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Quantum Error-Correcting Codes by Concatenation


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terms and things funny mutation it's great to be back here again after all 4 yields I thought this as to evolve it's completely different from all the experimental applications and I myself have eluded knowledgeable you can find the applications but it was all the sees some primitive so ideals which hopefully help to make the systems more stable the goal the goal is more to redefine base to construct Collett's then later they investigate how article what's could be performed in fault tolerance adjusted explore the landscape what evils of codes can we have in public law how can we come up with long all week courts the talk is largely based on Tor involved with bait saying actually Cloverway consulted for user goal yet to receive was 7 and some of you went look that could pull them before she was invited Spiegel Vice do not yield that's the
reason the Thanksgiving Day in U.S. general 2nd son Jonathan coming and she sends regards to all the audience all and I apologize that's completely new to me not the
baby so the main line of my talk I will but you have not already in parts talk the shore minded called a summary of concatenation but it comes already a particle of label and that gives fell behind you does non-local which might be the 1 to many of you will a cord of length 25 dimension 1 including 9 2 that's how we can get to this 1 and a talk a little bit about off coolants that although bits to add all ties many of you might know glass states that became belly popular but the which by Shing Mun all men far beyond that it was really a port including scheme and I will advise that little bit all in the context of the nation the nominal get channelize schemes for concatenation to go beyond we knew what the finding news on textbook I hope I have time to go into the do that thing channel and see hold the concatenation can head to get their local it's and while some conclusions so what we have seen many times he'll official call actually is based on a bit flip caught on on Facebook or just by according to Cureton different these but a point no as what are the effective just use the beach at Cold what would a similar to read I of call the egg this bitch adulthood it was and all code but they can be quite It's never actually if this phase walls if a is there was the the for all and it logical states civil alone but they give a phase tool all encoded once the and independent of causal which of the positions vector so the Bickford courts of sexually poop opuses the 1 pole positions we get rid of the egg cell walls but initially you may get encoded seal pointless India so what we have on a physical level is my idea all collection and the court we could be altered you literature which could have X Y and Z are also you get a channel that then there's only a phase alter and then by the next level of calling me can you was face court and we get this line once recalled important effect which I think is quite often not valid last unit scheme is not that it just had 1 called would be able to collect a phase should always in the novel for bitter dolls it's important the walls up obligated from the physical level to if-else included love and that the principle we relabel see for the improvement of the channel he can play along with that we can use 1 level syncording just to convey all channel in a channel bet on what to do with this channel and it's all something which was used in the fall term of by many cnil yet on the lowest-level testicle what file detection and then on the next level he would go on really file collection it'll color we come up with the
court 25 1 lines or including 1 to which a 25 and we can collect foils for school we can start as the best known for achieving the optimal of 5 fonts we call so including 1 can to 5 and viewing VN coordinative of 2 5 dual that's and what you get is just by the coordination and his life so the cordless in this and fostered of 5 copies of these 5 phones recorded therefore this that and I will think or what we have the 5 copies of the notions that Lisle kind of acting independently but then we put follow constraints according to the next level in Corning so that is we have additionally some kind of an included notion of to stabilize the whole which is called that's how we get a cold is all the news we have implies that we get the generally poor but that's even the public often a photon ones so now all fail since we have just Laureates symbol measurements and that introduces little but you also should north Portland's of course if you repeat that process of a general accord systems and just including 1 cubic should get difficult dimensions and this is the we get exponential woes boson in length and dimension but the keeper just in this film but a keyboard I mention fixed and that is something we're going to at advise if more shallow schemes that we want also to get the dimension of a local time of course maybe we have to make a compromise between the Court of two-dimensional and the court the distance but that is something which it was shown consternation scheme that's not really by us and all
aspect when we look at it was addressed in the 4th total installed but just look at 11 according so we have a clue what that can collect this is an SME it can collect 1 0 yeah mode section 2 0 point 2 schools that 1 more is that we just collect the also all the other mode is that we just try to detect the we detect also can detect well but since if distance 9 for the quarter should be able to collect foils or look at this 2 different all problems we have fuel the first one is actually an all problem of Sweden test the signal in each of the blocks and the other 1 has to well walls improve the blocks so total grateful so these are also just by the time it was the code to be collected what happened if it would separately for them so if you just dwell collection on those levels than we see in looking at just a single L in each block we can collect the next level doesn't see and they all of but you know we have 2 wells in the blocks so if if true well walls we saved collection this OK it's not too bad correction phase on this level but the next level to cope with but again since the field of DL collection on this block in all in the next level of and calling therefore again 12 so L. collection although this 1 on both levels became could not collect this people if you do it support could also if you like in accused him of Fort Collins would you was the lowest level only fold detection then for be detects also but in whole nuisance sweet detection of 3 also means it might be an ion collect welfare so before you for this 1 however all here we have 2 roles they can be detected now we know the position of the Elves so instead of laws we have the racial you can't just think off that approval of the blocks have been you waste and a distance we call what can cope with true well that known position so now we can collect the 2nd paper looks the bottom line of that as war schemes we have fewer which test group individually on the levels fearful sort they'll pick that bonds which are guaranteed to be decodable for the complete called soul what you have to do more if you went on to do optimally according we on a just to promote all the all collection color for all the levels but we all have to communicate patrolling the many that to get to that so we see that there are also some kind of difficulties and it by this concatenated schemes but of course you could just treat this orders at 25 1 3 quarters knowing at different levels and then they can really collection at the expense of higher complexity a concatenation of all ahead of us to break down on something into what you so what is this picture look like if you go to graph costs this is a table together with some of the I John people sure and beats and appealed in JMP at this meal and before I go to that we have to look a little bit at this graph cold and pass so favors the analyzer call what we have coming if the stabilizer court logical exons your people and we know that see if you pick all the logic of Siocon labels together with this the the lies all they all mutually commute so this is now a maximal commuting war and we have 1 particularly well Ivan stayed stabilizer states which defines the encoded C was they eat and then applying the logical labels to distill state the gets all including business we all know that or in the 2 tall we'll but you can map seen in Thom of Court of all finite fields and then actually the Logical Status corresponds to a set of to cold the basis states then correspond to call that off that whole and if you have a stabilizer or actually the union of honor cause that's 10 weights stabilizer called which sexually the courts sees style which and denoted would call what corresponding to normalize but the important thing is for every stabilize could if you fix the door to door playbills we always have a logical quartered states a state we can start and so that
is what was proposed India which showed PayPal by Schliemann Belmont than in the paper with if Martin whether laws in the audience and unfair stop nickel we short actually available pool was a little is nothing but a different way of presenting stabilizer independent though Tremont has shown that the silence and what we need for that to make it all the way national is that we can spot of assessed moot court that is what people label just the the name of 2 graphs they it's and we can put this into a stand of form such that the stabilizer matrix is my identity part in the model Paul so desired and part and just swapping the orbit responding to parts talk Connes 1 that all of the channel labels of all was the demise of his inflexible label because the code itself in order cosmetics asymmetric and a symmetric matrix can be interpreted as the edge of space and the matrix even the connections in the graph but that is only just the course 1 known so far all to a single state did last what we want is we really want to have or was it the logical operators coming to that picture and that is what was all that is constant this CWS picture all that essentially slaughtered global labels recall of the special form becomes phase on the openly so what the canon lawyers if you have to you with some always looked at all the labels and use K and additional input lot is to make the connection it's only abstract at
school to a concrete example we have fuel acute what would which means that all the skates can have different variants they can ever made 1 or they can have made from just by indicated by this problem an uphill we have these stand not form of the stabilize all fall included still state in symmetric matrix corresponds to the graph approval to science just checked the labelings but then we have also some logic open labels and since we can would you was by the normal abides that reliable is that they all C A 1 the Paltz and they come into play that we have an awesome input to all scenes and you connect them according to this 1 so although office from type of lot this sees it has some input Lotus's it at an altar plot disease and we laugh we construct vast states then by these connections from the input but this is too delicate vortices and if we pull some kind also tation inputs and after well Tifton measurements off the red ones to get rid of it and in the quantum sort that model again not the way of a chronic soccer tree you get some kind of my sockets we start with Fourier transforms all hot mothers and be correct case we have just commuting neutral phase gates bell articles pointing to the graph that they actually serve to other ones they go show you while the graph again so does that mean laugh dictates how we have these various gates then we have to open labels between those 2 additional input vortices to encode it but at this point of the socket makeshift problem since what we have is essentially the maximin entangled state left the island called its input sitting he'll and in input sitting there and then essentially techniques of measurement based on computation comes into play we perform a measurement in the x basis using the Harlem transform and then the condition on the measurement we would have to apply so collection of point what is in a measurement based on common the poly for it but if you want to do this call you until you've measurement the F. note to do put forward gave than the tones all at that actually the was gates they moral just phase gates but nonetheless there again the nice the pretation they tested logic to see a local labels corresponding to the X a point is that 1 we analyze and CORDIC Sulka tree but keep in mind you using input alters and in output focus so we're using more space for boys and however small concatenation conducted for the 5 fonts we call what you can't what spend the mom just 15 put all texts and the open focus you get know a nice picture lawful concatenation you just say OK all of these wet output what this is all included again independent the Kachin always describes no longer fits into or picture of the pictures that we had paid input and an output vortices but you know we have 5 additional vortices if to get rid of them again the will of measurement based on mutation tell us yes to measurement of them that's the recording will in the expertise and transform the graph but the catch you is that when the is measurements we have to do with sequence of local communications picking out you know what so we have ambiguities and you get 75 and we want to get in the quotation hold as this fly after removing about this is group so let's look at this graph again so we have this knowledge from them and we perform also in particle sequence the measurements and we get this off you think it's a mess it is as I mean but do they actually all factual Gaussian of which I will demonstrate a a different courts 1 if you just pick when the leader sequences for you all local complementation fully X measurements you wouldn't see any structure you you still have kind of a symmetry in that book and question is how to explain that's it so that it would have an example if you go for
this steam called this thing called actually 1 what inflation as the graph and told the of June and just 1 of these voters have the is the inputs the other of all put so we have this wet killed 50 input sitting he'll and all the all put disease if the small tubes attached they don't what find in the paper what measurement based computation and the class states to begin with but actually Norway has looked at what happens if you perform the X. measurement to graph how does it this they ask people on and you can to it's something we are
going to see maybe a little bit more off to prof you still see kind of this structural off the which a mole went went to laugh was connecting all of them no going on to the blue 1 that's becoming more busy but we have some structural and to
channel this animal you have an edge between the input and some of these optimal we know what's on the test would place this edge by edges between the input and all the neighbors of this 1 2 3 and similarly if you have an edge between them to connect l enable in this 1 this L would enable the want to be removed in the so it's a quite a nice move how you can get from the concatenation having the Aksa luminal it's not likely to the 1 devotee of still in all its again that is something avoiding the men in almost sure it's maybe the expense of losing this structure OK let's go
to how we can get more qubits into all system by concatenation I was tempted to try to explain it all the tone of 2 graphs but it'll the glass might confuse you want the movement so maybe I'm trying to get more back to which pitch office analyze the current what which you all the axiom is that allow the court essentially nothing but taking the whole space and decomposing it into the court in some of the spaces which corresponds to L to conclude so we can think all the have been labeling of all system and heightened measures of all say that I lost the label subspaces and if in review in each of the subspaces became when called so let's look Joel for this fight for sequel if no the altitude I mentioned and we can get a labeling that we have just the pool dimensional subspace that is all local would in many copy of that substance so we get 15 call a vote of 16 copies of all two-dimensional cooled each of them will be able to collect from 1 it's just by law Litella transformations they'll related to each other so we can define a new basis fall was biggest of component novels 1 numberless labeling which copy of the quarterly have which of these subspaces in this picture you have and then we
have the 2nd quantum of more that is the 1 being protected and that is you already kind of an idea of what we have to do we have to somehow protect all the higher quantum and how would we do this again by encoding the at another level of importing which test protects the normal in which subspace feel we don't trust skipping that and Thom
would the more Palencia nobody of looking at it so you have 1 quantum number and pulling all and on and this is actually telling us which of the closed sets which of the subspaces slow going so from the top we have formal almost a job not protected and we have the wet label the change that it's a protected and how can we get from that a better call of course be replicate it so what you do the clean it takes 3 copies of that the son hall have to make some correlations between the quantum numbers which about that protect that and the was all additional contemplated to get for free and if you do the analysis it told all of that is all the cold protecting this column of more action doesn't need to take care of you wants to can just take a classical therefore we can just take the repetition called in that picture and if you do the calculations in the end which to be
had already feel what you end up by this conquered adjusting the of a triple a petition court each of them has proved you it's to and cold and due to blue retention call what science is all a lot of work 16 and all all they get is far 15 7 to width recalled it's what we already knew so it's not
identical but actually because of that fact the torn timetable if GM Smith people Scholten small in all the painting we looked at this team and we fall and actually if you will you start again with the fonts called but no you take age and that and then you can find the nonlinear article would which is better than in the alkyl what and what at the end of the day you get instead of stabilizer corteges according to the balance at this possible including 81 June merits actually get a quarter including 81 comma decimal 8 2 5 2 so I see a lot of people smiling I agree for practical reasons we would not want to go into a fraction of all students that particular tool then go once that all you want to do computations or doesn't even need to have fraction of of and you cannot really do concatenation of fact that but if you really want to test how what can be quenched call with and what are the real balance for finite-size school actually get killed the Hamming on gives us find 9 1 it if it will yield when the bollards down 2 comma decimal 8 7 9 and all code action comma decimal 8 2 5 but getting already for this essentially 5 point 5 % fraction of a cube it close to the ball I don't know whether the bond is tired of child all they use all of this possible code but to get something back and you can play alone if you go to short trips the fuel or getting marginals since all we have action to go to a legs age home at fault according to get improvement we get improvement by 0 comma decimal 9 5 5 Q its three-dimensional systems and knowledge only proposes it's in part the we get better and that all but of course if you increase the dimension of Alsace limbs and this number scale so that this technique actually kill the gets that allows the calls and instead of just poorly also fiction here we go in Morrison's really so we have just a stabilizes the that is just a new 1 labeling of all bases then we move them into quantify mention fall and this this tool of course the whole space is only this before and 2 principle is that the know life to pay up this this this 1 quarter a distance fault work the distance to call would if it is too cold and the distance fall just if it is this 1 cold since all about this means you'd be just the minimum of 2 products of the universe and what he gained by that is again in dimensions and what the form is this the mentioned 26 and L. distance fall cord and before that my table did on a show this this week so with all its factual you ending you can find better call it and is it doing Dr. also you can always try to be more clever all about for the chocolate called to find lens 842 which quote is a paid if you going direct solutions maybe I you
want to use the adolescent you can also have different courts on the normal level and what this buys more is that you can easily get on court of different things income mechanics it's not for the concluding thought too easy to come up with codes of valley lengths just by dropping some of the Witten trading them for the mentioned but if you believe the local what's it's quite easy and effects adjustment classical ideal between from the edge of the transactions and from to become the although had some
ShoreTel the coexistence distance 3 and again we get some minor improvement so we get points for you compelled to the press the realize a cold reduced dimension 48 but the boletus comma decimal 7 so in honest pieces of monitor the school would you can play along but what I
know easily going to adolescence how old is actually had 4 intragroup and China so this goes now back to the ideal of the Shor cold this observation that using a coordinated lower-level these as an effective different channel on the higher level maybe to this audience I don't need to talk much about the unhcr Putin generals of yet just fill two-dimensional system that we can have a decay from the excited to the state some probability and just to make all channel a compute channel we actually get also at this the freezing together with so particularly have a channel that has a problem you go and it does not contain items it's quite similar what classically small as a sea channel the although we have won the 1 1 states being confused if Dallas state property of a long but if you know we have some additional because the quantum the feasible and we don't have identity no actually started to the only only by deleting likely as like tongue and the mortal Beck in mind to 7 maybe preprint was 26 and they came up with an encoding scheme for all this and the group that a channel using this particle called and instead of using your from conditions would ask kind for perfect collection the only asking for approximate which we'll have to talk to at least in the often what was in that by set and the so if a country utterly and by the end of the session have talks and discussion a bit more but it's sufficient if you just have both conditions FIL filled up to stop model that is for the quality called if only the high-order and to correct for fill stalled all people post this school and then there's been a lot of Law also reported by people surely can goal to do you off last times confluence vote was talking about what's to simply pretending channel but essentially it was always looking only for fast all and does not much known about final there's 1 relational informed then regardless months us if you expand was told the polytope away balls you actually see you have some as in the tree between the X. and the CEO wars in the end but if you really go for the ponies asking him at more since you don't need to collect for the excitation in that in your wake until the damn thing but not the excitation so you ask me to put forward more about but you can find the 1 thing is that this as symmetry Thom of the relation of conjunction all and then and then go to Muncie this cry plates chapel you find if you ever see is escort with sizes so the existence of 20 plus 1 and the same distance of T plus 1 then you actually get at people of so that's 1 of the benchmarks element wants us to ignore the dosimetry and that has and it turned onto his about simple ideal on the lowest level it it just you was what he called the quantum dual-layer called UN cordons you wanna once you can just think of 2 modes of abuse people or whatever and if you apply it into that general to this channel toward boosting it's what defined is actually a that the effective channel you little 1 with some probability or it decays into the states you was all which is orthogonal to the states had before so we can perform under lost level measurements detecting we had no formal and just by this trick the get effectively in the leisure travel and therefore we can just take any stabilizer called of distance to you and we get a T. called since small we know this higher level Enterprise of increasing length by cool how we do it results and you can gold tall eyes IT PayPal maple true the don't is there's a lot of set of poem you both and apart from small lengths although we naive constructions outperforms boys was to stabilize the Colts and as matrix is a school so the the value of a scheme of course you would expect that you could even battle but and investigate that much of so we have to
talk to what's the main idea of concatenation although why it is the worst fault on computation community and to get along the procedure all you have to have high Distin than would call what's long call what's and coke animations as the the 2 board this sort of if you go off on a generalized concatenation we can gain in dimension at the Prize of shuffling this loans but the good thing is that the although call what's out of class at a cost you can just go to the literature or what that's the cold they do even the linear line that is how we came up with this 1 additive called of its construction we can all the where we simply use the Comecon courts of different lengths we although this structured Proc arises according to fall be cold so we can you was at least all pilot sequences but we have to find them we cannot with the we use the valleys in devices that your point of different ones and what to me the most the important thing is that we can actually rule and L transformation of channels so we can get rid of these all of your hand just on 1 level we can transform differ more channel in the channels which ASK computer scientists mathematicians but failed to have more knowledge about the physic at all and have to Smale a porch and I think there is a lot of potential to use concatenation thank you questions so does this generalized concatenation approach also work for subsistence I don't know I haven't of that at that the thought about using the generalized codes for magic state distillation they might not be useful for it are easy to work with for other reasons that might be good for that new it's that it actually despite the fact that all the sum of 2 was cited been published last year essentially all do deli orders to you the goal you have not yet spent as much time as you to heighten the result of that that although just to invite the audience to look more at this generalize concatenation schemes it's all a question which some of you might have I have no clue what for Tones could come into play for that unit according you so the mass which was demonstrated already for this panel combination you have to interplay between the levels to get from all and the further questions that's the speaker


Formal Metadata

Title Quantum Error-Correcting Codes by Concatenation
Title of Series Second International Conference on Quantum Error Correction (QEC11)
Author Grassl, Markus
License CC Attribution - NonCommercial - NoDerivatives 3.0 Germany:
You are free to use, copy, distribute and transmit the work or content in unchanged form for any legal and non-commercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor.
DOI 10.5446/35321
Publisher University of Southern California (USC)
Release Date 2011
Language English

Content Metadata

Subject Area Information technology, Mathematics, Physics
Abstract We demonstrate how various concatenation techniques can be used to construct good quantum error-correcting codes for various channels.

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