Approximate Operator Quantum Error Correction

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Approximate Operator Quantum Error Correction
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Operator quantum error correction (OQEC) extends the standard formalism of quantum error correction (QEC) to codes in which only a subsystem within a subspace of states is used to store information in a noise-resilient fashion. Motivated by recent work on approximate QEC, which makes it possible to construct subspace codes beyond the framework of perfect error correction, we investigate the problem of approximate operator quantum error correction (AOQEC). We demonstrate easily checkable sufficient conditions for the existence of approximate subsystem codes. Furthermore, we prove the efficacy of the transpose channel as a simple-to-construct recovery map that works nearly as well as the optimal recovery channel, with optimality defined in terms of worst-case fidelity over all code states. This work generalizes our earlier approach of using the transpose channel for approximate subspace correction to the case of approximate OQEC, thus bringing us closer to a full analytical understanding of approximate codes.
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the what about a slightly different approach to approximate quantum error correction and 1 that now includes the framework of operator error correction as well and so I am currently active in into the mathematical sciences in India but this is what the bus dominance to that IQ right Captain and joint work with that we call so not actually
begin by telling you about this very interesting and useful quantum channel called the transpose channel which citric briefly mention install and we don't tell you model it isn't perfect and a collection 1st and then the the the review some of our earlier results where we approach the problem of approximate subspace error correction using the transpose channel and show you some examples of some good approximate codes that we managed to find using this approach and with this background I'm going to want to subsystem codes and and you about approximate operator quantum error correction and show you and that is in the task was channel provides a nice and unifying framework for approximate prediction so what is
this transpose gender so you many of noise Chaloner quote space are you can construct this quantum term which for so I could use is no good was on the cross elements of this channel I simply given as written down can be simply the projection onto the accord and i dag are simply the joint of the air operators of the noise channel and do editing down this channel in terms of its cross elements of this is that the channel itself was actually completely independent of the cross representation because it really can be thought of as a composition of the History Channel essentially the transpose channel is simply the adjoint joint of the noise channel with 2 additional maps the and and added along to make this channel to normalize this show is its trace deserving on the support of the action of the noise from the cold and so why should 1 of the interested in this channel this has been known for a while that before my son is perfectly correct a Blum Accord but then the transpose channel is indeed the recovery not difficult was the information contained in the cold now the wasted enough here it of course it's completely different from the rear 1 is used to seeing the perfect recovery not in terms of projections in unit trees but it's easy to show that these 2 months in fact exactly the same perhaps the rule of the
transpose German Perfect'' recollection will become more explicit but when we look at this point in that form of writing on the perfect interconnection conditions and so what we are sure a while back is that the minute long conditions of exactly equivalent to this other devoted conditions notice that the left-hand side of this condition simply consists of the Krause operators of followed by the and once we realize this it it is easy to see what this condition implies for this condition implies is that if this China's actually correctable on the cord then it means that the action of those complicit channel should simply be a projection on the code and so on this veil writing down the perfect pitch conditions in some sense makes the recovery operation very clear and what it helps us do it's also immediately obtain sufficient conditions for approximate connection resolving need to do now is to really put up this condition and what we can show is that the size of the perturbation is addicted related to the fidelity that you obtain on using the task was shown and because of this realization we now have a set of easily checkable conditions for approximate subspace correction so before I proceed as you did you a whole lot redefined of
section so city is already given a nice introduction and some motivation for the problem of approximate error correction but the measure that we use to quantify approximate correct entities something slightly different views something good views would be called the worst-case reality which for us is simply on the minimization over all states in the code space of this overlap between the initial state and the final state after the action of undercover and it's sometimes easy to think in terms of the fidelity loss of infidelity and really that state some of our bones in terms of this fidelity loss so it was we defined and we said the channels approximately correct and if all of this was his fidelity of is close to 1 of his is high so notice that finding the optimal recovery in this case so for this was his ability is that define it here is in fact not a convex optimization problem but of course 1 can use other measures of which make the
problem tractable the semi-definite programming or conic optimization and this has already been done and as was mentioned earlier that is being shown that of a channel which a very similar to what will be the task was genome and was written down and is close to optimal for the entanglement fidelity measure and analytically that we just saw that are close to optimal recovery maps goes to be constructed for the worst-case entanglement fidelity but what we are really interested in is the problem of approximate character with was case fidelity which really minimizes this reality function for all states in the cold space and the 1st thing of
the of the of the key result that we obtained here was to show that the transpose channel is actually close to optimum followed the whisky's fidelity measure that I just different so just a standard result more precisely on for a given code space of dimension d if eta all is the fidelity loss all for the optimal recovery map than the fidelity due to on using the transpose channel is close to the fidelity that you obtain the optimal not and it only deviates from the optimal fidelity by a factor of d so this corollary essentially sums up the fact that the fidelity loss of infidelity due to the talk was challenge is very close to the fidelity to the optimal not model this factor of dimensions and once we have this that this bound on the of the the optimality of the task was generally in some sense we essentially have Oman so we I want to make this point here that aren't documented it actually the though at so adopted fidelity loss is actually the role of Italy's goin' the setting of classic collection and then you see that the fidelity loss the transpose Jalili's exactly the dialog which once again inputs to the point that a particle collection on the task was channel is indeed the optimal decoding man so what I was going to say it was that this bound on the optimality of the talk was China's together with what I mentioned earlier about sufficient conditions but essentially gives us a somewhat complete picture for approximate subspace and addiction we now have a necessary and sufficient conditions for approximate correct ability and we can use these to actually search for good codes for specific noise models someone is sure you know and some examples of all that we obtained for the amplitude of the channel so what I plot here is the worst case fidelity as a function of the damping parameter gamma and so on there then go here with the process essentially the fidelity of due to the 5 monthly perfecting civilizer could but the black line cure the solid black line of the solid blue line on essentially approximate full cubit codes notice that these approximate focus with goods actually being pretty well compared to the exact 5 and corn and in fact the the blue line actually our outperforms the perfect by 1 who would offer somebody about so the blue light uses an approximate for 1 called and but uses a numeric and optimize recovery it uses on a newly at recovering the map that's obtained using semi-definite programming for the case of entitlement fidelity and the black line here uses the same approximate for 1 called but with the task was shown to cover so notice that the transposed recovery in fact performance almost as well as numerically optimistic of any and water these approximate codes actually performed very close to the perfect phi qubit code so 1 reason to was money to understand why this might be happening is because if you recall the recodify qubit corners oppose the genetic code which is constructed to correct for any single QB data but as these approximant golds at this particular approximate full 1 called by long and others of a wide back but this is actually been constructed specifically for the amplitude damping channel so in some sense what we're comparing is a generic of cord was says accord that is actually been tailor-made for a particular channel and that might explain why these schools actually outperform the perfect like you but called for some time but we did was to look for even charter quotes
and so using on the task was challenging covering we actually such at random for all 3 of the codes and to cubic codes for the after damping channel so what I'm comparing here this solid line is the perfect activity point of the dashed black line is a random of focus bit code I just imitated by some random numerical such of the task was challenging covering the solid blue line again is a random 3 chemical and this is a random to you would code and this other result was really surprising because what it's showing us is that even the random search can give you in fact very short words which don't before all that badly compared to the fight to but could and we managed to go all the way down to even to chords which perform well these OK considering that you've gone from 5 2 on the rate of 2 to 2 and it's over this sort of background of our results on approximate subspace codes the natural question was to go ahead and ask what about subsystem codes can actually and extend as a framework to include subsystem goods is been and how well does the transpose channel actually perform and in the case of operator quantum error correction so that
certainly talk about next year unfamiliar with the formalism of operator quantum error correction the idea is the that the physical noise and induces a certain partition on our system but and information is only encoded in a subsystem of this larger hilbert space so the recovery Mav or nearly colors on subsystem and this larger care about states in other subsystem B which I like to think of as a noisy subsystem and I like to think of subsystem as correctable subsystems so the 1st thing to do was to really you could call the task which I will do for perfect operated a connection and it turns out that a
it is in fact that the appendages are exactly identical to what we had in the subspace skis so that can in fact come up with alternate conditions for perfect operator error correction and these alternate conditions that again written down if you notice on the left-hand side of the gaming have trolls operators of the channel followed by those of the transpose channel and so in some sense but we now have a result which tells us
that the transpose channel is indeed the appropriate recovery Mav even for perfect operator data collection on the phone from sense subsystem meaning is perfect if and
only if but the action of the noise channel followed by the transpose channel is a simple projection on subsystem so once
we have that once we understand the role of trust with China for perfect operated connection and the next thing you know to do was to immediately obtain sufficient conditions for approximate
operator a collection so what we have so so the nonchalant he is epsilon correctable on subsystem in but collided the cross operators satisfiability condition where delta but is the is a perturbation of all size is bounded by this nice epsilon so the reason that I I call the sort of intuition behind the fact that these are sufficient conditions for approximate connection is this falling observation again the size of this perturbation out so this quantity of the trace norm of these delta S is directed to the worst-case fidelity on using the transpose chance are there was this ability measure is a little different here because what we're comparing is states in subsystem a tool the final state which is traced out was subsystem be after that actionable the noise and the task was China so these are sufficient conditions for approximate operated connection and once we have these conditions again is the checkable conditions so it's very easy now to actually much for good of subsystem codes is very similar to how the search for would approximate subspace courts but this is only part of the study the next question to ask is what about the optimality of the task was shown how well does a task was actually perform from a genetic and non perfectly connecting subsystem cordon noise channel and it turns out that we can only
answer this question partially in the sense that we can show that the transpose channel is close to optimal only for certain special kinds of subsystem chords some would just quickly run through what these special cases I for which we can show that the trust was shown as close to optimal for the of the community the case value you're my salary is just the company is coming the cup with that of the light is emitted the coupling between the 2 subsystems and it is easy to see that the trust was jealous of was approximately connecting because that is follows from a subspace result but the more interesting case may be sure that the transpose channels as close to optimal is the
case the subsystem being actually starts out in a maximally state the called that subsystem these are noisy subsystems a subsystem on which we're not interested you really don't care about what happens to the information subsystem B so this may not be that bad of a situation after all so of subsystem these sets on the maximally make state what we can show is that the fidelity due to the transpose challenge is very close to the optimal fidelity all is again the fidelity loss for the optimal to cover the as we defined earlier notice that the of dimension with only the dimension of subsystem of which is as it should be because any ball with which has the fact dimension of subsystem be appearing in it would be it would really not be very meaningful and if we try to just to the extent of subspaces out to the subsystem desire that I should tell you that invariably you end up picking up a factor of the dimension of subsystems B so this bound is something quite a nontrivial bound and whether it a sure this only on the system the starts on a maximally make state which is another way of saying that we just don't know anything about the state subsystem B I should do another interesting special case so if you look at just humint codes in an independent and model by which I mean that the errors of according to Pentagon each cube made an uptick in fashion and in the air operators on the noisy subsystem are simply scaled ball offered to us then we can actually show that the fidelity due to the task was channel is completely independent of the state of subsystem B it is an important kids because it pretty much all colors of the case of sup system stabilizer codes where really your a cube setting and their unit operators of skill policies and you are an independent IT of modern art sort of situation then this is a this is really nice is that as is that the fidelity to transpose channel is completely independent of the state of subsystem B and so are volunteers completely carries through so really don't need to bother about studying on the maximum extent on the so finally I scenario
where the transposons again close to optimal in the case is a sort of extreme case there on the nightstand completely destroys most of the information in subsystem B what I mean by
that so it actually the noise channel has the following property which I done in this last line have so the action on on subsystem the it's actually bring any 2 states on subsystem being close together what I've written down here this bound is specifically the mixed it but again can in fact the cases by any fixed date on subsystem be and this result so what so the idea is that the channel really contracts in some sense subsystem B and if that does this contracting of about a meter then we again have a near optimality designed but the visit with of course this additional battery to better figuring in there so again there's a large class of physical channels for which this is true so if the nice channels for example strictly contracted 1 subsystem B with the really smart data which means it's strongly contracting subsystem the that again we know that we can recover with high-fidelity using the task was general understanding can recover no matter what state we started out with on subsystem and of course know and that such a condition is also another way of saying that noise channel is close to understanding between subsystem in the so let me just sum arise
from what I've said here so what I need is a simple and unifying approach to approximate correction based on the transpose channel so by understanding the role that task was Shamir appears in perfect error-correction subspace and subsystems leads us to a sufficient conditions for approximate error correction and this was of this in this also leads us to a simple way to check then a given quotas is approximately correct about it helps us to find a proximate cause of even charge lens and the most interesting thing is that it compares favorably with cause it obtained by a numerical optimization so that brings us to our bounds on the near optimality of A transpose channel to establish the trust was young is near optimal for subspace courts for subsystem cause of your neighbor sure it optimality under some restrictive conditions but these conditions do not capture a large class of physically relevant chords and physically relevant channels images leave you with some open questions so it's a cause it'd be nice if we could actually find channel which get kind of this method of dimension of that sitting in our bounds of course if you're just interested in collective a single qubit others then this vector of dimension is just 3 and so that's really not much but if you want to look at large accords then maybe this vector of dimension is a little annoying but a more interesting question in a more pressing question probably is that this is how to implement this transpose channel in a more efficient manner what I mean by efficient also soft was it's all clear that it cannot be is implemented by the transposon and this is another definition here again it's it's given to go on his implement this of the some kind of generalized measurement followed by a condition unitary but but the question is is there more out clever way of implementing the transpose channel considering that it is in some sense just the ad joint of the noise channel itself so can we make use of some intimidation of the noise channel itself to implement but the transpose that would be a very interesting and challenging questions and of course there's always the notion of that you can use approximate error-correction as some kind of a 1st step in our fault-tolerant but thanks for listening and I'll be happy to take your questions questions is it easy to implement for a PAL the channel to recover PALETTEENTRY like certain channels do you know if it's He's implement the transpose channel for correcting a palette channel but then benefits all the channel I usually have probably are not interested in doing approximate the connection because of the state so that so that the the reason that 1 looks at 1 of the about you dumping out for example is that if you look at the end of operators of amplitude damping check then the early not skate poly operators so it's it's it's some OK is to actually do approximate collection rather than just use a stabilizer for channels like the amplitude and picture for the case of poly journalists and so when a channel is unit of which is what would happen poly channels I think it is very easy to implement the task was chairman actually but I think the more interesting case a non-trivial cases of channels like the amplitude of the channel that you really don't have to scale body of its lunar do for the amplitude I knew I had the I thought someone can of course and it is still use you can implement this was an example if you wanted willful Cubitt on it if you wanted to a for Cu off adapted up a channel you would just have to use 16 answer less and then you would be would implement this using these Apulian sign condition unities by yeah that's that's about it on the question ist squeeze in real fast at a question about what you mean by efficiently here that until of quantum circuit is that I mean after that but but hang on a 2nd if you could do that efficiently always could you do amazing things like solving in social problems at the end of ornament most paper on differ connection the subgroups cytoplasm cries and as you implement PG hands right and the idea is that it guys is exactly pretty good measurement and that the additional constraints may be in mind want to coach considering when you say efficiently yeah so what you have in mind is what I just told you that this is really the enjoyment of the moisture so if there was a way to do this so if you have anything if you have an efficient circuit implementation of the noise channel itself then we can use sort of maybe can if something bounds on how to implement the transpose channel piece of based on based on a circuit information implementation of the nice so the way I generally think about operator codes the noisy subsystem is something that you you sort of deliberately given up looking at in order to the you know you the gains some of some ability to simplify operations or or for whatever reason for because you can look at it so in that case generally I would if I were running to ascribe the state to it all I would describe the maximum of actually mixed say so if you're not doing that it seems like you may be kind of cheating right I mean if if you have more information about that system that is your in a state that's less than maximally mixed then you can sneak a little extra information about the error by looking at the noisy system also then you're not exactly doing an operator code your your kind of doing in a normal code but in in a somewhat imperfect way so it is so does the fact that that you can't show optimality for K in you know for for cases where where you can get extra information in this way just reflect the fact that it isn't really of acting like a proper operator of so there may be in part about it was actually the other way around so when you try to fix a state in the noisy subsystem that is you can actually think of what you're doing out simply in terms of subspace error correction because you be reduced EUR subsystem being told some kind of tribute to subsist system and so if I completely fixed in my state in the noisy subsystem then I'm not using the subsystem structure which is why we try to look for bones there it's not just the maximally mcstay but in the state but the other point is in that an ideal setting it's true you you I don't know anything about the noisy subsystem and so it probably makes sense to just model it but the maximally mcstay but in view of life in a dual system you probably have are going to have some leakage of information into the noisy subsystem and so maybe starting with any state in the noisy subsystem is a way to it is indeed was generally the thing to do in system and that's what we OK any more questions are common if not let's give a prob- on all the speakers of this week I think