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Realization of ThreeQubit Quantum Error Correction with Superconducting Circuits
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there's right you that that has a particularly right hand in repeat after me I solemnly affirm iPhone affirm that I will faithfully correct that I will faithfully correct all quantum America's all on the real or imagined real or imagined and will to the best of my ability and will to the best of my ability preserve protect and defend to preserve protect and defend the quantum computer the quantum computer congratulations thank you the authorities a kind of a reward and next speaker will be mad read 3 cubic quantum errorcorrection with superconducting qubits odds circus I think you very much so as I mentioned I'm I'm Maryam McGrath student in rather lab at at university anytime you of our recent results are demonstrating the 3 key code in SuperMac circuit so the general outline of my talk I thought I introduce us to bring concubitus generally because this is maybe not not in super familiar with them and talk about how we can do to Cuba gates in 2 very different ways wolf at 380 back interaction some interactions but we can use these these gates to make interesting states like GHC states of ends as I'm sure everyone knows that due states range thing because you can base 3 cubicle other than that will talk about how we can implement efficiency top of efficient toughly date to actually implement the bed and Facebook errorcorrection code and then apply to give a little bit of an outlook about where see superconducting qubits going the next couple years so I know a lot of different kinds of superconducting qubits the kind that we use at yale are called trans bonds which is sort of those maximally simple version of a superconducting qubit but in fact it's even just think of it as an LC oscillator of its own subtle harmonic thing good qubit so we add a nonlinear element that just injunction to make inductor nonlinear and so you have a Yemeni excited states but the the the the anharmonicity between the transition between the 0 and 1st and the and the 2nd of 1st and 2nd stage so we can use that the grass in the 1st excited state of documents he is a picture 1 it's kind
02:13
of traitor microns long so that's that's just barely big enough to see with your naked eye you know where to look and this sort of Bullwinkle structure this interdigitated structure is the is the capacities and resuming here we can see that this is the inductor
02:26
we have a little display just injunction so that you sort of thinkers as 1 junction sort of distributed so that we can thread flux through it so we can change the frequency of that with the applied magnetic field so we use this we use these trends ones
02:41
in what we call seek it's not cavity Quantum electrodynamics with country fruit of where we have a very high Q mirrors and we should Adamson them but instead we we use what's called circuit water
02:52
dynamics of which is sort of in analogy to this idea where we actually have a a microwave resonator that pattern on a 2 dimensional where you can have a standing wave of mode between 2 of breaks in a wire which can which a sort of analogous to Munoz and then we can stick some cubits and urine couple to the electric field this cavity that does many users many functions in our experiment all few here I 1 of them is just not protected from spontaneous emission from the Purcell effect what if we didn't have filter out the modes that the that the killer could see that the lifetime of our qubit to be on the order of tens of nanoseconds and if we do the engine and properly with our cavity than the result that isn't a problem along with there's there's there's no personal but we can put many qubits in a single in a single resonator and so we can multiplex the single qubit drive so we can have a single PortaDrive manipulates using frequency multiplexing which is very nice so we can through this sort of the the 2nd interact interaction of the qubit to the cavity in the cavity to keep it we can do to keep the gates which all all talk about in a moment and we need that read out through through the through the cavity by just basically measuring transmission to the cavity inferring the city that not a lot of that theories the audience public in trouble fight and have a least 1 equation so this is the Hamiltonian of our system was still the genes Cummings Hamiltonian it's a sort of you that you have a harmonic oscillator a spin and then coupling between them Molise exchange of interactions excitation so is that
04:24
helpful and and and we can see a way how we can actually keep it read with this with this system but if we were working what's called the dispersive limit on where the frequency transition between of the qubit in the resonator very different from each other we can rewrite this Hamiltonian approximately where we no longer have his interaction term but instead of others an effective us sort the dielectric that the cavity sees where the the frequency of a cavity depends on the state of the qubit and so that's on 1 day so if we allow this experimental data showing the transmission to the cavity when qubits in the ground state and if I were to do apply pulse and this experiment or something like this where we see that the primary transmission has has shifted quite frequency there's still some missiles and transmission here when we created the excited state just because if you had a chance to Cadence measurements so imagine that we send a tone of this frequency and we get a relatively high transmission we can infer that the qubits an excited state and now if we get relatively low transmission the casting I said the backwards and it turns out that this is actually very nice this is this is also the rotting for 1 qubit but the foot many cubits in single cavity the measurement operator becomes something very useful which is it answers the question oral arguments in the ground state and it turns out that this this measurement operator contains many basically all the power correlations the products of Eisen's he's so you can you state tomography very efficiently so by saying talking about that is what it looks like this it has 4 friends like it's coupled to a single resonator so you can can barely see that there's a break in this in the center would we will we wire here and here which are the mirrors that the coupling capacitors of our a resonator and then we have 3 qubits actually this picture the qubits what pattern yet I swear there they're actually we have at 678 figures and the a 4th 1 of the high frequency and which we don't use science experiment and each each qubit has these guys musical flux by signs which allow us to send send a voltage that down to 2 very close to the qubit and then run a current which change the magnetic field team but give it and we can use that to you it's very rapidly in fact in a nanosecond we can move them by many gigahertz
06:37
and this is actually a we do to Cuba case so the the kind candidates that we that we use in our system phase gates in the sense that they never cause changes in the excitation of this system but rather they because of various phase evolutions for the for the different computational states also in this notation the ground state will go to itself but the excited state of the city's top is perhaps a 9 gigahertz Argus itself but phase I had we get this phase is actually very simple if we if we imagine that we move this cube down for some period of time and then backed up that this phase will be basically the integrated the tuning is a function of time relative to this phase reference which is unlike with generator in the lab all you can imagine that the other qubit we would have the exact same the exact analogous phase but if we talk about the the 1 1 state it insufferable for these faces because it has it has a both for these kids excited but it'll have a special phase called that we call the 2 qubit phase which is associated entanglement and how we had because of something and it seems like a rather exotic thing will it relies on the fact that least in around mutation that are cubic isn't really Cubitt but it's really ammonia LC oscillator so it has a higher excited states so for if we do this experiment were removing this this talk it down in frequency the 1 1 state will go down the same slope as this 1 but this year to say what the slope and though being avoided crossing between the 2 so if we imagine if we imagine that going down the line idiomatically you'll see that the slope of our lines can change relative to what it would have been if this were a crossing work here and will have this parameter zeta which is associated with with the rate that were required to give face and if we make this to qubit phase if we if we do our excursion properly to give phase will give us a 2 pi i which will which we can then use to make a conditional phase gate where all the states except for the 1 1 state map to itself but the 1 once it maps to minus itself
08:33
so that's 1 way of doing it that's the 80 that equation I talk to you about of sort of of a similar idea but it's using the maximally different or approach which is we're going to use this a similar bordercrossing in there somewhere so if we move 1 1 state into resonance as quickly as they can to 0 2 then we will no longer be in an eigenstate it's as were of the the wave function having having had a chance to evolve instead we may be of superposition of the symmetric and antisymmetric combinations of the border crossing since these 2 since these 2 states don't have the same energy they're gonna acquire phase relative to each other and if we wait the exact grammar time meaning of full oscillation 2 0 to win back to 1 1 were have minus and so this is exactly could 1 way of making a sophisticated turned out to be about 3 times faster than and you some data characterizing supported but so what we do is we move in for some period of time to some of frequency location of basically you can think of this axis is applied magnet magnetic flux and we oscillate between the bright state is is 0 2 in the dark is 1 1 serine oscillate between these 2 states and if we wait until we do full oscillation come back to original state this is exactly want where 1 of you have a C phase gate and that happens in about 12 minutes I was like a point at this point that waiting 12 nanoseconds waited 6 nanoseconds we can very efficiently transfer from 1 1 in 2 0 2 and that's an important later on so that these he states the
10:06
interesting things I mentioned before that the sample but our talk about has 3 qubits so this is so this blue qubits at the gigahertz 7 and 6 so it's say we can make a bell say between the the top 2 top 2 frequencies also this but the 2 who to keep on the equator doing the phase gates so with this notation just means which phase which state requires our minus 1 face and then do it do I do about a final rotation induced state tomography and I can explain how would you stick to on the basic relies on the fact that this measurement operators this nice thing that contains all these 3 key the correlations so here's an experimental data this is a basically a nice way of visualizing the 3 qubit density matrix where we plot that the poly operators for all that all the different poly operators and scarcer 63 of them and so this is sort of a single qubit about how we operate operators it this the 1st qubit the 2nd year and 3rd year and we have to keep the correlations in 3 cubic correlations and we can really see we reactions read off the fact that this this graph this cube at this sort of this right you but isn't entangle with anything because it has an individual character it knows what directions according but these 2 cubits Ponting with each other they don't know where the pointing but if you ask them in a very particular way last not to Cuba correlations it does have an an answer answer so you can you can immediately read off from this plot that we have a bill stating in 2 qubits and we can also talk about the fidelity of this operation which is sort of 94 95 per cent we can we can call the logical extension of this a maker of 3 Cuban single status by involving this last year by the DAC phase on it as well and it was college it's the state where it's the maximum tendency of 3 qubits either they're all the ground state all the excited state and then we can we can visualize the density matrix see that all 3 qubits are entangled they don't have any particular characteristic them so strong 2 and 3 could correlations and again we can talk about the fidelity that's to induct product of this and we get about 90 per cent so RGC states as I'm sure everyone knows a very interesting because they have this property that all the AtoZ all the 2 keep easy correlations have the I plus 1 but in fact is a more general version of this alpha 0 0 0 plus B 1 1 1 also has a property importantly it doesn't by by measuring the easy correlations you don't learn anything about alpha and beta this is obviously a stabilizer states so if we if we
12:29
encode some information in this in this in this form and then we allow a single a single 1 over Q it's to be flipped will will have of if we measure the zz correlations we will have a unique our mapping of what areas happened and so this is the basis of the idea behind the bit flip code
12:46
as I'm sure everyone of us all to through this very quickly but this is actually the measurement free version and were so are gonna be dis encoding that so this is not anywhere close to being faulttolerant but it's something we can do and so if we if we have some state alpha 0 plus 1 that we can do is encoding I talk about where we can create this this nice state then say we allowed bit flip happen to happens milk you will a stable of all this if we dis encoded also decode the information on now the state is has suffered a bit flip but and so what cubits and 1 1 which indicates the milk milk you it has been flipped and if we if we do this last bit which is of a controlled controlled not basically it off with the middle if and only if this cubic and this qubit are excited but in in this case both qubit are excited so it'll flipped then we'll get our state back still will be in some state which I we can reset through some process but this is actually a this is actually tall order this day right here is actually very difficult but told topple your control control not gay and if you were to build out of 2 fewer gates you can indeed 5 of them which is a lot so is there a better way to do this is the question of course I wouldn't have answered I would ask you that if the answer was nest so
14:08
how do we make a better out tough we get but will so I already introduced how we to Cuba gate to sort of works on based on the fact that the 1 once it is in the computational state that can access the 2nd excited state of 1 qubit so you might expect that we can make a 3 cubic gate analogy by having the though the accessible 3 qubits talk to the extensive of which is not quite that simple although that is the essence of the gate and the reason it's not quite that simple is because these 2 states don't talk to each other they have avoided crossing in some sense us said we have to transfer the quantum population of 1 1 1 into a state that talks is using 3 in namely 1 0 2 and then there's no border crossing between to the
14:51
so what we do that had we do that change and I what I have here plot is a numerical diagonalization of the of of this system Hamiltonian with with 3 qubits and here's some data some time domain data characterizing this this is again we moved to a certain flux location and wait for certain amount time in which the oscillation between the 2 states so if we can if we saw in India was removes suddenly into this location where we exactly half a rotation and then moves on and then who suddenly further up and frequency and you know most this is actually exact same trajectory we would take if we follow this border bordercrossing idiomatically it's just this is a lot faster it's been so taking about 100 nanoseconds seduce transfer and back it takes more like a 14 nanoseconds which means it'll be highfidelity operations
15:41
but then once we are in this state is 1 0 to state we can now access the 0 0 3 state again there wouldn't be a lot of of the in the in gray here we have a lot of states that are on nearly degenerate with the states were top and most of them don't matter because there's a matrix on a couple of the states that are populated so this looks like kind kind the mass and it is but it's not nearly as bad as you might think but so what we can do is again we have some time domain data characterizing this well across so we create this 1 0 to state and move up as a function of of of flux biases and wait for certain and time and we can see that this this very rapid oscillations he would crossing are interested in so when approaches border crossing idiomatically because it's so fast we can be certain to it and again this line will diverge from the slope that it would have had if this a or here and this will give us the 3 qubit face so this is all these a lot of technical details get the street right but and also very them but so let's just talk about how to how we prove that the sky is actually doing we
16:38
think this so in the 1st thing that we can do is just measure the classical action of the game unfortunately is a phase gate and classical bits don't have phases so we have to dress up to make a control control not and so what we do control control not willing to put in a basis state and in this case it basis states that span the computational space when applied it to them and then the state tomography and the output and so that's what plotted here I have some inputs states in classical input state I take its up the dot product with the classical up Satan and and have the the result plotted here so the the in almost all cases nothing happens like 1 1 1 goes to 1 1 1 almost all time but the the 2 states 1 0 1 9 1 1 1 flat as they should be because we have that the 1st and 3rd beats are control that so will flip the milky that if an only if the 2 this to this to our a excited and so we can win this fidelity this science visits 86 per cent but this and this in the conference about
17:40
classical correction we really wanna know the action of the state on quantum objects namely the all the action superpositions so we do the same thing with it would increase in state active the operator new tomography on it but we have to do 64 basis states because that that been the focal public space of our computational basis and then as Ramon I mentioned we can we can convert this this the series of measurements into a kind matrix and here's what that time metrics look like it's in general it's a it's a complex matrix I match applauding the absolute value of here and there's some sort of law of order of operations here that's not very interesting but you have to know in order interpret this matrix but this is what it should look like and this is what it does but this is the external data and it's so you know it's reminiscent of sort of you can see that that kind of what's going on so you can take the dot product of these 2 matrices very at this actually a series of 4 thousand measurement account takes 90 minutes it's it's kind of you should be impressed that I can you measure this and then we define a fidelity to this case and it's it's a 77 per cent so if you if you lost me during the gate
18:47
explanation I don't I don't blame you but here's a here's a good time to tune back in but it actually do wanna correction so again we have this this gate circuit but in in expanding the show you on the next slide what we're doing is we're creating some state in this case the the plus X. state I were playing some deterministic rotation 1 of the bits and then we can measure the state fidelity of this middle cubit I'm compared to state that we created so all 4 show what that what happens if we don't do
19:18
error correction more willing so this is the case of nowhere corrections so if we if we don't have an air meaning a 0 rotation we sort of get the maximal fidelity but if we do a full pipe pulls them we're nearly a final to that state but if we do errorcorrection but it's it's no with all of it because that everything that has finite fidelity but all you can clearly see the amplitudes oscillations lots more also since we have 3 qubits now we need to make sure that all 3 robust errors so we can do that what's on the other 2 cubits and see that they all behave rather nicely so so that's a that's a nice is all it's sort of a null result we do something nothing really happens so how can you build up to companies are actually understand what's going on and so we can do that by actually looking at the looking at particular density matrix of this of these and so acute it's this this Jockers I label and we can't do this particularly easy to understand this after bit flip so these these these 2 it should be in in a computational products so there's no air but we should be in the ground state and as we are and so on and and we see that in in all these cases were were the you know we're always with some finite fidelity but were almost always in pretty much the system should be and you can also see very easily here the fact that we use me to full cubits to encode there's enormous Islam eat at least 3 but it will so this
20:46
isn't a very this isn't a very accurate model the Serre model of having a single on 1 qubit time isn't a very accurate model of all men actually have in a in a real on computer so let's instead of talk about having simultaneous errors on all 3 qubits with some probability so in this case an action you face of error correction just for kicks because it's you know it's pretty much the same it's adjuster just a matter of a single qubit rotations to do that and to their in their face and would go into the difficult and viceversa and what I'm doing is I'm doing is the gates on all 3 cubits of some number picture but and if I do if I do which is a little 3 qubits we actually have some probability of having had 2 or 3 errors which are uncorrectable but get this code can only deal a single air time so actually do something a little bit fancier and be doing instead of just state tomography process Meyer here the state so decreeing for our for input states that to spend that the 1 qubit basis and and here's the result if we don't do errorcorrection you see that of the state's go down linearly with his error probability of 1 of them doesn't care because it's an eigenstate of the of the rotation operator although we do errorcorrection we see that again we have 1 state that's flat but the other 2 states our nice incurred the the quadratic in her and the probability should be and we can process this data to talk about it said that the the process fidelity of the of the of the process and see that in 1 case we have a linear in the case of nowhere correction we have a linear dependence on the space the probability and he the case we have a quadratic dependence and is also you know there's some residual linear dependence that's sort of consistent with 0 it's sort of as Ramon mentioned it's 0 comma decimal 0 3 plus or minus upon the sex that so it's it's hard to say exactly how much residual by firstorder couple of prefer sensitivity we have but i it's largely quadratic also what
22:41
conclusions can we draw from this and we demonstrated that the simplest version of the the bits the of honor correction and both bit and phase flips to not fault tolerant because we're using gates instead of measurements were also on encoding that the process during the was on encoding that Canada process this it's based on a 323 cubic phase gate which show axis is the the 3rd excited state of a qubit 1 is a program available if if you're interested so I'd like to mention a few things about where
23:09
we see super enacting it's going on in the future so we miserable phaseflip corrections and so we can concatenate those too big for correction and maybe maybe we could we could think about having a having a 9 Cubases 6 per 7 qubits in a in a single resonator making a lot given in resonator and coupling but kept those those logical qubits to each other but really we we already know that
23:33
the the qubits that I used in this in this in this experiment are under now that not merely coherent but fortunately we made a lot of progress on that front in a in a parallel experiment where instead of having a upper plena by a superconducting device we actually have a qubit and in such a lot of this is sort of a cartoon of that it turns out it for various reasons but the the threedimensional architecture has about 40 times longer coherence time than the twodimensional Mitchell architecture and so on part of the reason this works so well because we took all we basically took everything out of the box it wasn't the superconductor made of sapphire at which which serve reduce the the ability control system but now now we think we know how to reintegrate some of those some of those are so not like the foot by signs and so on I'm aceMedia reset architecture and things us so weak so we could get the the capability of of the 40 devices are explained but with the cornerstone time of this of this new activity on actually this this appeared yesterday in imperial and Carol has a lot of viewpoint titled a superconducting qubits are getting serious which I hope you agree with on the other visitor
24:40
questions so to talk so wrote the same 3 dimensions is better than to the questions that strictly speaking you don't need measurements to do fault faulttolerant quantum computing excite obviously for the reality and but you do need to be able to say refractory said you're going to answer the cubist or were coupled to new ones I is here you have a way of doing that in your system test not absolutely there there are there are a couple ways that we've we've actually density experiment where you can reset units with very quickly on sort of the the easiest way to think about doing that is to have the cavity itself be very low queue so if you just bring a kid and residence with a cavity or swaps excitation in an and we very quickly there's also other ways that you can think about doing that and it is sort of 2 Cu sideband gates where you use what the excitation of the of the qubit again into the cavity but we've also sort of thought about more exotic ways of maybe we could recent acute through for itself beyond this is utterly capability that we were were need we are aware that we need and and we have some ideas about to have you are quantified to what extent the leakage areas that you get when you couple to the other 1 0 2 states and the 0 0 3 states effects here the Collection fidelity on that's that's all of this a little bit difficult to be quantitative about that because we need to we would we we need to do sort of state tomography on a manifold of much larger hilbert space but what I can say is that we know that those those the population left behind in those noncompetition the rather small because it would show up in a very obvious way in the way we do stick tomography and the fact that I hadn't should in this in this in this talk but if we look at the raw data that went into that kind metrics are there's a very very obvious signature of of leading competition state if you if you those if you look at this but it of what it looks at hand so you go through the trouble of creating a tollfree gate to avoid measurements and so it appears to me that measurements on the syndrome cubits would also affected data cubits and that's why you avoid them beliefs so what's the future I mean can you in the future can you incorporate measurements in your error correction procedure or in your circuits or is this incompatible with your architecture yeah but through the questions so the though yeah you're you're you're right that the reason that we sort of avoiding measurements in this in this guy's for 1 we're having a single cavity so if we measure measure all 3 qubits in which is undesirable the other reason is sort of a a more technical 1 which is until very recently that the single qubit readout fidelity of our time is rather low ET the rather low or a completely scrambles the qubits you can have high fidelity be stable qubits you lowfidelity but you're of on it but for so there's been a lot of progress in that in and so the last year where we can have by using very of a special with amplifiers or we can have very high fidelity and also QND readout and so we can we should think about having the SIL qubits coupled to more than 1 Resona so we have a center resonator for coupling qubits and then add an ancillary resonators for measuring single qubit time and that's that's a something of that word they were thinking about doing that we could be capability for with that brochure partly some implement phase gates using an on course in a bind to slimmer poachers easily lead to this last name if you can do to faster than they do in the new global you bloody about 30 influence but you seem to know your opponent's you could sign may be an optimal solution in the media the news Quillen Control did you think about yeah and so yellow using land as a kind of that is that was something that we thought about doing so in practice it sort of more trouble than it's worth because we can make these very we can we can make is very with this NS sudden dates so not only are they they rather fast so that they have high fidelity but they are also very easy to trap experimentally the problem sort of the land as he approaches that you sort of have a continuum of of parameters to tune and it's difficult to think of service report experiment to find the optimal solution for that is true that we know the Hamiltonian very well and the fact that for example this this this 3 qubits spaghetti mess that I showed is actually kind of exactly compatible with the measurement that we see but it's it's it's it's very sensitive to continuing our system parameters very accurately in in practice we just have to have experiments to to turn it up on the last 2 digits when the time it took me to give his 30 minute talk he could have been performing 1300 polychoral you better get back to New Haven because of that all right thanks to a highly disciplined performance by the session chair were only 5 minutes and some of it you so from that point that would be conducted by it thank you to the and that the and this the
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Resultante
Harmonische Analyse
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02:11
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02:38
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Titel  Realization of ThreeQubit Quantum Error Correction with Superconducting Circuits 
Serientitel  Second International Conference on Quantum Error Correction (QEC11) 
Autor 
Reed, Matthew

Lizenz 
CCNamensnennung  keine kommerzielle Nutzung  keine Bearbeitung 3.0 Deutschland: Sie dürfen das Werk bzw. den Inhalt in unveränderter Form zu jedem legalen und nichtkommerziellen Zweck nutzen, vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. 
DOI  10.5446/35324 
Herausgeber  University of Southern California (USC) 
Erscheinungsjahr  2011 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Informatik, Mathematik, Physik 
Abstract  Quantum computers promise to solve certain problems exponentially faster than possible classically but are challenging to build because of their increased susceptibility to errors. Remarkably, however, it is possible to detect and correct errors without destroying coherence by using quantum error correcting codes [1]. The simplest of these are the threequbit codes, which map a onequbit state to an entangled threequbit state and can correct any single phaseflip or bitflip error of one of the three qubits, depending on the code used [2]. Here we demonstrate both codes in a superconducting circuit by encoding a quantum state as previously shown [3,4], inducing errors on all three qubits with some probability, and decoding the error syndrome by reversing the encoding process. This syndrome is then used as the input to a threequbit gate which corrects the primary qubit if it was flipped. As the code can recover from a single error on any qubit, the fidelity of this process should decrease only quadratically with error probability. We implement the correcting threequbit gate, known as a conditionalconditional NOT (CCNot) or Toffoli gate, using an interaction with the third excited state of a single qubit, in 63 ns. We find 85 +/ 1% fidelity to the expected classical action of this gate and 78 +/ 1% fidelity to the ideal quantum process matrix. Using it, we perform a single pass of both quantum bit and phaseflip error correction with 76 +/ 0.5% process fidelity and demonstrate the predicted firstorder insensitivity to errors. Concatenating these two codes and performing them on a ninequbit device would correct arbitrary singlequbit errors. When combined with recent advances in superconducting qubit coherence times [5,6], this may lead to scalable quantum technology. 