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Bounds on achievable rates of sparse quantum codes used over the quantum erasure channel

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we'll talk about on some achievable rates of spots quantum codes over the quantum which the I rocker wheeze use and quantum LDPC codes all sparse print codes and I talk about the performance of this family of codes to be in wide use 2
dies is kind of a goes we have quantum channel which introduces the also and then we use error-correcting codes to transmit information the some procreation information series a determination all of the capacity of the channel it is the highest amount of information that can transmit banishing Europe really therefore we won't be irate and as we so this morning want an efficient decoding Hi Greece and Iverson Decoding and Encoding abreast for the right here I can relate and we cannot compute if you're a candidate to this fast decoding we use sparse codes they are defined by matrix of through weights migration of weight and therefore of this gives us an efficient decoding agrees but in compensation we note for classical sparse codes that we are be below the capacity of the channel and we we prove that this is the same thing for the quantum please use a quantum sitting we cannot achieve the capacity with quantum sparse codes all quantum leaps another advantage of this kind of codes it's more that realizes that realize chop the identity almost everywhere when we have the 2 because the world in a class in a degenerate see class we have seriality to become heroes and therefore we used to generate the this is necessary when we want to obtain good quantum codes for also they were raising channel for example
I will begin by the proof of the capacity of the quantum research and only for all stabilizer codes we pull using it combinatorial arguments that we have at most a tumor rate 1 minus to be for all the quantum eraser channel or for a after all will obtained using this new bond that the sparse quantum code don't achieve capacity and finally you thought within percolation theory using quantum information we obtain and a probe on the permeability in decoration cerebral autograph from use of graphs
what do we know about the quantum eraser channel its capacity is 1 minus 2 P that was formed in 97 by Benard Inzunza once morning and the pool is based on that is cloning hearing therefore we cannot improve this the probe on for a particular family of codes for sparse codes because it doesn't depend on the properties of the codes
know to be in what is study laser codes we so that we need to stabilize a group defined by independent generate tools and the quote will be the fixed the set of point also the sterilizer route the rate of the code will be and minus end and we know that we have and that from measurements associated we is a stabilizer codes it is at the center of it is a vector of F 2 to the air which is a long if n which is your if and only if the Europe commute with the ice the the the ice generator using definition can see that we can measure the syndrome and if 2 we rows d fair in the stabilizer of then they have the same effect on the print and code and therefore we can for and in the same way the quantum marries channel
because it's a regular and definition theory really using which is well adapted to the stabilizer from I'll using it is base and 40 operator or we consider that the committee erased independently with probability and then erased commits is subjected to run them for Europe or I x y o the Wisp or even to counter and the race position is known therefore on any qubits we do not by a vector of F 2 to the n the erased position distict always want if and only if z IceCube each is erased and we know this is a vector or the serious pollution and we know that in this case the states the quantum state is subjected to run them for Europe which is included in various position we can't yet 0 measure the syndrome on your way to a cure and the creation is that we the knowledge of the a-rays position and the syndrome what is your which of your what is the most for permeable Europe we
will see this on an example we consider this that relies on matrix we have 3 independent generate tools it defined a quantum code of predators 5 2 and we take and we take an erasure here is the 2nd in the cell can be I erase how many heroes can cure and how many of them and we correct we have to your race components and on each component we have for all different civil euros I X Y or Z the new this square different placebo euros how many of them can't it too great or we measure we know the various position we measure we measure the central and given the syndrome we associate the class of heroes therefore many difference and rooms are there we are interested in center of of Eurospeech I included in various positions and therefore only the use of blue matrix age is in the UAE's matrix is necessary when we look at this matrix we consider so the rule is exactly the produce of the 2 1st rule that mean the same room in the the cell components of the syndrome we depends on the 2 1st component for the 1st component we have 2 choices this summer will be 0 or 1 therefore we have to to to send wrote off your are included in these areas position we have a number of syndrome and given this and run all meaning you also can we correct how many rows are in the same generates the class included in the eraser to cease worry look at the red part 2 euros are in the same degenerate the class to we're also including in the air pollution they differ in the sterilizer art which is included in the new position the U.S. pollution that means that the stabilizer correspond to a relation between the role of the right the matrix in this case we can see that the 1st and the surrounding world are the same in the red part this the non-trivial correlations and his give us tool stabilizer of the identity and this budget of the true world which I included in 0 spoliation the flowing into degenerate see class we have to era that we can correct in the same way we can't therefore operates to salute to bind to the world bit in a long as it is the Foursquare was civil you also so far we cannot bring the Syria and we can repeat
this arguments for the G. AirAsia we obtain the number of Greek table Eurasia and function of the rank of the submatrix H a small so matrix the matrix of the area components and this the matrix of the known yrast complement the lobster matrix age using these we obtain a lower the Europe liberty and
therefore all we can see that the the maximum a tuba rates of a family of study laser gold is the you know 1 minus A B minus G of was a function g depends on this difference of rank of the the large symmetries allowed random submatrix and thus more and so matrix this give us the bomb which is below 1 minus to be recovered the bomb on the capacity of the capacity of the quantum eraser China which come from the cloning to but only from community relevant elements and moreover fall generous laser codes we cannot say something midterm at for the spikes if we are a family of sparse codes then we prove that this function GOP the can be bounded and we are here strictly you know the capacity no real teammates' function for the sparse matrix we
take and so matrix Ht generally for an n columns and when the grows we arrived through a square matrix in his position for this being we have a square matrix for Jeremy trace it would then this matrix HPE was that we either we are almost 4 right and we cannot say something for the bone but for the particle matrix of a sparse matrix we have a role of her late in the roles of weights straight and in the walls the carbonate it is the identity therefore we had the nonzero non trivial we have a non-zero probability to of neural wins the matrix Ht sets in this matrix and we can enumerates the number of new rules the expected number of new rows in the when them submatrix we obtain the function which is linear lines length this given this give us a momentary other Robohm on the function G and therefore it is proved that the C code and APIs quantum codes don't achieve capacity and in fact if we can improve these bone if we consider different case when we have in the same call on to the question from means of wait longings of symmetries well correlation between the 2 worlds this then as that is the role of this a matrix will not be for the rank of the submatrix will before and therefore we obtain an improvement of the preceding the preceding gaze using murals on they can continue using the relation included in 2 columns 3 columns it set up these viewers remark trade bonds and
this Blum looks like these we cannot see anything that we obtain at shear rates of CIOs' codes of type 2 and that means that the court on the columns are 8 2 and the roles of rate and without non identity coordinates on the sterilizer on the generators and if we have sufficiently alarming distancing the x patterns as he brought we obtain the and a prolonged when the rates of to see what is
his mom follow code of 5 to 8 we have the bombing of the chemistry of the quantum areas of China in right with this kind of codes we know that the rate is at most at least 1 minus 2 and over 1 minus 2 times 2 were 8 stimulus as your opened 5 so all using the capacity you can say that the maximum areas Europe threshold is 0 point 25 and using the rule Quran which is a cure when we consider only the near roles in the present we obtain sensing knew me bitter and in the black curves we consider the relation included in 1 2 was free to 30 kilometers this gave us along which is something like 0 comma decimal 2 which is clearly better than 0 point 25 falls a terrarium visual the I eyewitness thus the application in
preparations for it that eventually we obtain the same Saint Paul as yes this code of type n N we use hypergraphs for probity of hypergraphs to obtain this and we can do the same thing for stabilizer codes to Auburn creation and we do the same thing for the the pricing channel the clock and we have a new way to obtain to approach the capacity of the the prize in channel and the moon the bone is better for for all this year Cisco than semi-desert codes does that mean that we can construe beta studied for the year is official thank you for your attention are there questions so somewhere in your proved to do the fact that H should satisfy the symplectic scallop product at the end like basically that your stabilizer a British should commute to each other so H should satisfy this collective caliber we adopt needs of the commutation relations for all of us said
matrix we don't have a conversational relation between rural it's right I think it is but we can we can define the wrong of this matrix it would be the rank of the wall space without the commutation relations and we don't needs the commutation relation to than any information about the number of syndromes and the number of of fear or included in the original 1 last question here where was the last question looks like that it was hungry there's a final announcement from the word means and that's been produced few
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Formale Metadaten

Titel Bounds on achievable rates of sparse quantum codes used over the quantum erasure channel
Serientitel Second International Conference on Quantum Error Correction (QEC11)
Autor Delfosse, Nicolas
Mitwirkende Zémar, Gilles
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 10.5446/35292
Herausgeber University of Southern California (USC)
Erscheinungsjahr 2011
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Informatik, Mathematik, Physik
Abstract We study the performance of locally decodable sparse quantum codes. The most familiar example of such a code is Kitaev’s toric code. During the last ten years, a number of different constructions of these codes appeared, for example: surfaces codes, finite geometry codes or Latin square codes. These codes are defined by stabilizer group with generators of low weight. If the stabilizer matrix has row weight m and column weight l, we talk about a (l,m) code. Our main result is an upper bound on achievable rates of stabilizer (l,m) codes, as a function of m and l. Achievable rates are rates for which decoding is possible with high probability.

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