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Stabilizer quantum codes via the CWS framework
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the U.S. framework thank you all right so now I heard about these codes actually last last time on this meeting in 2007 and I sort was suddenly fascinated by them and I thought this is something that I should be able to think about because graphs I like and there are sort of ginger medical properties for them that are sort of very natural and I I just like him and so on and so you know 1
00:30
think we did was tried to come up with a way to error correct these codes and basically we failed so they're at the known additive codes and I'll tell a little bit more about them but you know if you try to error correct them the you know it's exponential complexity even in the classical case you can come up with tricks that are actually seem to be working and give you exponential speedup of this exponential a slow process but still it's not fast enough so you still get exponential complexity to decode the general CWS cold and then and so the the work then going to be talking about is trying to come up with actually stabilizer codes just by using this this framework and so I'll start by and this also start with this and it'll introduction but can about these codes in general and then sort of give you of our results so well this is sort of this
01:33
slide that should probably be here on this conference and basically I just want you to the 1 you want to remind you about the status of states so in a polyene and anchored the following group if you have commuting operators that don't include my I'm the a group that includes committed operate that generators don't include minus 1 the reason you know they will always that belies a unique stabilizes and so you can come up with that if you have some some someone is that laser called but you can come up to you know such things in a different way and so the quintessential example of course is the 5
02:12
1 trees that cold and that will be talking about such codes you know about this particular cold couple times they you know in a couple examples and so what you should remember is that of the
02:27
that well it's it's a small called ever you know there is a 5 qubits go just 3 and it encodes 1 unit so 1 of 1 of our results we came up with the Taurus we came up with a way to map this the surface codes this is sort of in the rate of interest
02:45
so so you know what the graph states the graph states is yet another way to encode this fabulous of states or you can think of this as just fix the standard for all 3 data so a graphs they user generated by these operators so you have 1 X. operator and then a bunch of Z is that all to the power of this matrix makes the matrix G gamma that forests adjacency matrix of a graph and so for example it doesn't have it doesn't have the diagonal and so important for this talk will be the notion of a distance of a graph state and this is the new way of full an element of the graphs that belies and so now for this case the you know the simple 1 of the simplest graphs that it's is this 3 Cuban graph this is just the analog and so the generators are acts and then neighboring Z Z and then X is easy and then X is easier so that 3 generators and so the grass they there's actually entangled states which is illegal superposition of all to do with the respect with some phases know because of
03:56
that belies codes were invented in order in 2007 the paper came out in the couple years later so know there were lots of reference for the actual there were lots of references to the preprint instead of the actual papers and so basically the these calls are specified by a graph which gives you the graph state and the classical cold and the classical cold can be additive or non adage basically you start with the graphs they and you will generate the basis vectors of the cold by this goal of word operators which is just products of is tool that you know race the vowel each codeword and so and so the beautiful thing about these codes is that any error can be mapped to a binary vector and so they might then is you just taken there which is a product of X is a disease and you multiply it by graph states that analyzes which have the corresponding access to it and so this adds this additional Z is but at the end you all know how error that is only easy operators and so would have a classical called the code words as specified in terms of this classical that and we also have the classical errors and so this is the map is basically gives you sort of an idea why the schools are sort of simplified so you have and so 1 sort of canonical example this is is the 5 6 to called this is known additive gold so it is generated from the 5 ring you know with these generators and then you just add this 6 classical called words and so you have this explaination Nacional called space which is more than bunkered it but war against what 1 and computers but less than 3 qubits so this is somewhere in the and and then now the problem is again this is to repeat myself is that there are no no known efficient algorithms to decode known additive to use codes and so what we're what we're doing we're trying to find additive calls which are equivalent to this vandalism cold so the subframe 14th includes all of this adolescent codes and so they before I go on I have to explain sort of the error correction or error detection condition for this cold and so basically it and the general error detection conditions that if you have an error and if you have vectors you know if you have evidence that form the cold the cold the cold space the span of these factors is the called the sort of this this product this matrix element should be a constant which depends on the itself bands the Kronecker delta with respect to the codon basically these very cold is either left unpatched war it is used to that of the null space and this is what we're after this is over for why we can correct the errors and so forth these in the body a skull there are actually 2 cases 1 case is the non degenerate case where the takes you basically to 0 this matrix elements is 0 the constant C 0 and then it for this errors you can just use the classical cold and you can just collect them correct and classical is so this that belies of that you would measure would be at the classical binary star otherwise and then have it and then you can sort of do it very easily and then the 2nd case is the generative where this error is not 0 and of course and the class this corresponds to the classical map of the our being exactly 0 and so in the quantum case if your contrived to construct the goal of this error must actually commute with every w open that means that these bind the scalar products of the binding code vectors and the fact that the that determines the error should be 0 and so drafted of simple bounds that you can come up with more suitable use codes but 1st of all if you talk about the that only involves that only involves involves only involves operators you know that this is just a classical error and so this error should be corrected by the classical code so the graphs that belies the elements are not involved in the use and that there and so basically there is a simple bound that the quantum code that you get out of a classical code the distance is always less or equal that of the classical code now if you have a nondegenerate CWS cold then you can also show that its distance does not in you does not in the the distance of the graph states that they defined earlier this is the minimum weight of this that belies however in the case of degenerate seduced calls in this situation becomes interest and so basically you can show that if there if there is a bet that is involved in the classical cold and that means 1 of the code vectors is nonzero on that this then the distance of the this goal cannot exceed the weight of the corresponding to the weight of the correspondence that the wiser and OK and so on and so if you combine these school then you sort of understand that if you have a graph with vertices of a given degree of then the distance of a cold does not exceed plus 1 and so in order you know these are all sort of the same degree and so if you're looking at code that are generated by some nice lattices like for example status you would know for a square lattice there are 5 of therefore neighbors so the rate of this as operator is 5 and so the distance of the CWS code that you can get out of square lattice can never exceed can never exceed 5 and so than typical if you if you want to find called a Flores distance then you would be looking at sort of very graphs have lots and lots of flags and then finally also that if I yeah this is sort of you get the bigger the distance cannot be bigger than the minimum weight of the
10:29
stability of the well just give you sort of a few examples to give you a feel on this cold so the fiveonthree cold is generated by this graph you know and then they are the only cold war this just 1 1 1 1 so the classical code is the repetition code and the graphs that letter generator at these and then of course you can you know this is a petition cold so the corresponding that the lies of generators of the you would be binary code and then knelt direct always 6 1 3 colds 1 you just add an empty qubits to the 5 3 cold and that corresponding graph has 1 disconnected vortex and sort of that you basically is not used in the cold but the 2nd 6 1 3 code is generated by this binary cold and and this graph and so this is called the school this gender and so if you take the product of this tool stablize of generators your I believe they share all of the all of their neighbors and so the product has you know just wait this is just little operator and so are the co generative above this graph distance tool but they just actual distance of the coldest 3 and then
11:46
this is 7 1 3 cold it is generated by this cold work in this graph and so I think it is interesting that whereas the code actually has psychic symmetry you know it's you know if you shift this is this that belies of generators psychically you at the business you know those would also also be members of this the blizzard wrote the nevertheless the ring is in and there is no explicit graph database this vicinity and there is no but binary code that is also symmetric with respect shown
12:23
with respect to cancelations so now we are looking for additive codes and so I'll just you sort of review a little bit this G of war maps of this vandalism codes and basic picture is that you have an error which is the the sum power X to some power and then you can map it you have to abide coupled binary vectors and further Tujia reactor which has this which has this 1 major and then the ball operators commute if and only if they are is this place product will become the corresponding vectors is is 0 and this is really really symplectic products and so 1 of the result is that if you are constructed and in the additive CWS called from ACE paddle as a cold and some graph there is a very simple expression for the correspondent generator matrix of this additive CWS golden remember generator matrix and you know the rows of the generator matrix corresponds to the generators of the growth of the study was the code so you have a parity check matrix and then you have graph adjacency matrix and it is very easy to chat you know that the graph adjacency matrix is symmetric so this is a simple graph then it's very easy to check that correspondent phrase product is actually 0 because you know it in just because of the structure of so this is so sort of the beauty of this decomposition is that you have a diction that is at the magical asked organelle so in the 1 of the biggest problem insertion quantum colds is ensuring the orthogonality condition and so 1 thing we
14:05
came up with so suppose you have a graph and it has big enough distance so that you can try to come up with gold and so so how likely it is that you can cut you you can get a good quantum cold out of it and so basically it turns out that for all of the Cook Cook calls that you can generate out of a given graph you control gilded where Shamir bound you can come up with the counter argument because that can be generated from a given graph just satisfy this in Gilbert version of and of course you know so this means that you can come up with codes you can always come up with codes that that are better than this as chairman that the upper limit on the distance is satisfied and short of what's nice about it is of course in is in some sense and state is a simpler cold than any Stadel as golden and so you can always come up with the graph states with which is has just those bigger than this and and therefore as a result this is sort of the divide and conquer strategy for fighting for fighting Sandel as a cold use 1st come up with a graph with a suitable distance and then you try to come up with the cold which has sort of building on the graph structure and well let me sort of in so this is the classical given so this is called rate this is just the straight and this is sort of classical Gilbert Parchem of all this is quantum field but quantum of want and this is a quantum field but version of Borland's force users coats but anyway so what's the point of this is trying to find codes which have you know good properties namely if you find it relative distance and find a great and so
15:46
now of course the easiest way to come up with colds is sort of start with a simple graph and codes which you can generate for on a lattice and so 1 of fine it's where years and this is 5 by 5 example you can actually very easy to come up with the cold and so the maximum distance of the code that involves all of the cube is for because of the edges and so you can actually make very easily come up with simple families that satisfied this so so this shows that Khot told of a cold 25 for 4 so you know and so this is so this the green red circles represent 1 of the classical so each work is accused the 3 year 4 circles represent the classical the classical code so this is the generator of the classical poet and then you shifted up left shift to right and you know and you can prove that the fate of a linear combinations is smaller than for just because you always have top bottom left right ages and so you can do it and then use says lactose is an infinite family of codes and then if you try to find numerical a cold then of course you can do better than that you but those schools stills are relatively simple and discuss struggle to
17:05
generalize this to an infinite lectures now where they looked at cycling codes and of course I too cold you generate from this psychic circulant matrix and the basic idea is that the the generators office that belies of can be sort of change cycler color you take the very the nth operator in the 1st place and then it's still be a member of this battle as a group and so the standard MAP foresight ecology will from circulant matrix to polynomial and then is sort of these polynomials are invariant under shifts sort of sort of you can you know for classical this for classical psychical this polynomial has become the all of of you invariance sort of the the sort of the the shift correspond to this multiplication by the apps and so for an additive CWS called you have this matrix and if you both P and R a circulant matrices that you can write most of them as polynomials and then the symmetry of this problem with circulant matrix correspond to this condition on the polynomial and so again this going to use the orthogonality and of course 5 1 called the polynomial P is 1 plus This is the parity check polynomial of the of the repetition cold and then the the effects is X plus X to the war this is symmetric with respect to this condition now you can think about aging what General psychic colds and in the paper g of codes over of for you know there is a statement that Janice have blue generator so this is a general case it turns out that there about generator psychic codes which have to satisfy this condition and are given just by the same equation just by a little bit less restrictive condition on so you have to multiply it by P and so this in class turned out to be very broad and so there are lots of wood codes inside of it and so what you can come up with is lover builder version of bound again for 2nd quantum codes because there are not so many binding not so many binary called look for binary safety codes it's easier to actually started the binary code and then look for a polynomial and so we can only prove would for a reducible polynomials and then 2 versions of it depending on whether or not the analysis synaptic or or
19:40
not and so then you can do existence proofs and so there are lots of calls that you can get out of the repetition Cold War generalized to petition to have this property and this quantum cold satisfied exactly the same bones and then there there are lots of gold the postponed to binary BCH goes so these calls actually you know provable they either saturate or you know possibly who you know what a process that they have parameters that the better than the than any than any colds that and also for to work all the same or
20:17
better and that sort of the last example is a psychic Doric like gold they promise show 5 1 3 cold is it is a is a Taurus and so basically if you enumerated it's like this 1 2 3 4 5 and so this is pure that 1 this is 2 3 4 5 then you can map this student on the plane and some useful thing to my conclusions and so there are families of such codes of how a positive part so the this article
20:48
thing doctors 1 question 2 questions so in the last slide you should at the at the bottom of this very wide class of codes of many different distances I was wondering if you could comment on this cold yes you I'm on some of the tradeoffs that come into play when you start looking at these larger distance codes all OK 1 tradeoff for sure is if you're looking for cold to don't have small rate but terrifying they 3 even binary codes get the penalty so you cannot have in in the sorry find that trade but small rate of this study lies of pride you these oppose that are easy to measure you know your opponent is example of course and stuff so voluntary offers that by having a bigger right codes it turns out you have to the rates that belies of generators there is no way out we don't have the bound that but but this is something that definitely plays out in the binary code and then you know over the world you know by this decomposition you can also show that this whole you know has been so for the quantum colds although sort of this probably even slightly worse so in you find a trade gold which you know K. over and you over and know he means fine they will have definitive have said the less of so for upper the of weight from for so this is 1 of the 3 these these are actually code are quite big it's a pleasure the 2nd speaker 1 which the
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Metadaten
Formale Metadaten
Titel  Stabilizer quantum codes via the CWS framework 
Alternativer Titel  Design of additive quantum codes via the codewordstabilized framework 
Serientitel  Second International Conference on Quantum Error Correction (QEC11) 
Autor 
Pryadko, Leonid

Lizenz 
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DOI  10.5446/35298 
Herausgeber  University of Southern California (USC) 
Erscheinungsjahr  2011 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Informatik, Mathematik, Physik 
Abstract  Codeword stabilized (CWS) construction defines a quantum code by combining a classical binary code with some underlying graph state. In general, CWS codes are nonadditive but become additive stabilizer codes if derived from a linear binary code. Generic CWS codes typically require complex error correction; however, we show that the CWS framework is an efficient tool for constructing good stabilizer codes with simple decoding. We start by proving the lower GilbertVarshamov bound on the parameters of an additive CWS code which can be obtained from a given graph. We also show that cyclic additive CWS codes belong to a previously overlooked family of singlegenerator cyclic stabilizer codes; these codes are derived from a circulant graph and a cyclic binary code. Finally, we present several families of simple stabilizer codes with relatively good parameters, including a family of the smallest toriclike cyclic CWS codes which have length, dimension, and distance as follows: [[t2+(t+1)2,1,2t+1]], t=1,2, ... 