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Design and analysis of autonomous quantum memories based on coherent feedback control

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the by the which stanford university of the sciences see something about this I'm I'm not as a book published quantum memories they should have thank you very much I should certainly start by thanking the organizers of this workshop for the opportunity to be here it's a given you this extra special keynote addresses there hope you'll be terribly disappointed is the stock will include neither group theory nor filter functions but all is is something else interesting instead so the topic as noted is design and analysis of autonomous quantum memories based on coherent feedback control into the thing that is just become a case study on how 1 can take the key ideas about quantum error correction they come from the standard formalism of stabilizes and you know coding unitary maps all that kind of thing that really push those ideas down to a physical level of modeling which is maybe a little bit closer to the kinds of things that engineers in experimental physicists I would like to think about and in shop We've heard a lot of about a lot of very elegant work on how to do that kind of pushed out of Kenner taking these ideas and going down to the level of Hamiltonian interactions of from these lattice sets of models which 1 could maybe say of couched in a language which is more familiar to people working maybe on atomic optical lattices or even these days in condensed matter systems and maybe you could think of the the idea of this work would be to do the same sort of thing except to really to connect those ideas to things that are a little bit closer to the current developments in experimental photonics and maybe some of those ideas can also be carried over into the world of Circuit QED and microwave quantum information processing so the me also acknowledge that the kind of precursors of this work was supported by the current and continuing work is supported by the NSF if I have time at the end of it in terms of conic contextualizing this work and connecting it to some of the things that are going on and they make mention of some stuff which is supported by by DARPA so is really the idea is to get to a formulation of quantum memory which is based on the usual ideas of coding standard detection and and restorative feedback but to really learn how to do that in a system which is completely microscopically described so I actually for the analogy alike to resort to the my career story of the the superconducting circuits your actual quantum dynamical stuff is done in the do or maybe even you got some preamplifiers down the doer but generally for talking about doing control or even a real-time feedback control you got signals have to come in and out of the doer that the come back up to room temperature for the classical competition and they go back down and this is all kind of a way to to short circuit that and just to all of the process and it's necessary for the feedback control for quantum error-correction down in the dual or in the photonics case to do everything really at nanoscale without requiring any and classical control signals coming in and the quantum memories in they're really 3 main ingredients to talk about in order to explain how the whole thing gets set up 1st of all introduce some simple ideas about how to do center detection in continuous time and I think there again our various approaches to doing this kind of thing all explain the 1 which kind of feels most natural from a quantum optics point you and I'll try to build a little bit of intuition about how things like signal noise ratio in a continuous syndrome measurement of the translates into the actual performance of an error correcting code all that talk a little bit about the idea of a coherent feedback control and the idea of doing the feedback without ever bringing the cut the signals up to a classical a level and then in order to actually construct an overall model of how the autonomous quantum memory is going to function all just introduce a few ideas from from control theory about what's called SLH modeling of quantum optical input output components and make some connections to to the to current things going on in control theory and all that will add up into a sort of a relatively simple picture in and how this kind of autonomous or embedded column record and controllers must work out but 1 which I think is very satisfying for us now not necessarily saying that this is how I want to go out and build a quantum memories but I think the idea is that are demonstrated their form a very nice no prior kind a lesson on how quantum information ideas can really be brought into the mainstream of what people are doing in a photonics so you know I mention this idea
about trying to take ideas from quantum error correction and push them down into the physical level of modeling in photonics and as many of you know you photonics is something which is very seriously being pursued not only in academic research groups but in industry so this is just a flash at the slide from our colleagues over at HP Labs which is just across the street from step stanford university and they have a fairly serious investment in what they call large-scale integrated photonics but you know photonics on 1 end connects to require mechanical things so there's a lot of work being done these days on embedding quantum dots or nitrogen vacancy centers or other sorts of adam like solids these into a man of photonic lithographically defined resonators so that you could you kind of a solid state version of cavity QED and that obviously it is highly analogous to the sorts of things that people have done in a single atom cavity QED things that people are doing in my quest circuit cavity Circuit QED at the same time I think that the investments that a company like HP is making in in in photonics is action not so much driven by the ultimate goal of quantum computing but rather just by the realization that as you look at the future scaling of conventional of computer processors to just digging within the informational paradigm of classical information processing there are a severe bottleneck so looming not so far down the road maybe 5 10 years down the road about how to really bring down that heat dissipation and had to solve kind of power bottlenecks both in computing and even just in transporting information around on a chip and it's about that and if you could learn to build a large-scale integrated photonics systems this might be a good way to kind of hybrid computer chips that might still do the computation using the most transistors there were maybe information is new ground in optical form so there's a large industrial base being built up for working with photonics as in information processing of physical paradigm no currently with a large focus on on classical information processing but as altered argue through the stock it's a very small step a to generalize and that's the quantum information processing there
was a I mean if you really take it to the the limits of of integration so this is a slide that I downloaded off of the Web up from a current project planning at IBM where they become very serious about large-scale integration in silicon photonics and so you know they can have these photographs like this of a large-scale wafers and using men and find that they're able to string together of large sequences a very complex of components in really make things that look like circuits for light waves as opposed to circuits for electronic voltages and currents so
you don't make contact to that world to introduce a friend the way to think about things like quantum error correction that very naturally maps onto that sort of developing fabrication technology I to 1 of the things that has to be done is to take that view of quantum error correction which has largely arisen in the kind of the paradigm of computer science in the Digital Signal Processing in really pull it back down to a kind of earlier more primitive state of Electrical Engineering which focused on analog components and in the hands of different kinds of circuit diagrams those at the very top of this page of written something which is it's a little bit weird but you'll recognize the notation as being that of what we aim at a cop the workshop like this typically when people say quantum circuit they mean something like what strong at the top but if you if you think about what the diagram really is the they in a talk history just put the labels on that there's time going across the horizontal axis and this kind of implicit one-dimensional space axis going across the top through the bad thing is not so much a circuit diagram so we might learn about in elementary electrical engineering courses in graduate school it's really kind of a graphical pseudo code that specifies an algorithm for what supposed to happen in contrast that to this diagram this drawn in the middle of the page where this is you've done some electronics this is very familiar to you there is an operational amplifier with the resistor feedback network providing negative feedback to turn this thing into an inverting amplifier and there's a big difference in that although this middle diagram is still something kind of schematic but the components are depicted in the cartoon the sort of form it nevertheless shows you what physical devices are supposed to be plugged into your circuit board roughly where it is supposed to be in relation to 1 another how at the outputs and inputs of these different components was to be connected and the idea of this kind of thing is that if you actually make such a circuit layout by c make the operational amplifier and 2 resistors or capacitors if you put them in this so if you lay them out kind of like this connect the inputs and outputs in the sort of topology once you've made that circuit all the work is done at the only thing that you need to do in order to have this little circuit actually function is you just had the power it and so that's I think that kind of paradigm which is familiar from the from the from classical Electrical Engineering in the question that we've been asking is what can you do a similar sort of thing to the idea of quantum error correction but but now adapted to the kinds of devices and layouts of 1 can imagine in the maybe not too distant future in banana photonics by 4 so this image that's at the bottom of the rest of my colleague Ewing of which which precious electron micrograph of photonic crystal membrane so this is the top view of the 2 dimensional thin dielectric membrane but a series of an array of holes have been punched in it so the little holes are the ones that are relevant and kind of you you had around some little structures so there's an obvious strike the goes down the middle betting functions as a as a wave guide for light so wire at the 2 ends it's probably hard to see but there is after a little bit of a gap is in the little can adopt there that will not functions as an optical resonator others another 1 of them down at the other end and so this is very simply divisory have to input output optical components connected by wire and so the the can a grand idea of this is to move toward the paradigm word but what we like to able to do is to specify at least in some schematic kind of way analogous to the op diagram in the middle of the slide how to lay out manner photonic components of a specifiable type such that you would go into your eating that and make a thing like this which had been on a photonic circuit which then just existed as a solid hunk of stuff and in order to get that circuit to perform continuous quantum error correction the only thing that you would need to do is to provide a power so the do a lot more like this circuit that's right in the middle of the slide and a lot less like the kind of algorithmic step-by-step synchronicity but that's implied by the diagrams written at the top so 1st the
money can buy these into this idea of going from discrete step-by-step pictures of quantum error correction into how we would think about doing error-correction continuous time without running in its into any sort of issues by columns in a paradox anything like that and so in almost all the talk and this can be focusing on the simplest possible corner action code so this talk about the 3 qubit bit flip code nevertheless all the ideas that are presented here generalizing a very straightforward way to state or other sources stabilizer codes and so all of these briefly show a diagram of a similar sort of circuit for doing it but they can sure version of the 9 code that the correction true single qubit error and so but to the bit flip code you know the only thing that you really need to be able to do is to make parity measurements on this the parodies of pairs of qubits in the register by a single logical qubit as the entangled state of through physical qubits in register logical 0 represented by 0 0 0 logical 1 represented by 1 1 1 and in order to read out the syndrome of of a possible error would you wanna do is read out of these 2 different parities in that in that register right you might look at as C 1 C 2 and also look at C 1 C 3 or you could you could choose any other pair like that Stokes so in order to think about how 1 does that in a continuous way as opposed to the usual kind of picture that would involve a controlled control not date and and and so on just consider the following idea which it is you know more follows from things that a previously been done in the literature and there's some alternatives to this very idea 1 of which is describe a Nielsen in 2010 paper but suppose I have a coherent state of light a single spatial mode of a coherent state so released in some approximation this is the kind of like the just comes out of an ideal laser so the core here is fully specified by a single complex number that's the complex amplitude the magnitude of the complex number tells me how much power is not been in the phase of the complex number tells me the phase of the coherence and so the notation here will be just in a direct had if I put the number alpha that's the complex number that represents the the complex amplitude and phase of their careers so in a certain scattering phase convention if I just take such light and I have reflected off of an isolated their rights the the laser beam comes in reflects off the mirror and said well I can't choose a phase convention in which under that type of reflection the face of the coherent state is unchanged now under that same phase convention if I consider instead that I have to ideal mirrors and place those face-to-face and make a little resonant cavity then under the assumption that the frequency of this coherent state is resonant to the spacing between the mirrors so roughly speaking of the wavelength of the light that's an exact integer numbers in the distance between these 2 reflecting boundary conditions then I'll be driving that cavity on resonance if I see that's an empty cavity and maybe it's a single-sided cavities of the light goes in and out 1 side of the cavity only then under the same kind of scary conventions of fairly good happens to the phase of the probe beam then rather than bouncing off of the cavity having no phase change actually suffers a pi phase change and so in order to put a cube into the story and talk about how this starts to be something like a sigma z measurement but we can choose as the physical carrier of a cube of information if we pick something like that a typical atomic level structure so this little thing inside of the strong inside cavities here we if this guy an archive of the 2 end mirrors of the cavity by my resonate probing beam is coming in this way if I imagine that inside of back cavity have this adam like object which has maybe 2 stable ground states and 1 of the electronic excited state I could imagine that the level spacing between this croustade that excited state is resonant with the transition frequency in the cavity whereas the energy between this lower ground state and excited state is so that means that if I have any of the ideal would be then to encode the qubits in the coherence of summarization of these 2 ground states so if the atom inside the cavity is prepared in this law ground state then it is there the transition that's accessable is not the 1 this resonant the cavity mode so if the qubit is in this state this probing comes by and grabs the cavity it still looks like an empty cavity and so we would get this high phase change on the other hand if the atom happens to be the other logical state then this accessable transition is resonantly driven by light in the cavity and because of the now familiar physical phenomena vacuum rugby splitting what happens in that case is the latest still perfectly reflected from the cavity but if you work out all the details than picking up a pi phase change it again picks up a 0 phase change so that says that in terms of a single qubit like this if we just take a coherent laser probing bounced off the cavity then depending on whether the embedded Cubitt is analogical 0 or a logical 1 but this Korean stabilizer pickup a pi phase shifter 0 efficient to make it very easily the idea about how to do a parity measurement so I just had the same optical probe beam of reflect sequentially off of 2 different cubic cavities systems right so there's 2 different qubits will be 2 the key bits in the register the cavities are there really just to enhance the coupling between the protein and the atoms but the net effect will be that if the qubits are both in the downstate then if a shifts picked up by the by the probe beam will be pion pi the course for an optical beam at 2 pi fish shift is the same thing as a 0 efficient so if the 2 qubits a in the 1 1 state then I'll get the shifts of 0 and 0 so that that's a reflection of the 5 1 0 or 0 1 then 1 of the reflections so good mother will give me 0 and so overall the fisher to the probe beam will be pie to an upset with something where now you've use this continuous pro game to to to determine the parity of the 2 cubits and the result is encoded in the out the phase of the outgoing Peruvian being either
0 or pi in is across kind a larger question a lot of these businesses of of trying to create physical realizations of the ideas of quantum error-correction especially tried engineer this very low level that all the stuff that goes on in stabilizer formalism right all of the trivial stabilizers are to body operators that you're trying to measure the sort of implicit in that is that will offer do this measurement some sort of indirect way where there's either and then Incirlik qubit that or a field like this if you are trying to measure of two-body observable then really the Hamiltonian that use was using to make the coupling is always supposed to be a three-body Hamiltonian but nature doesn't really give you knew those to work with so you have to synthesize and somehow so question you know there are lots of tricks that it had been invented on how to do that but so you very quickly realize that well by the Hamiltonian which is a two-body Hamiltonian but then if I think about propagators richer exponentials of those that I did this propagators and somehow pull out something nonlinear that's really what I need to do in ordered it synthesizes three-body interactions and here we had to use a very simple sort of form of the nonlinearity which is simply the fact that this kind of phase of of of a coherent beam this on a circle rather than on a line right so that's what's giving us the nonlinearity that allows us to make something that really looks like a two-body parity measurement that you know as opposed to something else so you're just to show that OK so well I think that you can do to analyze exactly because on obviously this is the simple picture but you would really like to see how well does this kind of scheme perform under a more exact and physical model you could write down the James Cummings had an interaction Hamiltonian between synthetic cubism the cavity mode yes a nice quantum optical input output theory about how the Prodi really interacts with the whole thing anything go by using some recently developed kind of optics theory is that in the limit where both the Robbie frequency that couples to keep to the cavity and the cavity to KeyRaider both large but where the ratio of those 2 things is fixed at that some more or less arbitrary value then the model the input output model for this kind of a scattering system goes over exactly into a using measurement as a as the marina come back to a few more times in constructing neural circuit model which is that although we understand how to write down the explicit physical model for these things in terms of microscopic physics like the James Cummings model on it turns out that for the kinds of components that we've designed if you consider this limit where both the Robbie frequency and the carry decay rate when those both go to infinity compared other parameters then be effective scattering models for all of these components will greatly simplify and that's really the key to allowing us to construct an overall circuit model that this this quantum memory will work using an actual can choose time differential equation of sort of model in that particular limit I think is a very natural 1 for this man of tonic setting see you then get the idea that using the beam lithography other such techniques you can make very very small resonators in in this context and so as you go to this small volume limit the thing the very naturally emerges for cavity QED type models is that G is very large and capping gets very large parts of the vacuum Ivory frequency in the cavity to carry it naturally both get large at the same time in a small volume on it and that's exactly the parameter limit that we wanna consider of for the functional models of these sorts of devices that we want is basic business of designing a autonomous quantum to confirm that this kind of thing can work reasonably well using c currently accessible experimental parameters if you just mark up in a contradictory had of simulations situation like this where that atoms couple the 2 cavities and we allow a laser beam to scatter off of them and then we do something to the precarious state for these 2 bits which is factorisable so we have 0 1 asserted 0 plus 1 tensor 0 plus 1 so that's our initial states but then return on the sort of continuous parity measurements and then what should happen now after some time has gone by right is that by detecting the phase of the outgoing beam better projector the answer 0 or pi which is then project out either the even or odd parity component of of the of the initial factorisable state so should actually creates an entanglement there but there's little bit of a question about how that dynamic selection looking time so here's an example of a of of a contradictory simulation which shows the projection of the conditional state as a function of time on to either the even or odd parity components and so they start a both that have but you see that they can wiggle around for a while before settling down in this model we've also got occasional bit flips so there's an a that there's a bit of that occurs here in this particular of simulation the parity measurement catches up and realizes that happened but you see that's the kind of by construction in a sort of a a physical model of the parity measurement the measuring the parity of the 2 D that's takes another time but so this fire rated retraction information it's very easy to see why that should be but because a year for those of you who were used to thinking about coherent states of of light beams are or harmonic oscillators that the vacuum state of that something which has a 0 complex amplitude but it's got a certain quantum uncertainty to it and so now if we talk about making up you had state that means we displays that that the disco on some directions and that would be maybe like the 0 0 phase version of of the current state amplitude if we give apply fish if that means we bring it over here but in order really be able to distinguish the 0 and hyphae states very well fiction it's it's separate the amplitude by quite a bit right because otherwise the uncertainties are overlapping can insert a picture that just means is that the flux of photons coming from the lasers is finite yet that you know it when you just turn that thing on an only like 1 foot through in scattering off of both the cavities you that single photons worth the amplitude doesn't really distinguish those 2 states have that as more more photons come through then these sort of effective amplitude of pulls out and so gradually over time durable distinguish the fact that the scattering phase-shift overall as your part right so that kind of good the drawn out in time nature of this kind of measurement is very naturally captured in sort of physical model and so you say you really
wanna really all of this stuff back to what would this look like in the usual sort of a quantum circuits setting and I think I've actually drawn on the top there sort of effective model for what's going on and so we would still be the case that about the the kind of 3 register qubits which of the 3 lines at the top will be to Incirlik cubits running along the bottom representing the outputs of the 2 different parity measurements that you need to make clear a 1 to parity and won the very end of if you use this kind of circuit we had control controlled lots and out 1 iteration this thing you could perfectly measure this part is that you are after but if on the other hand you consider making those controlled controlled knots and you can them a controlled controlled rotations where the rotation angle was something very small but then you increase the frequency with which you do these things ready you smaller and smaller conditional rotation angle but you do more more of these measures for unit time that will take you over to this continuous measurement limit right so where just a single click in the in the detectors that are looking at the Incirlik that's a single click doesn't give you any definitive information but the parity if the conditional rotation angle is very small but if I do this over and over again in too many of them eventually I can add that up a statistically average out the noise and I can determine what the parties are into this optical version of this talking about phases are common measurement whatever this a lot like this kind of a set up where the the power of the probe beam and other things related to the strength of the of the of the vacuum of frequencies in 1 of those go into determining something like the product of the cation angle and the and the frequency with which those measurements but so this a notion of information per unit time which in the optical limitation has mostly to do with the strength of the probe so I mean imagine if we really were
trying to just do this in the usual error-correcting sort of set up so we would literally as as suggested in this diagram back here you know we're allow this probing to bounce off of a cavities occur in the qubits we would do our best job of trying to measure the phase shift of the outgoing probing and say we tried to take that noisy signal and do something with it and tried to use that to determine when a bit flip that occurred and then I went back and to perform some restorative actions will that overall how that overall system performance
Bellamy introduces kind of notation we've used in some some other publications on that on that topic where but so in this kind of setting you can imagine that while we're trying to do is estimating state of the code based on these sorts of continuous measurements and so initially when you're register as just injured initialized in its logical state in in a very short time through pretty sure no error has occurred you can say that the state of the code is III in the sense that no error has occurred and then if some time goes by then there's increasing likelihood that you may have suffered a bit flip on at least 1 of the cubits so you might have to consider the states XIII XIII acts in the obvious kind of annotation likewise to errors would be represented by things like XXIX I x i x and eventually you might get to the to the state where all of its foot the triplex but of course you know since these that thought to happen is a random process so I kind of any of the transitions that drawn by arrows are allowed rights of phi star from I make a single that provided by X and maybe the last you look like you apply expects that I I acts of little bit flips again presidential dynamics of the errors that the code is a random walk on kind of a graph in really you can view it as your job in electrical engineering you sort of ways so that these noisy continuous time measurements of a pair of parodies so I'm trying to do is to use that information to optimally reconstruct a posterior probability on this graph right so at time T equals 0 I'm sure that my register still in a perfect state so I signed a probability vector that has value 1 on I I 0 on all the other rooms but then as time goes by like in a small time step or maybe 1 error has occurred if I don't do any detection in an hour or anything like that then that problem listeners start to smear out to that 1st column of different states and you know if the errors are happening fast compared information read my measurements that probability will continue to smear on this path and eventually screwed but you know if you're a measure if your information gain rate is fast enough compared to you the reader witches a recurring then they conditioning on the measurements that you receive allows you to kind of keep the probability mass localized mostly in the someone in these areas states and you what you're trying to do is sort of construct a model that will do that the optimal way see is that this business of looking at random Markov random walk on a graph are based on measurements of continuing dancing noise of the optimal state estimator for such problem was actually done in my around in 1965 like in 1 so you want the 1 filters actually optimal numerical procedure for doing that in a recursive way we're trying to do this in real time and such that the problem of how to do that is something that's all the stuff we can take that for granted consider he started in simulations of that sort of process so here or doing as we take 3 cubic code initialize it in some logical state and then we simulate in a continuous time sort of way both bit flips that might be acting on the key bits of the code in also simulating these continuous noisy measurements of the parties and so it's a drawn in the simulation plot at the top is initially we start with the probability value 1 assigned II and we're going to look at the probability values the poster probabilities that are assigned to the EU single there's detection of time so up on the top there time is on the horizontal axis is see the there's a black trees that starts out at 1 and in the red blue and green traces start out as 0 as time goes on we have some we we features at horizontal green dashed line there in this particular simulation of before actually occurs on the 1st 2 it to the actual state of code ghost XII but then the twister reconstructions of that state as determined by the 1 filtering equations are shown that they have a little bit of of latency it's a because of the fact that you have finite information rate coming out of your measurements it takes a little bit of time to notice that that that's what is actually heard but eventually in fact the green probably does go up close to 1 the new ones come back down close to 0 that's 1 feature of the sort of system and then also notice that actually even before that that for the occurs this is funny little hiccup about 4 comma decimal 3 right where there's a little bit of a glitch where the probability of 1 good standing in 1 go up so generally in these sorts of systems we don't have instantaneous perfectly but rather you have these continuous noisy fire-rated information comes in measurements is latency in detecting errors and there's also some of the false alarm rate you think that you know people studied in control theory for ever but these are the kinds of things that regardless of how we actually implement the error-correction or their tracking whether we do that using a classical computer propagating the 1 filtering equations or whether we do that using a core here feedback set of of the type that I'm about to describe we should imagine that because of the finite rate of information gain the parity measurements right there always will be these non ideologies and eventually those things will catch up with you is eventually factored heavily in C and the fact that there's a finite rate of false alarms to student you know like anomalous sequences of the random measurement results a unity your fidelity will decay in in an irreversible fashion that kind of the better your code is is the better your of your memory is the more you will slow down the actual on the decay of fidelity as compared the situation we don't know OK so so so little prior continues and detection and if move on to talking a little bit about the ideas of coherent feedback and how you might use this kind of approving set up to actually diagnose and and diagnose the syndrome and try to correct the errors without ever actually sticking afforded sectors set up there that bringing these signals necessarily back up to a kind of macroscopic controller
so this is just a starts to make some interesting connections between the whole business of quantum error correction and kind of a major theme in quantum control theory at that's developed over the past few years and so I think by now many people will recognize that kind of set at the start of the right hand side of the slide where we have some quite a mechanical input output system which is the system were trying to control the control theoretic jargon usually call that plants and so for many years people have been considering situations where some probing goes into the plant may be so these approved so the laser scanners off of the the things inside the plant the states become correlated laser beam comes out of the plant you sent into a photodetector right so that destroys the outgoing propane you convert those entanglements intercostal correlations between information in year electrical signal and what's going on with conditional status plants based on that information you got some kind of classical circuit or a computer which tries to extract updates about what's going on with the plant dynamics possibly with provision dynamics in the classical control make some decisions about ways to act back on the planet to try to correct the way that it's evolving and typically those corrections are done by either altering the way that laser beams are going into the plant or maybe modulate other things like magnetic fields electric fields or what have you so that that's a picture that now we would call measurement feedback control rats was really real-time feedback control in the sense that within the coherence time of the plant dynamics were trying to execute many such loops of detection and feedback but real time but measurement based in the sense that when quantum fields come out of the plant we detect them and then really only propagate classical information for a while and then do something that acts back on the planet so in a certain sense you could maybe just say that the feedback loop itself as 0 quantum capacity by construction and on the other hand 1 can obviously think about a situation of more like what's on the left this is what will now call coherent feedback on control it's not a new idea you can find it very interesting papers on this kind of thing in the Electrical Engineering in quantum optics literature going back into the least but to here you have the same sort of situation where there's a plant which is the thing that you're control some laser beam goes into the plant scatters becomes correlated but now in that laser beam comes out rather than detecting it you know rout that output laser beam through another physical system which kind of processes that information in a coherent sort of way and then that a laser beam moved backing its reinjected into the planet the interconnect closing a control loop which is of fully caught in the sense that we never assume that there's any measurement going on at least not necessarily so from experimental point if you really want this this is you're just making some kind of a China from the but the I think what's been interesting in the control theory literature recently fled if you're gonna think about designing large controllers like this where there is a 1 piece of the inner from the which is given in you're trying to design the other piece of the N from there so that you control the dynamics of the 1st part in some desirable way that's now the business of the coherent feedback control synthesis and so there's been some interesting work to relate that kind of sign problem 2 methodologies that are already known in the context of Central and you know from a very fundamental point of view what's interesting about that maybe is that then you can have situations where the quantum capacity the feedback loop is nonzero and it is interesting to ask how that changes the game I will know ahead of time that in the coherent feedback controller that I'm gonna show you for the error correction codes we don't use that quantum capacity but is may be an interesting future of how could you know would there be any advantage lecture trying to design a point memory where you grew did take and so some of you may recognize that this kind of idea of using 1 quantum systems of control another quantum system that there's a different way of formulating the sets of Floyd and some others have looked at and I think the main distinction is that where there they tended to sort of take a system splitted into parts and ask about designing the interaction Hamiltonian this is a little bit of a different story because we were really assuming that all of the interactions between the time the controller are mediated by propagating electromagnetic fields and selection makes for a much more experimentally friendly and more engineering from the kind of design paradigm which is much closer to the kinds of things are used to dealing with an electrical engineering right so we don't assume that there are any direct Hamiltonian interactions between the controller system the plant system rather the only thing is really happening is this kind of funny loop scattering involving coherent optical or would you like to feel and so he is a schematic
example of of what we would call a continuous bit flip quantum error correction system and so this is kind of schematic is meant to be completely analogous to the simple op-amp with the 2 resister a network that was shown on the characters slides so the items that are acting on this diagram so there's a Q 1 Q 2 Q 3 so those are meant to be the 3 qubits bits of Europe of the register and so the logical state in the memory is encoded in an entangled state the street you that's on as I had earlier mentioned we assume that each 1 of those qubits is coupled locally to its own optical cavity and that cavities they're in order to enable this kind of scattering readout of parity now in addition to those 3 cubic cavities that form the register a roster can have 2 devices of a of a kind of recall really and I'll show you a kind of zoom in their push down into what that looks like in just a 2nd but then all of these things really can a cavity QED this devices so they have very mind optical input and output ports and with the red and blue line show you is really how those input output ports are supposed to be connected right so the idea is that if you're a in a brilliant advancement photonics engineer and you understand the basic principles of the cavity QED that goes into the design of each 1 of these components then it's pretty simple for you to translate this cartoony diagram into an actual schematic for the devices that you would fabricated by the inner lithographically patterning some very thin dielectric wafer maybe that had to do some fancy things about and planting quantum dots the more the sets of of of impurities like that but if you could make that sort of thing right where the connections among the components are are realized as by fixed wave guides at you fabricated into this chip but the claim is that once you make such a thing in orders a candidate this thing to go and do continues quantum error correction the only thing that you have to add is yes the power it and so in this case what that means is that there are 3 input ports which really drugs too so there's a blue input for all the way out and you're supposed to inject laser power at a given wavelength into that port and in these 2 other the radium purports marked data of obviously those could come out of the beam splitter from a single red input port but that's a place response to inject a different wavelength laser beams and so the in order for this thing to work there's a certain parameter hierarchy that should be respected so alpha should not be much larger data and beta and alpha should fit in between some other parameters what action gets built on the circuit but a very important thing about this is that you don't have the fine tune their values all right so the overall error correction circuit will work for a very wide range of this actual parameter values and also i they don't have to be perfectly constant time you don't the clock them and you certainly don't need it to make them conditional signals are coming out of the device you're the put power and from from 2 lasers right so that regard we feel like we get a lot closer to the kind of familiar situation from electrical engineering we just got ICCC you just look up a power supply in the thing goes so that's really the the ideal that were trying to use it in this sort of the design of enough time this 1 end so in order to show you have finally how we really the rap all of that stuff out into an overall model of how this circuit works and the the essential idea is that once you have defined input output models for each 1 of those different components that we can sort of abstracted to where say you know these boxes that are marked the 1 of the 3 5 there's devices every beamsplitter has if we just think about unidirectional flow it has 2 input ports and into up reports was represented by left and right circles likewise the devices that have called relays have 4 inputs and 4 outputs in the efficiently there the cubic cavities and of having both scattering input output ports that for the parity measurements and they're also the back porch to drive run on transition the atoms was also drawn there but sort of what you have those individual components then would you would like to do is build an overall model for the circuit or reconsider the interconnections of say is the you take the 1 of the outputs of this splitter and you rotten into some specific input in 1 of the 2 bit components so what to specify that connection topology we use this so called circuit out concert in algebra and most recently in detail by John Govan and James then we can take those individual components which are mathematically represented in a certain way of by and will be 1 2 2 1 2 and 3 and the really the connections of but this this serial connection to specify by all triangles on the Xbox symbols indicates something which is kind a like a tensor product right but we now have a way of understanding how these input output models the connected in order to construct an overall CA model and so that's something that we can now do in a relatively straightforward way using developments in quantum optics circuit theory which of the world over the past few years so as for those individual components have also we've already talked about how to model the parity measurement by this cavity QED scattering kind of set up that that strong in the upper left hand corner at the back operations on the individual atomic units can be implemented by Ramón transitions so G H here are like the the logical states before the atoms inside the inside the the register qubit Adams if we wanna flip the stated that we can do that by closing optical run on transition in the solid transitions are things were the only if they're sufficiently to to and from a virtual excited-state then those run transitions are only driven if so both of those different proteins that are drawn by the red arrows of they're both present at the same time so that is a a kind of an ending a capability that 1 needs in order to transcend into specification of which can structures supposed but and India really is a somewhat complicated kind of thing and again involves an atom like a meter impurity now suppose we couple the 3 distinct cavity modes but again you can write down the kind of microscopic description of what that thing is but in terms of the cavity QED model I you can simulated at the microscopic level and verify that it works reasonably well in excess of different regimes but what's perhaps more important for a really describing this overall circuit and doing simulations or master equation innovations of the circuit now if you think about it here if we've got 3 cavity modes in minute meeting of 4 level description in order to get the microscopic model of the relay then you probably talking about a thousand copper space dimensions to really do that the sort of in the familiar sort of way but within very nice is that we can make use of a limit theorem there is the recently derived by found them and the fountain and the once-over fob but such that in this small volume limit that I've talked about a couple times already which is the natural 1 for now in photonics in reducing the gene kappa for the cavities both get like same time but that thousand dimensional image kind of microscopic model for each relay limits to a very simple model that has a two-dimensional internal to the qubit space in where the ways that external field Interacter that internal state can all be specified by scattering matrix type thing of which is like the usual sort of scanning matrix except the matrix elements are operators the of projectors onto the gene H this of the relay in these are operators that's what the state of the really so having done that the relays can have you replaced by two-dimensional sorts of models rather than thousand national sorts of models and it becomes reasonable to imagine that in building up the circuit model for something like this we've got 2 dimensions for each of the cubic cavities and we've got to mentions region relays and so this is now something whose overall dimension is small enough that we connected you explicit integrations of pain we can actually look at the master equation tell what's going on on the other hand we can have some confidence that that's a limit model the really controls the behavior of the true microscopic physical model for such a thing in the spring were gene Kapor both large that's been in a very important kind of modeling and analysis tool that we've brought along in in conjunction with doing this work on quantum error-correcting
codes so paint the the day and if we now apply eliciting their stuff and apply the concert in algebra what it will spin out for you at the end of the day's natural master equation for the evolution of this autonomous some of core memory and so would you load and it is that users 1st so the overall form as the usual sort of master equation the top left hand equation wrote he that's the density matrix as a function of time that's the density matrix jointly on the internal states of the 3 qubits in the register and of the 2 realize that the usual then blood forms of this Hamiltonian H this set of land let operators of which out here written for any of the 3 years simply the bit flips that act randomly on the 3 register qubits and so that such a 1st look at what the jump operators are so 1 if we look at this thing with that look like so that one plus Z 1 Z 2 it's a C 1 and C 2 uses of the the how these observables for the 1st and 2nd it's their product is the key to longevity parity so when the qubit 1 qubit parodies even napping takes value 1 and so 1 plus 1 is 2 so that term is operative and so the operator that multiplies that is sigma Hg really 1 right so that says that when the parity qubits 1 and to his even there's a kind of a decay term which tries to decay the state of really 1 from the state and state and in hand in L 2 we've got the opposite sort of thing where since this is 1 minus the ones you to what that turned us is that when the parity of key this 1 into his odd this a term that tries to decay this data really 1 from H and G right so really a tried this term the tries to force the state of the relay into something that logically reflects that the parity of a pair of qubits in about the complementary thing going on in L 3 no 4 its 2 and 3 and the state of relate to it then go back and look and see what's happening in the Hamiltonian we find their terms like projector onto g of really 1 times projector onto H really 2 times X 1 so that term in the Hamiltonian says I'm going to induce a Robbie oscillation two bit flip Cuban 1 if and only if the state's really 1 into R G and H respectively at 2nd term will try to flip qubit 3 if the 2 real H and G and the last entrust afoot-the the state give to the relay states' GNG and obviously if the states of the relays R H and H then there's no feedback applied written the overall logic of decoding syndrome table the Chairman which 1 of the key but this was be corrected that's just directly written to what's going on in the Hamiltonian and so the the action of the probe beams in terms of now driving the relays to correctly diagnose the syndrome that somehow encapsulated in his land black-tie proposition giving it decay and you know this when this this really so did these equations popped out of the candidate 20 page the calculations that start from really microscopic models of the cavity QED dynamics of all the components that were designing to the circuit model that after applying these limit theorems but this is the thing they just pops up on its own as the dynamics of the of the quantum error-correction circuit in the small volume limit is very nice to see this whole thing come out in this area the answer to easily interpretable kind of way and I think I'm sure I I know that other people of looked at running down master equations to correspond to something like a quantum error correction that that this is a this as far as we know the 1st example of actually microscopically deriving 1 starting from a completely specified in reasonable underlying physical component models now the thing that I think is attitude about this is that you know since we we had these things were there a lot of terms that are kind of like the decay terms it points out that some doing classical control theory you get used to the idea that you use real-time feedback bird kinds of purposes so 1 reason why you would use real-time feedback as to accomplish disturbance attenuation so if there's some disturbances acting under planting you wanna suppress those real-time feedback is the only really powerful way to do that when you're in this kind of Markoff limit for for the for the dynamics of the plant versus the the correlations of your of your disturbances but the other reason why use real-time feedback is that by doing so you can actually tailor the effective dynamics of your plants so class the example that is if you have something which is a nonlinear oscillator asec squared plus X the force so like that but if you have a measurement of X is a real function of time and you can act back on it with an arbitrary force wall is obvious if you act back on with a force that looks like a negative your measurement to the 4th power you try to cancel out that money at nonlinear part of the of the oscillation the real time feedback is also used to synthesize dynamics in a certain way and so it would again here's a funny I would like to come lectures take some time to look at this but have really done by by use of these of the
scattering loops maybe we don't have any direct physical interaction between any of degrees of freedom that are here right so the register qubits talk directly interact with each other they don't directly interact with the really states all those interactions really carried by this field that we go over into this so strong parameter limit we find that the overall equations of motion have a kind of a ferrimagnetic sort of flavor to them and maybe have terms that are really trying to keep the states of the relays and the Russia's as correlated as possible in the event to kth types of terms to this somehow what you're doing all of this and maybe this is just a feature of the for error-correction code mission of synthesizing a ferrimagnetic interaction and then try to get the thing to stay cool and you're using cohered feedback is a way to do that where there's interactions gets emphasize that the wind Latin in Hamiltonian operator level OK so just a last
a couple of minutes they just mentioned that there is having written down several master equation you can do numerical agree with a range of different parameter values here may get some single parameter that captures the overall strength of the of the optical probes in the feedback loop in you as that increases from 0 up to higher values you see that the fit the decay of the fidelity could achieve it slows down more and more but I will this so another thing which I think is interesting about this is you know that the the way that the parities of things and and the relay states appear in these but operators if you have a bit flip operations are acting on the relay states those also get corrected by the Korean feedback right because when a parody of Cuba's wanted to even that you know it tries to push the really once they from G. H. now if the Perez don't change but some environmental error comes in an accident leaflets set related we will try to get pushed back by that same term so in this feedback the sort of the cube correct the relays in the same way through Rec the Cubans so that's a nice property the code is about
things like propagation losses so in these sorts of non photonic circuits if from taking light from 1 of these devices propagating through wave guide and trying to inject into another obviously the transmission of light through that is nearly perfect answer 1 wants to have modeled includes propagation losses but b it is straightforward but extremely cumbersome to do that but users work workers having done by Cook also whose here somewhere in the back and of really taking apart the network model inserting a whole bunch of beam splitters to model losses sort of distributed everywhere throughout the
circuit and having done that 1 can then go back and take that composite model includes the losses and look at the performance of the vocoder save face what could as a function of the increasing of loss parameter I should say that in this kind of a
circuit so in this circuit layout and if you just change some things about the polarization of the light in the way that the logical states are defined for exactly the same kind of atomic level structure you can implemented they would instead of that that's covered if that's what you wanna do it and that gives you the idea that what you really ought to be will construct sure like and so in fact in the a
paper from earlier this year a new new Journal of Physics we sorta lay out the picture of how you would construct a 9 qubits that Beacon short tight code where the actual implementation that we looked at the only way that it uses the subsystem structures that it's reduces the cheap and easy feedback method disorder corrects some particular qubit in each row and column rather than going back the exact ones but you can further make use of the subsystem structure by breaking of the measurements and things like that so I think all of those basic ideas that come up in the usual world of quantum error correction they propagate very easily into this continuous time so it just as a last
comment into this idea of now being able to view quantum error correction is a form of of core and feedback control and mentioned on a couple of occasions that cohere feedback control is the theme of increasing interest in the climate control of community as a whole and something we've been particularly interested in is that if you think about the industrial laboratory efforts that are going on in places like HP and IBM you there really plunging ahead building and photonic systems for doing classical information processing but if you listen to the kinds of performance schools that they have they're always talking about trying to get into regime of Abergil switching energies and pick a 2nd switching times so that system is added jewels in the optical is a handful of photons pick a 2nd as a small-time compare to the coherence times of things that we know and so like it or not but if those your Goals for your classical information processing you will have to deal with quantum fluctuations in the optical field at a proper your circuit the course if you wanna play the usual tricks that people have about sort of taking NAND gates and making the latches or all that sorts of classical logical stuff if you don't push them onto a photonics implementation is added people who pick a 2nd world you will have top model those things quantum optically or you will not get the classical information processing performance so large part of our groups effort is now that the DARPA-funded thing that I had mentioned at the beginning is on trying to take cohered feedback control theory for quantum optical systems and to recast it as Circuit Theory for possible men photonics as a lot of work that has some
fault is that kind of building all system
we can do schematic-capture using graphical symbols and using your mouse to connect the inputs and outputs of the symbols doing that using a common tools like she scans so so commercially available or even open source freeware kind of schematic capture tools we can compile that into a sort of a text-based description of what the actual quantum optical circuits are in terms of how the
connected and then there's a of very large Python code that will go through and compile that and spit out at the end of the day at the end of the day a completely quantum chemical model and so were were naturally to scan it sort
of merge these streams of thought where had to to think about Quantum optical circuits for more complex codes with the without propagation losses included do those calculations by hand but there already extremely extremely cumbersome and so on but you know any of this methodology that we're building to make 1 of optical circuits for classical information processing therefore Kwan mechanical the carol entailments all that so you can also use them for the quantum
information processing and so you know just finish
up me emphasize that maybe is the main thing that we that we like about all this where by taking quantum error correction union as a form of cohered feedback control Indiana photonic circuits it somehow now appears as 1 end of the continuous spectrum of things that would do in the world of Men photonics maybe Circuit QED and I think that a sort of very simply connected to the to very large scale and of industrial efforts that are going on and maybe helps us see how quantum engineering really is just a kind of perturbative evolution of present day electrical but the thanks therefore division is so this novel by security ovary Christians several my question may not be well form so bear with me but what I'm wondering is how does this Q C career feedback control depend on this physical system being coherent so for example H Boeremag being greater than K T or coherence time is much greater than operation time I guess what I'm getting at is how far can you extend this idea beyond cavity QED perhaps of this so I think the general approach is certainly extensible and in really the only had been the main idea that happens here is that we try to take the logical operations that are usually done in in co in decoding the syndrome in determining the feedback and we try to implement that using parts that are the same parts that we use to make the qubits we try to do that as simply as possible in the end of the molecule looks like this no obviously the cubits a register those things happened if you actually look at what happens the sort of operation is that since the feedback in regular quantum error correction is the the classical but would you find is that the relay states internally never have to be coherent and then you know the only other place to really make use of coherence is that the way that the phase shifts in the probe beam to rent out is essentially by closing in a kilometers so these beams letters that appear at the rate just on the input 1 this sort of an interferometric way of changing the phase shift on the probe beam into routing of the probe power into 1 of you to the really or the other but still think is in some sense kind of and from there so we do rely little that optical coherence but otherwise pizza because the basic idea is a curious ovary classical and the diagnosis and feedback this circuit itself is not particularly coherent as a community by accident other you can ask and say things I mean the germ going way outside my my my realm of knowledge so this may be a completely stupid solution but you could say if you're going to do all this feedback for here the you could rely on coherences entanglements of the relay is indeed if you did that you could think about codes read didn't have to have a stabilizer generators commuted I know you know so maybe there's some room for using this kind set up to generalize a little at the way that you think about this on a point raised so at the beginning in your talk you said there weren't going to be any filter functions of my question is well why not I mean it seems like you have enough tools here to ask the question that you can usually answer and q we see is that how's this work as the speed of the bit flip there's or the noise that leads to the bit flip pairs sparse that's spectral density changes it seems like you could do that analysis pretty you could sort readily that's the only limitation would be in the dimension of the numerical model that you end up with in that we go to a limit where things are Markovian and you effectively in this small volume limit on there are lots of things that happen that this this states of the relays change with scattering right so that lim allows us to get a green America Compaq model but if you go back to the full model then you can insert any kind complex might energy was is just that we look at how the media how the performance degrades as a function the various you know kernels of of whatever time you would then just be you have to deal with the model that was dimensionally much more complicated but you could sort in the interest of time I worked for others just you 1 question and the short answer the in figuring out what the right the back is you could imagine a a wide range of of modeling I mean in the in the classical case up to the point where you actually kind of kept up a model of the state in the exact state of the system at all times if you the right thing down to just from your instantaneous the measurement result immediately feeding back something and it's a simple function of that how much aeration do you have to do or how much disk simplifying cost you performance yeah so the the the way that the coherent feedback thing is implemented the logical processing is kept as simple as possible so just using straightforward if you take this sort of set up we imagine that the conditional probabilities and the error state are accessible the control and you could do anything that you want you can use the classical framework of stochastic hybrid control to define things like cost functions and you can do optimizations and I described a little bit in the 2010 new Journal of Physics paper or would you do is you define a continuation region so you plot the conditional probabilities 3 areas best now point that evolves in a three-dimensional space you can you can derive surfaces where when you hit 1 of the surfaces that's when you take the control actions and the locations of the surfaces are determined as a function of from cost functions specified so that there's room for a lot more complicated stuff OK I'm afraid in the interest of time you have so close here thanks again you from I stopped
Bit
Prozess <Physik>
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sinc-Funktion
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Modallogik
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ROM <Informatik>
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Metadaten

Formale Metadaten

Titel Design and analysis of autonomous quantum memories based on coherent feedback control
Serientitel Second International Conference on Quantum Error Correction (QEC11)
Autor Mabuchi, Hideo
Lizenz CC-Namensnennung - keine kommerzielle Nutzung - keine Bearbeitung 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt in unveränderter Form zu jedem legalen und nicht-kommerziellen Zweck nutzen, vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/35297
Herausgeber University of Southern California (USC)
Erscheinungsjahr 2011
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Informatik, Mathematik, Physik
Abstract Key ideas from the canonical theory of quantum error correction via coding and syndrome measurement can be "pushed down" to the level of physical models with stationary Hamiltonian couplings in various ways; in this talk I will review our group's work on an approach inspired by emerging ideas in coherent-feedback quantum control, which seem well-suited to the implementation settings of nanophotonic cavity-QED and superconducting circuit-QED. I will introduce some basic ideas of continuous syndrome measurement and embedded control dynamics, and discuss a class of intuitive master equations for autonomous quantum memories that can be derived via a modeling approximation that we call the small-volume limit. I will discuss some computational challenges involved in constructing and integrating such master equations for memories based on complex codes, and close with some general remarks on coherent feedback as a tool for interaction synthesis and disturbance attenuation in quantum engineering.

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