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Entanglement-assisted quantum LDPC codes from combinatorial designs

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so I'm going to talking about my family assistive often LDPC codes and their relation to a community design and it is that's going to work with this with that 1 of the things she's staying somewhere in the our ultimate goal is to construct a quantum error correcting codes which had univariate error-correcting components he ended we want to construct a called people out by its range of heart and Maria slang and write rates and so on and then he wanted to call them they're very very efficiently and when we construct them of we want to share explicit constructions so to me that how to achieve these a tree to try to achieve this the the Goals of we the fuel that so why these so called entanglement insist that they either only which is a searing vision of of and Sundance thing of what was and we apply it to a certain classical close called known and the predatory call I leaving the core of pressure in the now we see so between them on that which is ridiculous to combinatorial design it and the Ethernet familial with a desired theory and here is the the deviation on pairwise balance of design PVD for short
of parameters the K and 1 in which is 1 here quite datasets the you know aware of the needs of instead called points and being just the family of subsets of B and that's on each call it a pair of distinct points is complaining that we 1 block and so and the media said to the other replicate if each point appears in that way 1 ought number of blocks and these are very fundamental upon the interior combinatorial objects in the classical example have come from wearable find geometry such as project in space and our minds space and so on in the 2 PBDE generally it's the only other designs and have this idea for a long long time I think it's an since the 19th century so we have a lot of mathematical tools to to study them in the 2 and partly to try to achieve musicals we use in intimate they lined up it and they knew me l LDPC codes in what on any PC goes in and when they are I think other people here all with the formula then let's let not only be sequels are a 2 the special kind of linear called which can be in the cold it's very efficiently by using a certain very efficient suboptimal cold then the point is that I have this LDPC the calls are kept an about the Apple watching because at least in the graphical Turing so that the if you contractility LBP think of carefully now you can almost achieve the showman and then you can also the called builds a culture where in linear time actually in because the EPA Ajijic calls just linear close and they can be represented and so on the my provide graphical tunnel graphs and it is known that and if you want to get a better error-correcting performance you need know is minutes you have got us all the time and graph the lord of the manor or equal to 6 so basically what I would call cycles ended by the length of the in vision so we use the lift departed chick goes as ingredients of a stabilized called and normal hours follow the same stabilize full 1 is only y yeah the so-called symplectic often knowledge and conditions and so statewide engineering of commutes and so while we can employ the following year goes in their way but I know we cheese and the range of the classical world we can think about this olf is limited but the and the generalized version of Sable lights is and can we move the this alpha will orthogonal condition my assuming that we can share that we can we share so tiny and wantonly and so the thing there and the the worst share in its in the light how might this assumption and how we can think of a montage of 80 our winery or potentially called and think we only consider the 5th constructions and this pressing the case when we the in that same called our for correcting a bit with anaphase there it's so the had check matrix and must look like this ancient H is the operatorship matrix of the ingredients in the sequel and in this case if we have a classical your called Opava DDoS in carrying the then I had the we can always get quantum LDPC cos of length and in dimension 2 K. minus in our book and seeing its the rank of part check matrix science it it and in this city that happens to me exactly the same as the amount of pressure and entanglement and that means and that because the which is the required number of if to make of and because I mean quantum LDPC codes in the same kind of what the call the we want them to have a lower score and also on using a bunch of immediate we know the practical so and we know it wants you use only a small number of periods so in this talk we only consider the case when that NBC cause consume only 1 here which is the smallest possible and with the largest possible God so let well we prove it that's on this such an LDPC cos and consuming only 1 gave it away as the the longest possible Gorus are actually the Cleveland it's young the mental committed called on elliptic it PvdA dates and especially if the ingredients in the the close of regular which means that party check making things are of concern Conway can the come from wall weights then and they are equivalent to the so-called China to design in Europe from here where wintering design theory base just so as to give the ways of the way in smart overcame and long being in so what we actually prove in balance and the ingredients our partnership matrix the class the party checkmate X must be an easy this matrix of the PPD ways in the x y and there are other implications number so my disagreements only leaking out we now characterized the higher the PC close consuming only 1 it so that we can now derive a certain bounds on the minimum distance you mentioned involved and what have you and then we can also gives unnecessary and subject the cornerstone for their a good this and is in many cases enemy can also be done and of saying a bunch of explicit constructions so these are assembled mean of me obtained so far in we I just copied and pasted them in these in here out from script in their political well on them as they're feeling on the when the pope face that the condom LDPC cause who who was Wisconsin 1 1 in a bit and it's of length and has the largest possible Gorazde and is of course on the way it goes and the color way to mu the the the the it only 1 this number that happens to be an odd integer and in this Commission of its natural nothing Tuscany and sufficient which means if on the other mean this is large enough then we can always construct an entanglement of the the point LAP the calls constantly only 1 in as low as long that's this number is on the integer and of course an extended without 2 on the real case which means an no possible and can't wait what long weights and now this season example of bound on the the major not alone quantum LEP coast of causing constant mean only 1 in it in the typical and government of the a quantum melody the coast not consuming only 1 in handles at very low cost weights so meaningless dimensions so if you want to construct our hierarchical then making very good news but for them
so but this mean being the news if you want to construct an causeway minimum distance and the so this is but it use them both of our simulation results over time that deep learning China in and then guys marked as Beijing are consuming a bunch of areas in the end the guys on marked at the PGA only consumed 1 it in as long as the following I'd seen you know so and they became quite similarly and so this and this and that you may need a lot of it is to of taking that to achieve the high you're targets and look our and this is a better I think so be found the between call them out effect and kind and the community of designs and light that the equivalence we can because we were able to bits of the wheel we don't like that in that and so the next question might be but if we can characterize the LEV thing called quantum relatively in the course which news of the points cream important for and where training error and the errors and another question might be what if we are multiple in and how to construct from them and include the calls in under that function that we have lot phase error or more so here what you have this kind of problem thinking that thank you the questions something to the idea that the encoding circuits for these codes if you can do them sir progressively like Daniel Garden was asked for his
time and know we have look into the encoding and the other question I have come this if there is noise on that 1 the bit who how is the performance affected he study that that numerically a numerical and of with this do the simulations account for noise and the that's as well I I thought that no knowledge of the units must be there we thought we knew I in this there is nothing to with a series of depending on on it right but I guess I'm asking how they perform like with the whole thing just collapse if that is
no good what it sort of gracefully declined the roots of quantitative question whether well we'll have the turn of so these this that it would buy a little quickly I wanted to ask you to clarify what was that in your results that you said it is too indicated that there it might be hard to find families of codes with high distances also worked well was that limitation some the Willard limitations in that I at which we as sume and that we need to see if this construction and we did the same the parties chipmaker for X Arabs and the Arabs and then we also assume that we only consume 1 ended man the predator committees and must be of certain kind of in the design and in that case the as the main junk-a part of being a willing proto language and that kind of and the peace equality is known to have a very high rate in the classical domain and because the other class come LDPC codes and the call by the peace called pretty much share the same parameters so we have the general concepts the long-awaited polls only
when on only a limited its
parameters a scientist agreement
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Decodierung
Klassische Physik
Familie <Mathematik>
Abgeschlossene Menge
Schreiben <Datenverarbeitung>
Kolmogorov-Komplexität
Low-Density-Parity-Check-Code
Physikalische Theorie
Netzwerktopologie
Spannweite <Stochastik>
Maschinencode
Quantisierung <Physik>
Zusammenhängender Graph
Kombinatorik
Maschinelles Sehen
Konstruktor <Informatik>
Fehlermeldung
Relativitätstheorie
Paarvergleich
Mathematisierung
Systemaufruf
Paarvergleich
Bitrate
Quantisierung <Physik>
Summengleichung
Druckverlauf
Verschränkter Zustand
Verschränkter Zustand
Codierung
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Inzidenzmatrix
Matrizenrechnung
Bit
Punkt
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Element <Mathematik>
Extrempunkt
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Natürliche Zahl
Familie <Mathematik>
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Regulärer Graph
Fortsetzung <Mathematik>
Ungerichteter Graph
Extrempunkt
Low-Density-Parity-Check-Code
Raum-Zeit
Service provider
Gebundener Zustand
Gruppe <Mathematik>
Maschinencode
Konditionszahl
Skript <Programm>
Kombinatorik
Maschinelles Sehen
Gebundener Zustand
DoS-Attacke
Parametersystem
Konstruktor <Informatik>
Dicke
Physikalischer Effekt
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p-Block
Kommutator <Quantentheorie>
Frequenz
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Teilmenge
Arithmetisches Mittel
Druckverlauf
Einheit <Mathematik>
Ganze Zahl
Physikalische Theorie
Konditionszahl
Gerade Zahl
Low-Density-Parity-Check-Code
Projektive Ebene
p-Block
Decodierung
Gewicht <Mathematik>
Orthogonale Funktionen
Hausdorff-Dimension
Klasse <Mathematik>
Zahlenbereich
Abgeschlossene Menge
Kolmogorov-Komplexität
Räumliche Anordnung
Physikalische Theorie
Ausdruck <Logik>
Steiner-System
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Jensen-Maß
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Paarvergleich
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Chipkarte
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Mereologie
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Kantenfärbung
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Bit
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Wellenpaket
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Extrempunkt
Geräusch
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Steiner-System
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Konstruktor <Informatik>
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Metropolitan area network
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Metadaten

Formale Metadaten

Titel Entanglement-assisted quantum LDPC codes from combinatorial designs
Serientitel Second International Conference on Quantum Error Correction (QEC11)
Autor Fujiwara, Yuichiro
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 10.5446/35309
Herausgeber University of Southern California (USC)
Erscheinungsjahr 2011
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Informatik, Mathematik, Physik
Abstract The entanglement-assisted stabilizer formalism is a generalized form of the stabilizer formalism. This framework allows the code designer to take advantage of a significantly wider range of classical error-correcting codes by using pairs of qubits in a maximally entangled state (or ebits). Low-density parity-check (LDPC) codes are among the best known error-correcting codes in terms of error correction performance and decoding complexity in the classical domain and can also be imported to the quantum domain in a simple manner through the entanglement-assisted stabilizer formalism. From a practical viewpoint, it is desirable to rely on fewer ebits while keeping the error correction ability inherited from classical LDPC codes. We present necessary and sufficient conditions for the existence of quantum LDPC codes consuming only one ebit which are obtainable from pairs of identical LDPC codes, and show relations of entanglement-assisted quantum LDPC codes to some fundamental classes of combinatorial designs.

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