I excite one ion and then I turn on my interaction well if I decompose this one ion excitation in terms of the natural quasi-particle basis of the system then this is just some superposition of all these quasi-particles so I'm exciting all these normal modes of the or spin waves and then at some point what I will see is just the effect of all these quasi-particles being excited at their natural frequencies which are given again by the strength of the interaction so the question is okay like now how can I characterize this quasi-particle so this spin waves in my system one way to do it is precisely this I excite one ion let it interact so that all the modes get excited now bring it back to the ground state and detect at the end the excitation of all the ions and measure that as a function of time so then I have here for the seven ions in my chain as a function of time the populations of the excitation or the expectation value of the spin along the z direction or the x direction and this forms some time trace and then if I know Fourier transform this time trace and I prepare my initial state in something that is close enough to the eigen modes of the system so the quasi-particles see for instance that I prepare my state in something that looks roughly like this doesn't need to be exactly the eigen state of the system but I prepare my system I excite these ions a bit and this spins a bit more and at the middle the most and no nodes in the middle then if I project this into the true proper eigen states of my system I have quite a considerable component there and when I evolve and Fourier transform I see a peak appearing at some frequency which is the frequency of this eigen mode and then I can because I know how roughly how these eigen modes look like I can create a state with one node and two nodes and so on and so I can repeat with a different initial configuration and study all the peaks that appear in my spectrum just by Fourier transforming the time evolution what this tells me now if I create a superposition of say the k equals 1 and k equals 7 modes is the frequency difference between these two modes and now I can plot so I'm doing spectroscopy of these quasi-particles and now we can plot the frequency as a function of the k number and I get the dispersion relation so I'm understanding now how these quasi-particles propagate in my system just by doing this spectrum spectroscopy technique which is really a very powerful tool so right so with this I have said all I wanted to say now about simulations of spin models I mean of the icing model in particular I hope that I could convince you that you can study things like phase transitions you can characterize and manipulate the range of interactions by using very simple experimental tools like the tuning lasers that you can perform spectroscopy of your quasi-particles you can study entanglement in your system you can study velocity of propagation of information this leap Robinson bounds there's really a lot of information about your spin model that you can very readily access by these very simple and now mainstream experimental tools and so this is the analog you know you have some model like the icing model that you want to study you figure out okay in the lab if I turn on this bichromatic beam and at the same time enough resonant beam then I have exactly the same Hamiltonian as the one I'm trying to simulate that's great but this is not I cannot repeat this trick for every possible Hamiltonian because at some point if my if the physics that I'm trying to study are complicated enough I will have nothing in the lab that actually completely resembles what I'm trying to look at so that's what when I need digital techniques and that's what I would like to talk in this last half an hour and as an example I will discuss how to simulate a problem in particle physics and and the model we are going to look at is so called lattice gauge theories studying gauge theories on a lattice but I will just say what I mean by this in a second so to be a bit impressive you know like in CERN they build this multi-billion dollar particle accelerators to figure out the new physics no one is trying to live CERN out of work but I would like to make the contrast that you can study the same kind of physics both with this multi-billion many kilometer machine and a humble very tiny ion trap with the difference that in here you can actually go and discover the actually what's the Hamiltonian of my system you know you come here as the like Ranschen of your particle physics and then once you know the equations of motion you can come to your quantum computer or quantum simulator program them and see what comes out which is a non-trivial problem as well so this is a view through a window of a vacuum trap the vacuum chamber the vacuum chamber looks more like this has four windows to the sides this particular one it's an ultra high vacuum so this is 10 to the minus 11 millibar which you need because otherwise the ions will collide with background gases and get kicked out of the trap so the ion life time is mostly in the trap not the information but the actual lifetime of air in the trap is mostly limited by background pressure with a good vacuum you can keep an ion trap for months without it escaping the trap it's at room temperature you can do such experiments in cryogenic setups which has advantages when it comes to noise but it also complicates and heating up but it also complicates your experimental setup quite a bit so this particular experiment setup where the experiments that I'm going to discuss happened is at room temperature that's what you would like to know so so then star model you know that the interactions between you have a number of elementary particles that describe the matter in your system and then the interactions between them are described by engaged theories which are I'm not going to get into detail because I don't know the detail but they are described by theories that have certain symmetries certain mathematical symmetries and these theories are quantum mechanical theories at heart so they are difficult to simulate with classical computers and thus comes the idea well perhaps we could simulate them on a quantum computer and now so of course the long term vision here is to simulate the stuff that is actually so for electromagnetism for instance you know that the electromagnetic interaction is weak enough that you can treat it perturbatively and there has been huge theoretical success in the past century in solving this problem analytically or with analytical calculations however if you want to treat engaged theories where the couplings are strong like in quantum chromodynamics that studies interactions between quarks with a strong nuclear force then your perturbation theory doesn't work anymore and then you need to at least for some phenomena and then you need to do non-perturbative simulations and one way to do it we'll see however that's the most difficult problem so that's exactly why we're not going to start there, we're going to start again by the simplest problem which is quantum electrodynamics show that and show how we can simulate this on a quantum computer not because this is an unsolvable problem analytically but rather because this is the first time ever that we are trying to do this so we might as well start with the with the first step okay so this is like an experimentalist view of quantum electrodynamics I have charged particles, electrons and anti-particles that interact via electromagnetic force fields particles and anti-particles can mutually annihilate and one prediction of the theory is actually that if I can if I create a strong enough field then a non-perturbative effect that happens is that particles and anti-particles can actually get created out of vacuum fluctuations and this is a so-called Schringer mechanism and it has not been as far as I am aware, experimentally observed yet but this is a very strong theoretical prediction of the theory and so how can I okay so this is the conceptually for very simple people like me the view of what we are trying to do and now how can you simulate such a theory okay so in reality you would say I have all these particles floating around or interacting if I want to reduce the problem and make it simpler I can discretize space and so divide space into little boxes and each of these boxes can hold one particle or one anti-particle for simplicity we are just going to consider one spatial dimension okay and so we have reduced our continuous three-dimensional space to a lattice and then on each side of this lattice we have actually let's say we have blue sides and red sides on the blue sides we can either have a particle or no particle and in the red sides we can either have an anti-particle or nothing and this is a discretized version of the of the actual space that we are looking at and in the limit where we make the spacing of this lattice go to zero then we recover the continuum version of for instance Dirac's equation this you have to believe me go through the math and you recover quantum electrodynamics in one dimension just by letting the lattice spacing go to zero this makes sense so far so each side then has two possible states which are to be full or empty and to encode whether there is a particle there or not and so then I can encode this as a spin model which is the whole point of any quantum simulation usually is to have some physical problem try to reduce that to a spin model a model where you have spins up and down interacting with some Hamiltonian and that we know how to write down in our quantum computer so then if a side is has an electron or a positron hole then I'm going to call it a spin down and if a side has a positron or an electron hole then I'm going to spin up it's just two level system per site and then of course I have also superpositions so I encode this in each of these sites in a qubit or a two level system or a spin one half this makes sense because this is if this is clear then the rest follows yes so you know that in in the continuum description sorry in the ideal continuum one dimensional model that we have in mind then in each point of space we have a field that has two components like in three dimensions you have spin up and down and then you have the four components of this field per positions you have a space dependent field in this case you have a space dependent field that has only this the particle component let's call it an electron and the antiparticle component let's call it a positron and now the mathematical trick that you do is to split this now you discretize your space such that in each site you have this two component field and then a mathematical trick that you place that you break down break up your lattice into even and odd sites and you put the particles or one component of the field in the even sites and the other in the odd sites in the original stringer paper they literally say we do this because it works so in one dimension let me see that you have no spin actually because there's just one dimension so you only have two components so this is a case where you have like spinless fermions and then a step that I'm not for those of you who are more theoretically inclined a step that I'm putting under the rag is that of course you have fermions a fermion operator per site and here I'm just drawing spins what you do in the middle is a Jordan Wigner transformation to map these fermion operators as spins and it turns out that the one dimensional problem is so nice that the presence or absence of like the value of the fermion operator and each site can be mapped directly to the expectation value of this spin up or down per site but if you don't know and don't care about Jordan Wigner transformations then you can trust me when I tell you that it is enough to write down a spin up or down encoding the presence or absence of a particle or antiparticle per site does this make sense it's like the the separate basically when you look at it like if you look at it from far away and let the lattice spacing 10 to 0 then you would group these two sites into one physical space right so that you ask okay like do I have a particle and if I have just one particle then this would be this would be full and this would be empty and if I have just one antiparticle then this would be full and this would be empty and so on just if you look at it from far away well you don't neglect it actually rigorously speaking one dimension you have no spin degree of the particle right yeah I mean you do the most you can do in one dimension so which is admittedly a very simple toy model does this encoding make sense because this is like pretty much the last conceptually complicated thing that is to come anyways you can stop me later so now once we have this spin chain we also on top of this we have interactions which again are given by the gauge fields so the way we encode this is we put a gauge field on the links between the sites which is the electric field and the expectation value of this field of this field on the links between the sites is the strength of the electric field between the at one particular site in space and then the effect of this gauge field without writing down any equations is that a neighboring pair of particles and antiparticles can get created or destroyed and then since I have to respect Gauss's law if I destroy a particle antiparticle pair then I also have to remove one unit of flux of electric flux between them just to keep the balance of charges and fields constant okay so that and now a nice trick we can play is that if we know the boundary conditions of the electric field say the field is zero here then I know that I go through a minus one charge so the electric field is worth minus one then I go through a plus one charge so it's zero again zero zero zero minus one again so by knowing the boundary conditions and the distribution of matter in my system so the charges I know the field everywhere so I can trace it out and what I do when I trace it out is I get an effective so I have removed all these gauge fields and I'm left with an effective spin Hamiltonian where I have this long range interactions that come from having eliminated the gauge fields and if you look a bit and the shape of these things have is that the interaction of a spin with the ones to its left is constant which makes sense because this interaction was just given by removing the like by propriety in the boundary conditions and to the right it decays linearly or increases linearly depending on the sign and this is exactly the form that Coulomb law has in one dimension it's a linear so the potential of a point charge in one dimension is the potential increases linearly and the electric field is constant up to infinity and that's just how electrodynamics in one dimension looks like so we do nothing strange here we just recover electrodynamics in one dimension from eliminating our quantum gauge fields but the whole treatment is still quantum and now the physics of my problem yes yeah it is right yes it is an appearance you do retrieve the exact same results if you start from the left or from the right it's not immediately obvious but you do I mean the procedure the physics is left to right invariant the procedure that you do to eliminate your gauge fields is not like you break the symmetry somehow so then you end up writing a Hamiltonian that that is not left to right that is asymmetric which I agree with you is like super anti-intuitive but if then you see look at the physics that this Hamiltonian describes and they are the same how you started from the left how you started from the right right so then the dynamics that is going on in this model is first of all I have this creation and annihilation I can create or annihilate neighboring pairs our intuitive idea of electrons coming together with positrons and annihilating each other then particles have mass so it takes energy to create particles there is another term the self-energy given by the mass of the particles and finally I have this long range exotic asymmetric interactions coming from eliminating the gauge fields which is the Coulomb interaction and this is exactly the kind of interaction that would be impossible to do in an analog simulation because this is a very asymmetric counter-intuitive looks very artificial it is what comes out of our mapping but this is something that you would never go to the lab and find naturally a kind of physical interaction that has its shape so that's why digital simulation is great because now we know that these are the three terms that we have to simulate so this sigma plus sigma minus interaction that flips neighboring qubits plus long range sigma set sigma set interactions which have these coefficients that decay linearly to one side and state constant to the other side so a very exotic thing plus sigma set terms which correspond to the particle masses but each term in this interaction we know how to do because for instance this sigma plus sigma minus term is equivalent to if you go through the math a memory servicing gate around the X axis followed by a memory servicing gate around the Y axis so just two of these entangling gates that we discussed before the break around with different faces one after the other give me this nearest neighbor interaction this spin flipping term so I know how to do each of these terms then each of these terms get the sigma set sigma set interaction which is actually the same as the sigma X sigma X interaction only rotated by pi over 2 so I can sandwich a memory servicing regular memory servicing gates around some rotations around the Y axis and get exactly the sigma set sigma set interaction and finally the single qubit Z terms are nothing else but rotations around the Z axis which are produced by off resonant laser pulses that you address on each qubit so we have seen how you implement each ingredient of this full Hamiltonian and now because of the power of digital quantum simulation you can just break the whole thing in steps and apply each of these terms one after the other repeated times and look at the time evolution of the full thing yes absolutely correct absolutely correct what we let's go over this so what we do is first prepare some spin configuration and I know that because of my mapping an arbitrary mapping but because of my mapping this corresponds to vacuum vacuum vacuum so I start with the state with no particles or antiparticles present in the system and now turn on my evolution ideally if I turn on my evolution what happens is that particles get created annihilated which in my encoding means that spins get flipped such that if an even spin gets flipped then it corresponds to a creation of a particle for instance and a not spin gets flipped corresponds to the creation of an antiparticle and so at some point if I now look at the state at each point in time and count you know which spin corresponds to the presence or absence of particles so you count how many particles antiparticles do you have how many particle pairs do you have and you plot here the particle number density which you can instruct just by counting spins up and down it is very simple and then you see that you create particles antiparticle pairs and at some point the density of particles antiparticles in the system which here is pretty small it's so big that they start annihilating each other so then they start the particle density goes down again so you get these oscillations between the vacuum and the excited states in your system with particles antiparticles ok this is the theory now we are going to apply the Hamiltonian step by step and this is what we get so we start in the vacuum and then follow the ideal evolution quite well until at some point because of this the discretization errors you start deviating from it but you can still see the oscillation of the particle antiparticle numbers trying to understand the performance of the experiment on the red curve we have the expected evolution taking into account only discretization errors this is like we are making a finite step size instead of making the steps infinitely small so how much of the error comes just from making this course time steps it's actually quite a lot and if we then add to this a model where we take into account the experimental errors so the coherence and mostly the coherence then we get a pretty good grasp of what's going on you may ask why don't you make the time step finer if you have this because it means that to cover the same time span you need to do more time steps which means a larger number of gates and in this case you already have something like over 200 gates being applied to the system so if imagine if each gate has a fidelity of let's say 99% or 99.5% and you apply 200 of them then the fidelity of the total operation decays exponentially with the number of gates so this is why you have to find this balance between the step size and the discretization error yes yes well it's more gates per step you can evaluate let's say it's not conceptually hard you just have to evaluate whether the added number of operations per step justifies so you will get less discretization error but also less fidelity per step because you're doing more gates and this is a non-trivial experimental question like given the performance of the gates in your system how easy it is to add the additional overhead to make a higher order total decomposition well in this particular one we chose the size of the total step in such a way that we can see this is really like pushing it this is at the very limit of what quantum processors in general can do couple hundred gates of idea of 99.9% per gate let's say so in this case we chose the total step in such a way that you can cover an oscillation because less than that doesn't give you much information and not make it so large that you kind of overshoot and don't even see the oscillation so we aimed at something like four or five steps per oscillation now you can compare the so one nice thing is that now you can tune the parameters in your system which is after all why you're interested in doing simulation because now like now that you have everything set up then the mass of your particles just corresponds to a sigma set term in your Hamiltonian which is the length of some laser poles you can make this easily longer or shorter and change the mass of your particle so you can repeat the time evolution which is on the horizontal axis as a function of different particle masses and what you expect to see is first that the higher the mass the faster the oscillation because you are further detuned from your vacuum state and your excited state and second that your oscillations have less amplitude and this is very understandable because the more massive your particles are the more energy you have to inject into the system so the fewer particles you can inject the fewer particles get created and this is exactly what you see here this is the experimental data and this is what you expect from the in the ATL case you can also do stuff now you can study quantities that you cannot even look at in a particle accelerator so you go to CERN and basically what you can do is count particles, energies and directions that's it it gives you a lot of information but you don't know if your particles were entangled here you do because you have access to the full web function if you are smart enough at studying your quantum state you can learn anything you want about it perhaps not all of that at the same time so here we study the vacuum persistence which is basically how much of your web function stays in the vacuum in the initial state and entanglement across the system so I divide my chain into halves and then study the entanglement of the left half of my chain with the right half of the chain in this case quantified by the lower rhythmic negativity and you see how the same thing happens the product state the vacuum then you create entanglement and as more particles get created across the border at some point they start annihilating each other and entanglement actually goes down and you create more entanglement the less massive the particles are because you have to it's harder to create entanglement when the particles are massive and it's also harder to create entanglement when the coupling to a field and to create a particle you need to pay the energy price of creating the field that comes with this charge I think that's all I wanted to say about this experiment yes I had a couple slides left on quantum chemistry simulations but I think we're closing to the end of the talk so do you prefer to have like five minutes for questions or the last topic anyone has a question so far yes the ground state of the system is an excellent question it's not the initial state so in this particular experiment we start with the vacuum so we start with the absence of particles and the particles we start with not the ground state of the system because I have interactions so in the presence of interactions my ground state is an entangled state actually not the simple product state your question is how this ground state looks like it's some entangled state what I would say I don't have a good do you have any intuition there will be some in the loose like an antiferous magnet so the vacuum is not at a trivial state like the one and actually like the experiments that they're doing at the moment are precisely how do I prepare the actual ground state of the system and not the actual dressed vacuum and not just this bare vacuum and this is non trivial because a complicated interacting Hamiltonian finding its ground state is a non trivial problem and one way to do it is I'll say what's going to explain next but I'm not going to have the time for instance with the I would not like to start something that I can rush through it if we're out of questions here then I will quickly explain that but the basic idea then for your next question might be if I want to find the ground state how do I do it which I cannot just turn on my computer and then you can create some approximation to your ground state apply your Hamiltonian and measure its energy experimentally measure all the operators that give you the expectation value to your energy and then try to change your state and minimize the energy experimentally and then at some point your ground state is a state that minimizes your energy can you do non-abelian gauge theories can you conceptually yes there's no no you will need so per site in this discretization step of the encoding you will need per site a more complex quantum system it will not be enough to have just spins up and down because you will need to also the color degrees of freedom and let's say the different flavors right if you are if you wanted to make like an SU2 theory for instance so you will pay the price of having a more complex quantum system per lattice site you will need many degrees of freedom which you can implement with higher dimension spins for instance with more electronic states of your ions and also your interactions will be more complex because now your spins will interact in a non-trivial way however I mean this does increase an experimental complexity but to that extent there is no conceptual problem in repeating this with a non-abelian gauge theory David concerning the easy model we were speaking also before if we use a bosonic atomic species or a fermionic one there are differences in the experimentally controllability or feasibility yes no you are thinking of you are thinking of for instance of a cold atom experiment where my bosons or fermions are mapped into bosons or fermions in my lattice you know and then absolutely the bosonic or fermionic nature of your atoms are on the statistics of your problem and you have to use fermions to study fermions and bosons to study bosons in this case the spins are given by the internal electronic states of your of your ion some s orbital or d orbital in your valence electron and the physics doesn't care at all whether your ions are bosons or fermions they might be bosons or fermions they are exchanging them because they are very strongly confined so there is no interactions between the ions coming from their bosonic or fermionic nature in none of these experiments all that matters is the electronic degree of freedom and whether it's in one orbital or the other so you encode that in a two level degree of freedom and there is absolutely no interactions apart from that when you turn off your lasers there is no other physics going on is it possible to overlap two traps I mean using two lasers using different tuning frequencies and shifting them for example linking some valley for sides and in the other case for edges you mean to do what to probe their bosonic or fermionic nature no I mean this is another question just to use two alternated lattices linked by the laser trap oh no because the lattice is not given by the external laser but rather by the coulomb repulsion between the ions so you put two ions together and then they will repel each other because they are both positively charged and then if you put a third one then they will also like repel each other so you have three ions and so on and there are no lasers involved so far you don't need any lasers to trap the particles you need them to cool them down but this is a separate thing but the lasers get the ions get trapped in a crystal just because of their mutual coulomb repulsion so no later no laser lattices there in this experiment there are no laser lattices at all so when you introduce the mapping to take care of the gauge fields does that mapping also include the sort of like fluxes going through the entire system or like flux loops like coming from well in one direction taken care of I mean so in one dimension I have no loops or I mean if I understand it correctly or you mean the like say I have periodic boundary conditions and then is that what I mean I don't know so I was talking about the actual flux loops because they would be contributions to for example a ground state the actual ground state of the model what do you mean by flux loops in this context so on every link between sites you would have a gauge field basically going through and that doesn't violate the Gauss law but of course that's considered in the mapping the degree so in each link between my sites I have a gauge field and in one dimension the only possible values that this the expedition value of this gauge field can have are integers so corresponding to the electric values of the electric field are discretized so these are really all my degrees of freedom in this system I could explicitly encode them so this is another in quantum link models I can explicitly encode each of these gauge fields into a higher dimension spin say I have whatever spin 10 here and then I have 10 possible values for my electric fields and then I can implement all the interactions and not discard the gauge fields however in this case the tracing out of the gauge fields is an exact thing just mapping them to this exotic interactions so I have a question about the ion the ion trap so you mentioned that when you're increasing the number of particles that you have in the trap then you have a lot more phonon modes which are closely spaced so why don't you why don't you use less massive particles so that you can increase the space in what the choice of particles given by many considerations one of them is a mass but there's really a lot of things that go into choosing one species and it turns out there's not so many good species for working with to begin with you want to work with something on the second I wish I had a periodic table you want to work with something on the second column of the periodic table because you want something that has two valence electrons I mean it's a requirement but it's nice because then when you ionize it singly then you're left with one valence electron and this has a very simple electronic structure so then you can find an optical transition that is your qubit you can find an optical transition that allows you to cool down the ion and this requires a lot to do this simply on a simple electronic structure so this restricts you to the whatever seven elements that are on that column of the periodic table are not radioactive if you don't want to get into complications and so these are really the few popular choices for trapt ions people do work with other elements and then from that you can definitely work with heavier particles but usually you may pay the price that the wavelengths you have to work with are more less comfortable so for calcium you work with red light for the qubit laser and blue light for the cooling transition which is quite nice to work with in other species you may have to use ultraviolet light for for manipulation and that is technically quite challenging so and are there ways that you can shield the Coulomb potential in some ways such that you can bring them closer you can always increase the Coulomb so you can bring them closer but this just increases but up to some extent because at some point if you confine your ions too much then they don't so if you're external there's two energy scales here the external confinement and the mutual Coulomb repulsion and if the external confinement is a lot bigger than the interion Coulomb repulsion then your ions will not form a crystal anymore and will collapse into a cloud and then you cannot manually address them anymore I think we're kind of running out of time I don't know if we have another question some extra question extra time bonus question