 OK, good morning. I'd like to continue basically where we stopped yesterday. And before we go to some new things, I'd like to summarize briefly on a few slides what we learned yesterday, basically. So we're talking about, so the theme is interacting ions. And in the first lecture, you learned that they interact via the common vibrational motion. So that's our quantum bus. So we need to be able to couple the internal dynamics, the qubits, to the emotional states. And that's what we did yesterday. So we looked at emotional modes of harmonic oscillators and of internal qubit states. And we learned how we can couple them by shining in appropriate, retuned electromagnetic radiation. And so that's how we make ions interact. And I want to summarize quickly what we saw yesterday. So we looked at our qubit states, two level system, like a spin 1 half. And these qubits are confined in a harmonic oscillator potential described in this way. And then we shine in radiation. The radiation has a frequency, linear momentum, detuning, and intensity. And these parameters you can use in a proper way to control this interaction. That's what we saw yesterday. And I just want to summarize this now in some pictures. So let's look again at our two level system, which is now called G and E. And I just saw in Moro's note he also changed between G and E and 0 and 1. In any case, so that's our qubit G and E here. Then we have our harmonic oscillator states. And they can be the ground state, first excited state and so on. And now this is our exciting electromagnetic radiation that we shine in. And we can tune it. And we tune it now to the resonance frequency of our qubit. So just that you have in front of your eyes the definitions of these quantities, I quickly draw again. So omega is the splitting of the qubit states. Omega L is the radiation. And then we had our harmonic oscillator. That's the frequency splitting of the harmonic oscillator. What else do we need? Omega is the Rabi frequency, the strength of the coupling. And then we have this phase phi of the radiation. OK, so we tune the radiation to the resonance frequency of the qubit. And then these transitions are possible. We saw that in this case, our Hamiltonian, which is a pretty long and not so easy to handle expression, shrinks down to a very easy case. So it's simply rotating the qubit itself around the sigma x-axis. So we do not change the emotional state when we tune to the so-called carrier transition. But we do change the state when we, for instance, tune to the so-called red side band. So our electromagnetic radiation is now red detuned with respect to the atomic resonance. And then we saw yesterday, under this condition and doing the rotating wave approximation, we find this Hamiltonian here. And here you see now that internal dynamics and external dynamics are coupled. So we create an excitation. And we destroy an excitation in the atom. And we destroy an excitation in the harmonic oscillator. And this thing is just doing the opposite. OK, and the coupling strength, please observe this. It's no longer the Rabi frequency. It's eta times the Rabi frequency. So remember, eta was this lamb-dickey parameter that measures basically the momentum of the radiation of a photon, the linear momentum of a photon. OK, so that's the important parameter. If this lamb-dickey parameter is 0, then all the fun disappears. There's no more coupling between internal and external states. Then you're just left with no excitation, basically. OK, and then you can tune to the so-called blue side band. So that's the other Hamiltonian that we saw yesterday. So this is called the chain's Cummings Hamiltonian, actually. So this was first discovered in the context of quantized radiation interacting with a two-level system. And exactly in the context of cavity QED. So when people quantize the electromagnetic field in a cavity, then it's described by such a model, a two-level system, harmonic oscillator. Here our harmonic oscillator is a mechanical harmonic oscillator. In the cavity QED case, it's a field. So we can realize the same Hamiltonian that appears in cavity QED here using a real mechanical harmonic oscillator. OK, and then you can have this anti-chain's Cummings Hamiltonian. I wonder what this wiggling is. Is this due to the connection here? Or is this something intrinsic of the? I'm afraid we have to live with it. OK, so that's then the so-called anti-chain's Cummings Hamiltonian. And what you see is you create an excitation in your qubit and at the same time create an excitation of your harmonic oscillator. So you go, for instance, from the ground state G with zero excitation to the excited state E with one excitation of your harmonic oscillator. So by changing, by flipping your spin, you can, at the same time, induce an excitation of your harmonic oscillator. In this case, all you can lose one quantum of excitation in this case. Of course, the opposite is always also true, so you can also go backwards. So this you lose one excitation in your atom and you also lose excitation in your harmonic oscillator. So that goes back and forth. It's a completely, it's a Hamiltonian with a unitary time evolution. OK, so now what we learned, we learned to couple a harmonic oscillator to the internal qubit states. And what we really want is to have several ions talk to each other. And I showed you this picture already, so we do what I just explained. And now this is supposed to move. I don't know why it does not. So we excited this ion, then the whole ion spring moves. That's our harmonic oscillator now. So the common vibrational motion of all ions, the normal mode of n ions, that's our harmonic oscillator. And we can excite this and then couple internal. And as I emphasized many times now, this is the important parameter that tells you how well this works. So if it works at all. OK, let's look at the same thing in a slightly different picture, which turns out to be very useful for more advanced quantum gates. So the picture is simply still a two-level system and our harmonic oscillator. But now the harmonic oscillator is depicted in phase space. So this is the coordinate in the position coordinate, and this is the momentum coordinate of our harmonic oscillator. And what we do now when we excite this qubit or flip the spin, whatever way you want to think about it, then what we do, we give the harmonic oscillator a momentum kick. So we displace it from its initial, from its equilibrium position. And then the harmonic oscillator starts to oscillate around this equilibrium position. You convert momentum into position into momentum into position. So yeah, yeah, so how do I do this in the lab? I'll show you. We need a harmonic oscillator. So this is a harmonic oscillator if you only have small angles of extension. So you have your harmonic oscillator. And what you do in the lab, you give your harmonic oscillator a momentum kick, and then it moves. So you turn on your electromagnetic field for a certain time, and then you transfer momentum, and then your harmonic oscillator starts to move. So the kick is provided by the momentum of the photon. Yes, everything we assume so far is completely coherent, yes. OK, so when we look at this harmonic oscillator here, so just to illustrate this phase space, so we give it a kick, so we give it momentum, and then it oscillates, so it has momentum, it moves, it comes to a rest. That's then where did I put this? That's at this point in phase space where we have maximum displacement from the equilibrium position, and zero momentum, and then the game starts again. It's going backwards, back and forth, and so in momentum space, this is just a closed circle. OK, so I'll come back to this picture later on. I just want to mention this here. Now, and what we learned is basically the Hamiltonian, the interaction Hamiltonian, that's the interaction Hamiltonian, and this eta measures how well you can displace your, in this case, an ion from the equilibrium position. OK, and now we come to some real, so this was so far completely general everything, and now let's look at some real experimental implementations. So what do you actually choose in a real life? Also, a precious question, how do you do it in the lab? So you need to choose an ion first, and if you want a qubit, you typically, so the first qubits were metastable states of ions and hyperfine states of ions, and I will illustrate both cases. So for convenient ions are barium, calcium, strontium, etherbium, where etherbium, not so much for this, but more for the hyperfine structure. So, but calcium is a very popular ion these days, and it has the crown state, which is stable by definition, and then you have a metastable state that has two quanta of orbital angular momentum, so there is a total angular momentum of 5 half, so there's no dipole allowed transition here, that's why the state lives for a long time and is well suited as a qubit state, because you don't want to lose your coherence by spontaneous decay, of course. OK, and now you can do everything I showed you by building a laser that can excite this spot-report transition and because we want everything to be coherent. At several places, I already pointed out the face of the light field must be kept stable, otherwise you just lose your coherence very quickly, and now if you look at the frequency of such a transition, that's a typical optical radiation, 10 to the 14 hertz, a few couple of 10 to the 14 hertz, so that's a lot, and let's say you want one second coherence time, then you need to keep your face fluctuations stable to one part in five times 10 to the 14. So this is a formidable challenge for an experimentalist to stabilize the frequency of an electromagnetic field source to one part in 10 to the 14. So this challenge has been taken on by several groups, most prominently the group of Reinerblatt in Innsbruck and they did great experiments using these transitions here. OK, so this is again our Hamiltonian, now only for cubic rotations and you see that's the face where it appears and if on every operation you do, you have a different face, this just amounts to defacing what you just saw on the blackboard in Moro's talk. So you start with a density matrix that has off-diagonal elements and if you average over many off-diagonal elements with different faces, they just disappear. So the coherence disappears if you do not keep this face stable. And this is, like I said, a challenge and has been tackled by the group in Innsbruck, so this is one of their first demonstrations I believe of this electric quadrupole transition using a highly stable laser which was not that stable at this time but I think today that's about the stability they have, the relative stability. And if you look at such laser light sources, they are very impressive. So you need to keep, like I mentioned, you need to keep the face stable, so you have to have a high absolute stability of your frequency. So not only face fluctuations, also frequency fluctuations, which in the end is the same, have to be suppressed and then you need a high amplitude stability because in your Hamiltonian you have this Harvey frequency and this depends on the amplitude of your light field. So you have to control, you have to have exquisite control of all parameters. And so you need a very good beam quality of your laser beam. It's not sufficient to have some fluorescent lamp focus or something. So you need very well spatial profiles of your beam. You need pointing stability of your laser. So if you look at this slide beam, this has not a good pointing stability. You see my hand is shaking and this is amplified over this large distance, of course, and if an iron were sitting there, I would have trouble hitting it exactly. So that's what's pointing stability and then you have diffraction effects on all your optical elements and on your vacuum apparatus. So this is, to do this with high fidelity, it's a great challenge and this challenge has been mastered by, for instance, this group here. And, yeah, and I want to show you one recent experiment from the vinyl blood group, from the INSPUR group. They did implementing Schor's algorithm. So that's the slide I showed you in the very beginning of the first talk and before I show you this, I want to point out that they do not use this Xerox. Xerox-Zollar gate that I introduced. So this Xerox-Zollar gate is very useful for pedagogical reasons nowadays to explain how ions can interact but the actual implementations today usually use different gates, different types of gates and one of them is the so-called Mermer-Sørensen gate or Sørensen-Mermer gate and this works in a slightly different way and I will not go into details. I just want to point out that this is the case and say a few words that illustrate the fundamental mechanism. So you have two light fields now. So you have your two qubit states, both in the ground state. So this is both qubits in the ground state S and your harmonic oscillator both in state N and then you can excite using two detuned light fields, this red one and this blue one, going to this intermediate state where one ion is excited in the D state, the other is still in the S or one is in the S, the other is in the D state and those are the harmonic oscillator quantum numbers and then so you can excite this way or you can excite this way or this way and this way. So you have four interfering excitation paths and the interesting thing about this gate is that the excitation of your harmonic oscillator basically drops out of the equation. So the nice thing about this Mermer-Sørensen gate is that you're not have to cool your vibrational motion to the ground state. So that's what we always assumed for the Zirak Zoller gate so this is the advantage here. Okay and so now the basic operations that people in Innsbruck have is a global laser beam that addresses all ions and does global rotations of all ions. So that's exactly what we had on the blackboard yesterday. The second ingredient is okay, so this is the operator S5 and this is a global rotation. So this is the time evolution operator using this Hamiltonian here and then the second operation is a local phase shift. So you locally talk to only one ion and this is a detuned light field so all it does it shifts the energy levels and thereby induces a phase and that was a question yesterday how can we do set rotations? That's one way of doing set rotations. So you simply change your level splitting by the AC start shift. Changing the level splitting means you change the precession frequency of your spin relative to what to your lab frame rotation and this induces a phase. Okay so you can do global rotations of all ions then you can do individual phase rotations and then you do these murmur serons and gates so here you see an S squared so it's the interaction between ion I and I and J so you get all kinds of cross terms so it's an interaction term. Okay so I'm not going to go into more details here so but I will explain to you the spin-spin interaction later on. So just as an example of state-of-the-art experiments so that's what they did. Now they recently they implemented this short algorithm so they factored the number 15 which is not easy in the lab with a quantum computer so and let me quickly show you outline this algorithm so that you can understand what this experimental result means so if you want to factor number N so I just give you a rough outline of the whole algorithm so what you first do you choose some base A which is some number between two and N minus one and so you choose this randomly. So random choice then the second step is you calculate the greatest common divisor between A and N and if this is not equal to one then you already found one factor of N but so the question is is it equal to one and in a typical case it is equal to one so you did not find a factor already by this random choice and then the third step is and so this is all classical stuff so far and now the third step is you calculate A to the X modulo N with X being a natural number of zero and then you find the period of this function in X so you find the first, well I express it now in a not exact mathematical language find the first X large or equal to zero such that such that A to the X modulus N is equal to one again okay and then and this is then the period of so you calculate this then you find the sequence of numbers and they repeat each other with a certain period and this period, so this first X gives you this period R of this function here and then you calculate, yeah so and that's basically it so that's the and then you have more classical calculation steps and so the real quantum thing is happening here so that's the where we need a quantum computer this calculating this and find the period so these two things that's what we need a quantum computer for all the rest is classical computation okay so the number of qubits you need is to factor N, if N is a N bit number then you need for the usual algorithm you need three N qubits so this and each qubit is a very valuable resource so it's when you calculate this go to the next larger number then it's becoming difficult experimentally because of this scaling which is linear but still challenging so yep but let me since you asked this question let me point out actually what they did in this experiment so first let's look at the experimental result now so what we see here is exactly so this base two, base seven, base eight that's this random number they chose so A is the space here and then they look what their task is to find this period of the function so they look at the output state of after they did the QFT the quantum Fourier transform so to find the period you do a quantum Fourier transform of this function and then they look at this output state and you see this has period two period two, period two, period four, period two so that's the output at this step here and then the rest is like I said classical computation again and now coming back to your question so they did not implement this thing here this is the classical Shaw's algorithm this is a variation proposed by Kitayev where you take advantage of the fact that for this algorithm your QFT doesn't have to be coherent you only care about the population in the end so you only measure the population but not the coherences so this is particular to the Shaw's algorithm when other algorithms that incorporate the QFT they rely on a coherent QFT so that you really keep track of the faces between your qubits here you don't have to keep track of the faces and that's why they could simplify this so instead of doing the QFT on two end qubits they do a QFT on one qubit two end times so they do what amounts to a QFT which is just basically two single qubit gates two conditional, well it's conditional gates in the end but you just do these two gates and then you do a measurement and then you do gates again and so on so this is a simplification that allowed them to reduce the number of qubits and now you're asking about a perspective that sounds easy doesn't it in the lab it's not always easy I mean for such an algorithm you need complete coherent control at any point in time and space and this is difficult with each additional qubit but a way out is thinking of clever ways of replacing algorithms by experimentally simpler algorithms and so I would guess that the prospects to extend this to more qubits is pretty good okay so the question of scalability that's a whole lecture in itself I would say so okay now I'm a little bit behind in time but we saw this electric quadrupole transition and now let's look at a different thing it's using hyperfine states and hyperfine states that's a very smart choice because they live forever basically on any timescale that you have in the lab typical hyperfine states simply do not decay so they keep their commitments forever and there are several ions that you can use for these hyperfine states but then you use Raman transitions typically and for these Raman transitions you have an effective Rabi frequency which is proportional to the Rabi frequency of this beam times the Rabi frequency of this beam divided by the detuning and then you have to take care that your that this condition is fulfilled for the wave vectors of the two beams so that you can impart momentum on the ion and this has been done for instance in the group of David Weinland for the first time and now if you think about how well you need to control your laser fields you get rid of this requirement basically because you only care about the relative frequency difference between those two light beams and that's much easier to stabilize than the absolute frequency but you still are in trouble with all these other problems so this is also quite a challenge to do quantum algorithms using such a scheme here but this has also been very successful for a recent example from the Monroe group where they implemented a five qubit programmable trapped ion computer so they have these five ions sitting here and then they have the new thing here actually it's a crystal optic modulator that allows you to focus your light beam onto individual ions in an arbitrary way and then you also collect fluorescence using a multi-channel PMT so that's the two new elements here in this experiment and using this they also implemented a QFT with five qubits so that's the classical QFT a coherent QFT where you do all these C0 gates here or not necessarily C0 but conditional quantum gates between two qubits and then at the end you measure them all okay so that's a very recent example and now I want to finish this part by looking at a slide that has been prepared by Dave Weinand so this is a picture of the lab at some point I'm not sure from which year this picture dates but so that's where some of these great experiments were done and you see lots of laser stuff and lots of optical elements so some of you have probably seen such a lab who is an experimentalist here by the way okay so we are the minority here clearly so you probably have seen such a lab in atomic physics in cold atoms also they look on first sight on zero-order approximation they look very similar so you see lots of laser light optics and stuff and so this is a pretty huge table with all this stuff and then you see this iron trap apparatus far back there so it's a large effort to make these experiments work and these groups that I showed you here they mastered these challenges but when you think about scaling that's your question then it's kind of difficult to imagine how this thing can be scaled from two five qubits to two thousand, fifty thousand so that's probably not scalable in this way okay so yeah but you can never say never of course but it looks like an even more formidable challenge okay so and that's where this new concept comes into play this magnetic gradient induced coupling that I want to show you now so that's I introduced this at the beginning this nicely abbreviates to magic and so I'd like to show you what this magic stuff is and allows us to get rid of all of the lab basically that you saw, not all of the lab I'm exaggerating but all of the laser stuff that you saw on this not all of the laser stuff but 90% okay so the important thing is we want to get rid of our lasers and what I showed you is you need lasers because of the diffraction limit because you want to address individual ions and because of the coupling between ions and emotional states so you need this lamb diggy parameter to be different from zero if you go now, if you drive a hyperfine state directly which has a frequency splitting of let's say a gigahertz or ten gigahertz or whatever then the momentum okay so we calculate this for optical radiation what we end up with a typical value is ten to the minus one to ten to the minus two you don't want to have it too large because then your expansion doesn't work anymore you don't want to have it too small because nothing happens anymore so this is in the optical regime if you look now at a microwave transition here a radio frequency transition for instance in the uterbium you end up with a value of ten to the minus seven basically so this is for all practical purposes this is zero so this is RF so RF means anything from kilohertz to gigahertz okay so that's why people did not use microwaves for this purpose here but you can use them actually if you modify your trap a bit and this is shown here so we start with our two states here and we have our two harmonic oscillator potential so spin down, spin upstate position coordinate and now we apply an additional field for instance a magnetic field gradient that changes its strength along the set axis so if we have magnetically field sensitive states this one might be have this dependence as a function of position and this one has this dependence and now if you add your harmonic oscillator what you notice is so let's, we start in the ground state we excite now the ion and so, yeah actually that's the wrong sequence of my slides so what I want to show you here first is so you have these spatially dependent states and now you add your harmonic oscillator potential and a linear potential a constant force plus a harmonic oscillator simply gives you again a harmonic oscillator with shifted equilibrium position so what you see here you have this equilibrium position and you simply slide down this potential hill slightly in the upper state so now you have displaced equilibrium positions of your harmonic oscillators so when you do this excitation you see that you start at the equilibrium position but you end up at a displaced position so your ion will start to oscillate around this new equilibrium position and you did not impart any momentum to your ion so I'm not talking about any momentum here I'm just flipping the spin and I have this state dependent potential this starts to oscillate around the new equilibrium position and then you can show that this if you go through the mathematics you find you can describe this coupling by an effective lambda-dickey parameter which is given by the shift of this equilibrium position in units of the extension of your ground state wave function so that's the set here okay and now I look at this phase-based picture so the same thing again in phase-based so you have your spin one-half with these position dependent levels and you have your harmonic oscillator and what you do now is you have your radiation that might carry some momentum but doesn't have to so you displace your harmonic oscillator along this coordinate along the momentum coordinate but what you certainly do in this case you also displace your harmonic oscillator along the set direction and then it will start to oscillate around this new equilibrium position and so and this equilibrium this is the shift of your equilibrium position so that's a very simple expression here so you basically you have your harmonic oscillator potential is given by one-half mu squared d squared where d is the extension from the equilibrium position and then your force is given by minus the gradient of v of d which is just given by m times mu squared times d so and this force is equal to the force on the spin which is so this the force on the spin is also minus the gradient of h bar omega h bar omega is your splitting between the two energy levels and this is now also spatially dependent so I'm just looking at the one-dimensional case now so the gradient just amounts to a derivative with respect to d in this case along this direction which is called set here so you just have h bar and d d set of omega and when these two forces are equal again then you find a new equilibrium position so the equilibrium position is just shifted by this amount here and yeah and then you can excite your harmonic oscillator simply by flipping the spin and you don't need momentum anymore okay and yeah if you go through the mathematics that I'm not doing here you can find this in these papers for instance then you see that you find a new effective lambda parameter which is called eta prime here which is the usual lambda parameter that makes this displacement along the momentum coordinate and then this kappa thing here which makes describes the displacement along the position coordinate and this kappa is this displacement in units of your harmonic oscillator ground state wave function is extension and then you end up with this Hamiltonian and this looks now very similar to what we had before so for instance let's take the limiting case where the usual lambda k parameter is 0 then this red thing is simply 0 and then you just have a face factor here this I and this minus sign and then you end up with the usual Hamiltonian that you had before except that you have no momentum transferred at all so and this allows you now to do the same thing how much time? finished? okay so let me finish by giving you a physical picture of what we did here oh yeah so before we had our harmonic oscillator and we use a laser beam to excite it to give it a kick and now we use a different physical mechanism we have the harmonic oscillator we don't give it a kick what we do we displace the equilibrium position and it also moves so if you look at a pendulum that has been described suitably for a few hundred years now this seems rather straightforward but to realize this in the context of trapped ions that you don't need necessarily laser light was not completely obvious okay so let me finish here for today and then I think there's one more lecture right? okay thank you for your attention