 Okay, so I would also Like to thank the organizers for inviting me here giving me the opportunity to present my work Actually, it's a spin qubit in semiconducting another structure But I will have some superconductors in my talk. So I thought actually you anticipated that also that I was supposed originally to teach in the School last week and I couldn't come I had to go to San Sebastian and shift my talk now here and so I have a little bit of a hybrid which is Part of this school lecture, which I wanted to give but I have only half of the time. So I will speed up I'm not sure I will get through with my material and Here is a quick outline So my my goal is just to give you a little bit of an impression what the spin qubits are Because I looking into the program. I saw there's not too much on that topic And it's quite of a big effort in the field on building a quantum computer so I introduce briefly what the spin QBD is in quantum dots and Then I go on to a few Specific topics and I will show a lot of experimental progress in the field Okay, so let me start with a table here which summarizes What gives you an overview a little bit where the field stands the field meaning building or aiming at building a quantum computer Which in the end these days needs to have about a billion qubits 10 to 8 10 to 9 qubits and that is believed to be the kind of numbers Which we need in order to have a scalable machine and there are several approaches They are superconducting qubits probably it's fair to say that they are the most advanced ones Trapped irons. They're also a pretty advanced and then cotton dots different materials here is a silicon cotton dot listed and the new Runners here are topological qubits also listed and maybe diamond vacancies and you see they are kind of numbers characterizing the systems and a very important number in the field is this longevity basically the coherence how long can you keep a superposition of an up and the down spin in its coherent state and The longest one you fit finding trapped irons, but then you have to pay a price that the system is slow So every kind of proposal here has pros and cons Maybe the biggest pro for the silicon quantum dot structure is that it's very small and very fast and that it Might at some point be integrated into CMOS technology and this has actually been shown in recent years since one or two years there's kind of a second I would say Revolution in the field because it was possible to use CMOS technology to define a spin qubit And that has raised a lot of interest from industry These numbers they also they get outdated very quickly for example Just the fidelity has now also been proved here in some structures in topological qubits We don't even know whether we have a qubit or not let alone whether we have actually knowledge about the coherence diamond so forth But it might come at some point Okay here a historical remark the spin of the electron is a very obvious candidate for a qubit In vacuum, but if you put it into a solid state Then there are of course many other items and interactions which will then change the properties of the spin and so it was kind of a long effort to get isolated like spins in quantum dots and The timescales have changed almost with Moore's laws over the beginning of the field from nanoseconds So actually even below nanoseconds up to seconds now these days and the spin degree of freedom has a much better Property than the charge degree of freedom and that's a kind of a natural choice In this field so the idea goes back to our work here almost 20 years by now and The idea is that basically everything is now called the spin tronics scheme You use electric fields to control the spin and not magnetic fields because electric fields can be switched locally and very fast and Here the scheme is that you can do a read-out of the magnetic moment by Electric field spin charge conversion. I'll show this later than the exchange coupling which generates entanglement can be done electrically by tuning gates and The spin rotation can also be done by tuning gates Very fast and then you can assemble hopefully a fundamental gate like the exclusive or gate Which is a fundamental gate to build up? Quantum algorithms for two qubits and single qubits in addition So just to show this work is still Being considered by people a serious front runner and I think it's for one of the highest papers in cotton computing Now what is the key idea? Let me go through steps here and then show where the experiments come again the key idea is all electrical control and That we can scale this in some way So here is a for example a structure of a hetero structure where you can find electrons to a two-dimensional plane and With gates you can find them further into so-called Quattendots the size of about hundred nanometers that can be smaller ten nanometers in silicon these days and You control them also the overlap of wave functions between quattendots with electrostatic gates and the idea is to end up then with a Hamiltonian which couples this left and right spin in this way here and local seaman fields so you can also think if you if you are not very familiar with exchange coupling, but you might be familiar with the chemistry of an artificial hydrogen molecule if you take two hydrogen atoms together you form a molecule and What happens then is that in the ground state you get a splitting of the ground state property and the lower state is a singlet Two electrons spins in a singlet, but they are not sitting on top of each other So this is the difference between a helium atom where you also have a singlet in the ground state But the orbital wave functions are basically Indistinguishable here in this structure They are separated but still in a singlet and the first excited state is a triplet and that defines the coupling between the spins That's called exchange coupling and the coupling can be controlled Not by controlling the Coulomb charging energy, but by controlling the barrier in between this barrier height is controlled And by gates and that leads to an exchange which is In the simplest approximation of t squared over u t is the hopping back and forth and you the Coulomb charging energy now you don't only need to control dynamically the interaction between the spin but also You have to care about the so-called decoherence how long a state can actually be maintained and that turns out to be a very very rich Subject by itself. There are literally thousands of papers on this question and The simplest description is an exponential decay which you get by a Markovian master equation and in those systems most Often this action not appropriate you get deviations from our Corvian because there's memory in the environment in particular if there are nuclear spins Which are unavoidable in some materials and then you get deviations from that So let's look now at one specific example, which I actually like to show which distinguishes this system quite Distinctly from let's say superconducting qubits, which are a much bigger system in this system We have a lot of information about the microscopics. We know the band structure We know many many material parameters This allows us to calculate to predict very precisely what the influences of the environment is and compare this with experiment And find strategy to improve these kind of scales So just as a reminder There are two times which are important one is called the relaxation time if a two-level system Let's say you spin down can decay into a spin up and that Takes a certain amount of time. There is a decoherence time when you start from a superposition of up and down and you ask how long does it take to decay into the ground state and In a in a solid state in a semiconductor environment the most important source for that to happen is coming from spin orbit interaction It's a relativistic correction very strongly enhanced via atomistic lattice structure in the system and Lattice vibrations phonons coupled then to the spin wire spin orbit interaction So let's look now at the simple setup here. So a general spin Hamiltonian Describing this process can be described in as is written here where the second part is a fluctuating field and this fluctuating field We want to determine it's determined by intrinsic fluctuations and Basically, there are two contributions one which Contribute to the relaxation process and these are the fluctuations perpendicular to the quantization axis So if you have fluctuations a perpendicular in two directions x and y you pull down your spin that's giving you a decay rate and that's written down here and Is the value is given frequency given by the same on splitting and then there's the second part Which is called the decoherence party to part which has also a t1 part in there because we start from a superposition And that basically is half of the state as if I would be up And that's accounting then for this factor of two But in addition there is also fluctuation along the field and that leads to a pure Defacing and that's the fluctuator which describes that along the set direction, which is the quantization axis Okay, so now if you go into a quantum dot and model it Then we have a lot of knowledge about microscopic quantities like the spin orbit interaction So here's the simplest one, which is called raspa spin orbit interaction And the dressel house spin orbit raspa is coming just from the gates you apply and then you break the inversion symmetry Dressel house is coming from intrinsic Shifts if you have two atoms in the center then they cause an electric field and break inversion symmetry of your crystal Like in gallium arsenide and then that's referred to as this one and the quantum dot is the model Let's say with a parabolic confinement in two dimension the third dimension is just quantized And we have the same unsplitting we have spin orbit here And then we have a coupling between the electron degree of freedom the charge and the phone all phonons are lattice vibrations And you couple to that that my electron phonon interaction. This can be a complicated beast Typically it will have so-called deformation potential. So if you just push your crystal You get some deviations from that and it starts to have harmonic oscillations but they are more complicated ones, which are called piezo electric ones, especially in gallium arsenide and They need to be taken into account. I will not show you the details So there are then certain parameter regimes, which are also Defining then a weak coupling regime such that we can do controlled perturbation expansion And then we do an effective Hamiltonian derivation. That's a Schriefer-Wolff transformation usually Which leads then to an explicit expression of this fluctuating contribution and Here it's written down. It's given in terms of the external field plus some internal field, which is then produced by spin phonon or spin orbit interaction and electron phonon interaction and the interesting Aspects from this here is that there is no fluctuation along the field because this fluctuation has to be perpendicular to the field according to this construction and that's basically the Upshot is that you have no pure defacing and the t2 time can be even longer than the t1 time in these systems So just to flash you How an expression will look like in the end which you have to compare them with experiment There are many many material parameters, which you have measured in different experiments. Then you calculate those times also the Power law dependence on the magnetic field, which is applied So for example here is the t1 rate or the inverse and as a function of magnetic field And you see here in the log scale that it has a power law behavior Which is given by the fifth power and this for example has them been observed many years ago almost ten years ago at MIT and You see down to the quite low magnetic field you get a B to the five power and all the numbers times they agree actually quite well, and it was only up to very recent Actually, this is a preprint Going out now very soon that people have come to even lower magnetic fields Which is difficult to measure because the splitting gets smaller than and then you have to fight against Temperature effects and the cooling becomes very important. So these are experiments at the temperature of about 60 milli Kelvin and The spin one half in a gallium arsenide quantum dot has now a record time of one minute And this is about eight or nine orders of magnitude longer than it was let's say 10 15 years ago Just to tell you a little bit how the progress is in silicon It's about 30 half of it 30 seconds So which was recently measured by Michelle Simmons in Australia different structures there's also a very strong angular dependence in which direction you apply the magnetic field and in which direction you align the quantum dots in your structure and Knowing that you can make predictions then how it depends on the angle tether out of plane and of the field in plane with phi and These two spin orbit interactions. They can actually interfere and if you choose a certain field direction you get in at least in leading order no effects from these sources and That leads them to predictions like this curves here when you change the direction of the magnetic field that the t1 time oscillates That's a experimental in our confirmed also in this recent work By some will group in Basel Now there are more sources which act on the t2 which are not described by this mechanism These are nucleus spins and that's a very very long story again many many papers have been written on that and This can be also filtered out to some extent and here is a recent Filtering experiment by the Copenhagen group Charlie Marcos and Ferdinand Krimit and here Basically, they obtained the record time of t2 now being 800 or 900 microseconds reaching about the millisecond Which is sufficient? Given the speed of switching which we have so how do we get? switching ideas In the system or estimates so the entanglement as you all know is crucial for quantum computing For example, you start with a product state you want to end up is an entangled state You can do this with a number of these exchange coupling Effects so you can turn on then this exchange coupling and you go then from a product state into an entangled state Can estimate the time scales and so forth and this is an experiment From 2005 which demonstrated for the first time that this entanglement can be generated as a function of This coupling is a double dot Here's the picture and then the coupling of the exchange oscillates here and that gives you a timescale of about 200 picroseconds And that shows how fast the system can be so now we are talking about a few hundred picroseconds Compared to the coherence times which are hundreds of microseconds So this gives you a large window to do many many operations which is needed then for fidelity or for the threshold of the surface code Okay, one can do two qubit also single qubit single qubit There is a list of methods used in the field standard ESR as you know from textbook But in a solid state you have more knobs you can shake The electron and because of the spin orbit the spin will then also rotate when you shake the electron and that Leads them to a number of so-called electric dipole induced spin resonances And they are also very very fast much faster compared than the conventional ESR techniques so there's an entire collection of quantum dots all over the planets in gallium arsenate and Where as I said, you know these record times now have been reached Probably the best structure at the moment in gallium arsenide is the one by Tarucha in Tokyo Where they have demonstrated the full two qubit operation a single qubit operation switching times on the scale of megahertz entanglement on the scale of gigahertz and This year is now actually Not very fast. We hope that this will get improved this was up to two three years ago basically the driving field in gallium arsenide and Now more recently people basically all groups are working now also on silicon and the silicon has the great advantage that you can get rid of the nuclear spades and that I Will show now in a few moments So the spin qubits from electrons the simplest one is a spin one half but in a solid state you can think of many many more qubits I have here incomplete list a Favorite one are the singlet triplet qubits where use the state of a singlet and one of the triplets as a logical zero and a logical one It has some advantages in terms of control you can even take free spins and Form logical qubits out of it as shown here So a logical zero is a singlet two electrons plus the third one with the spin up and the logical one You can post them out of the triplets and the third one up and down and so forth there are many variations on the same here and a new Rather new now for experimental from an experimental point of view are holds spins Not only electrons can be used but also holds can be used and holds have very nice interesting features Which I will discuss briefly in a moment So this is basically from one to many qubits which we need to go just to show you a few structures It's possible now to get the linear arrays with quantum dots. That's not so difficult anymore here is a published data by Thunder Sightman's group in Delft Where they have full control over four quantum dots with the assigned gates here Then the most advanced one is in Princeton by Jason Petter where they have about 12 quantum dots which they can fully control and Basically having as a partial scale up in one dimension this type of a scale up is not sufficient we need a two-dimensional array and That but at least for some near-term applications You can think of getting 100 qubits with such a scaling but not let's say you cannot go up to a billion as I described before So people are coming up now with a lot of architectural ideas in particular Churaks group in Sydney is one of the silicon players in this area So there are many candidate materials for spin qubits so far. I showed you mostly gallium arsenide you can go in this so-called To five materials indium arsenide indium phosphide and so forth, but probably The better choices now are the nucleus been free systems or at least you can get them nucleus been free and this is silicon Silicon or silicon germanium combinations of these two Have become now quite the focus in the field And in those systems, it's also actually easier to get holes Compared to gallium arsenide. So what is what is a hole hole? It's not only a missing electron. It has a different Origin in the band structure. So here a flash a band structure in a typical semiconductor and that is what we call a Parabolic Dispersion as a conduction band. They're higher conduction bands. So typically in gallium arsenide. We take this one But then there is a bank gap and below there are Bands which have opposite curvature and these are the whole bands or valence bands and the big difference between these two bands is that the atomistic contribution from in the valence band is coming from p-wave orbitals oh Okay, and up here as wave orbitals and these p-wave orbitals. They have many good properties You can couple better with electric fields to them So this has been looked at Early on and there are many experiments. Given the time I have to switch a little bit. Maybe I jump in my talk here I wanted to show you This year. So this is a quite recent experiment in done in Grenoble and what these people did Mark Sanke and Silvana Franceschi Basically, they took a CMOS structure off the shelf just a transistor as you find in this chip and Rewired it a little bit and could redefine the structure in terms of qubits and do qubit type of measurements and This is a so-called the CMOS silicon structure. That means it's industry compatible It's basically what the industry is used to do like Intel and many many other chip producers And that has Attracted a lot of attention as you can imagine because now for the industry It's not such a far step anymore to think of building spin qubits in their structures to have already and in this structure They managed to get the ten holes per dot about and control these holes to spin read out and spin manipulation So here for example is a spin flip they can control it in two directions and the scale is rather fast here By now now, it's even larger on the order of 150 megahertz And so this is a very important development now if you want to go further then you have to think of very large and dense structures With such a spin qubits and if you make an estimate how big a size you would have for silicon or for for Spin qubits in silicon or in germanium or in gallium arsenide Then you have to think of about ten to eight ten to nine qubits and For this it's known that if you have a gigahertz clock speed then you can factor a 2000 bit number in the RSA key. So this is one of the kind of holy grails for a quantum computer in about a day and The speed of course is important, you know, if you just go down by a factor of thousand to megahertz then of course It's a thousand days. So these kind of scales are there in the end and also important and with the size here Let the size of a micrometer You can place this ten to eight spin qubits on a square centimeter That's about the size of a chip in your computer, which also contains a billion transistors I mean just to give you a little bit the relations actually, you know it from a from a Intel manufacturer point of view This is no longer such a dramatic Quest here because we are not talking about more integration, but what is more challenging is that you need to address more Refined for spin qubits or for qubits in channel compared to a transistor and that has now led to a lot of Activities to figure out how to get all the wires into the system and you need to go out Definitely from the two-dimensional into the third dimension. You also need to have more spacing between the qubits such that wires can be brought in there and this also has triggered many Kind of ideas how to make an extension of this exchange coupling. I showed you in the beginning. It was nearest neighbor They had to be rather close. Can you make this a little bit further away? Can do this with cavities? That's where superconductors start to play a role with floating gates or with edge states in quantum hole edge states so I think I Done some nice work here and that allows you to give you more room for Building up just to give you some number if you use cavities standard cavities like introduced by the Yale group and very much developed also by Valerov then you can couple such hole spins to cavity electric fields and Such cavity electric fields. Let me switch that take allow them to Control the spin qubit on a rather fast scale and you can turn it on and off How would you scale then up such a system and that's something which is the last few minutes of my talk? So here we looked at an array of a superconducting cavities So there is one cavity line here and another cavity line here another cavity line here and they form plaquettes and The cavity lines themselves. They are coupled to each other capacitively So this is well known has been demonstrated experimentally that this can be done Now we have here at these points at these points at the blue and red ones. We have let's say such a silicon or silicon germanium spin qubit line here and They form then the X and Y or X and Z qubits in the surface code And here's a typical Hamiltonian with which I would describe then the system So I have then a coupling of the electric field to the spin of my qubit Just the cavity photon and then the cavity photons of neighboring cavities is coupled by this capacitive Effective exchange coupling. So here's a little bit of a blow-up of this Let's say if I zoom in into such a qubit Then I envisage a structure like this where I have these Gates here and in between these gates. There is a wire and Here electric fields allow me to control then the spin to turn on and off the coupling to the spin and the interesting feature of these devices and of these qubits is that The electric fields they always couple you cannot really turn them on and off But you can go on and off the resonance point and that you can do very efficiently and very locally That allows them also to shrink the system to a rather small size and if you go then through the Calculation integrating out the cavity modes you get an effective coupling between the spins Let's say between this spin and this spin so you can do the nearest neighbor coupling of all of them all nearest neighbors can be coupled and Just by turning on and off the coupling locally of the spin Basically making it resonant to the cavity and then Two of them are resonant and they talk to each other Okay, good. So this gives you a little bit of a outlook. So this is also where many experiments These days now are working. I mean just this one here that is now Worked on in many labs Which work on superconducting devices and try now to combine Spin qubits in there in a hybrid structure Okay, so I hope I gave you a little bit of a Impression of the field and with this I would like to end and thank you very much for attached