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Approximate simulation of quantum channels
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and we heard about this topic before I think this
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morning briefly and Cédric Finney is going to talk about approximate errorcorrection this time thank you so I think that did it still might be because that's the red tide designed myself that's where that so actually I don't my answer some of the questions that were opened value by that so what is so it's
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Chinarat reserve onion which is here so what is
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approximated you see in weight why should you care about that so let me 1st explain restates exact quinacridone so 1 way to look at it is the following we are given a channel which describe the happening to a physical system it that you into used to encode or trust me it's column information then the goal of the actions to find encoding genetical C and according to the oxygen what should and you know happen you correct and should the test the identity of code so this is exactly the because this inequality sign so he and the focus on is the easy part of this problem to not part is to find a that would the body is given a cold testing whether it is correctable meaning testing whether this collection exist OK so I will where arsenical is given I only care about is a supremacy from finding was underscored is corrected so society because it's been solved by in a different conditions they talked about debt that is easily whether or not a given code is correctable for a given nice no new however we can never really expects to live exactly say that in the code so this the lefthand condition which is proven to be a necessary is exactly is but buried in the research is where we for know smaller so that's 1 reason why you might want to look at this it's a case of course is a reason is also to the control on as a possible electric arrows imaging when you errorcorrection but the major sports this strike important is that this new language and in the real world map is just too strong you might be saying of cold will actually work so 1 indication that this is the case is this current
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thing so if you use only Latham condition you can prove that he she eventually got qubit then it is more easily and for the more in the but there wasn't and to read and Get corrects begetting put any quantum information whereas this work His work showed that but should have 4 voices Moerner which tends to 0 as goes to infinity then you can correct almost so that's that adopted failure of the different conditions so therefore the modulation of the maybe you mention a few best work
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which relates to whether we talk about so in this work he has a study the listed in the privative QC so it's approximately where the goal is that there is a the the most about what bromide Europe and the goal is to find constant for correcting its up just in order in the small parameter and the show that the new which is actually sufficient in this context to could talk to grow to a given order if you we applied to the right and the patrols I will come back to this later in this work it is shown that the toast portion which is called of version of the Bayesian inference channels if you wish is is near optimal for approximating QCD if you measure the error in the movement and once you do so again this in the next finer this is the just was also user He's a couple of an exact core errorcorrection to as the next Oculus bookable source portion of local too much more with this and on which was was who gets a similar about just enough to the 1 that will I would show using a very different to me so before I continue I would actually look at the premise is much more general than the error correction so again this is error correction finding see in Austin that is the identity I will replace edited by a nation so it's a lot more general so the goal is and given and defined according according to us to see me at in this way this stems simulation has been used in the in the field of 1st Shannon theory licensees annotated instances error correction if musical starch and this is trying to make a private school and the families of died if is a deterrent then this is trying to find it ethical code youth correct ConAgra fact all the indices of subsystem the so is original so again here we at look at that problem some given C N E so I cool see equally and because it will never separate them as it will just to find even and ends find was a exists at a and so this is conclusion addition mean that I do and then the to be equal to segment so this is exact national at 1st as the session and then later we'll do the approximate from which we all the same so I will do some editions of the postprocessing order so if if exist in the question and we say that n is more effective than N and write that and it's not and more informative know indeed I'm not trying to look for an encoding fast extends this this aggression is interesting in itself it the it's a situation where we have no access to the encoding so we only offices it according and the idea that you are trying to do use offer physical of you we are determined to get us some information from something that's we observe in it can only try to recover as that I began societies got this audit activities what kind of information the channels of preserving above the source so what I mean by a more informative of what is a partial order because sometimes and the truth that all in this sword of also I will say that this as I can if there are each more informative and then the other so in this OK we only care what channel up to the solution because that's about what kind of information as they trust me so word that introduce the contagion
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also well educated is if a channel is given by U 2 interactions environment chicken was due to the testing transformation then the resisted of the environment at the full truth which they please as a confrontation the Richard Corps and had so when a worker to hats that an example of a call rotation to to the so for instance if n is unitary so preserves all information then and hats is a constant from what we the phone corporate Ethernet you make also part of the Ivan a 1 way to the domain and had is to to see that is due out applied on metrics element will these are basically the space then you get this demi adiga EJ which is a term that and other news from collisions and that's that's no coincidence received so
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1 can't resistance and so I will I will give an approximate version of this and we were the proof for different initiative prove that lecture so if a that's end is more informative and if and only if an hat is less informative and and that's so this is a definition of reality that that it was talking about between the system and the environment for instance so this is a generation of nephron condition for instance issue this and but he attended the river that finding whether n is more informative than and is trying to ask whether an is correctable because of course what s also lessons what is and the identities of these just asking whether n is similar to the identity and this say that is true if and only if and hats is less intuitive and the trace of addresses a Gaussian channel with upper dimension 1 so you could have any constant in here and of course is asking whether or not it is acidic of traits and as the city readied dying component is an 11 college again it says you can correct the channel if and only if the environment gets no information that you would see so as
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we would at the end yeah no if you take to solve this problem taking whether any is more committees and M then this doesn't look easier because been to arbitrary channel or relief and has a nice that you're like being for effective meaning that if you buy and that's why Sigurdur sensing as being at once then again so the right hand side so as we see what it means for and that's to be less likely than and that so we will define these and it turns out that's going to waste it ought to be and that's that the we solve this problem so this is the tests to check whether and is more workers and this is an great it is we can convert it directly free new and and as soon as a form of another way of writing and 11 couldn't indicate or easier it more generally example aware and interpret dollies when and projects on another Russell where you started writing universe space you which and I wish for the country and is that the case then that projects on the committee so if you don't know it's not important this is an example so for instance if you had a right is a fact meaning that it acts only on 1 such systems then a this condition gives user condition for a
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place for a subsystem errorcorrection was the transition of in Latin collision to such a simple so
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normally going to separate the addition of these present of so 1st I would have to corner 1 definitively between channels that 1st and to define the the the state's Soviet given to states role in row prime then took over the city the in that in that it will fit so I you I proof I rule into PCI in raw prime to 5 prime and then I computes as the overlap of the 2 states but mathematics old tariffication so although a segment that you auditory acting on the system that I don't care about so this is your thinking in but we as as secret or as a dense to if you wish so fiction this 2 channels so article by the so 1 way to do it is to take states the role put it into both channels and then call brings you which states with the date so it's almost worked and here except that's the putting oppressed it's the provision of role and acting on 1 side is was channel and then could bring with the this is the same thing as a primer the to the state that obtain reserved directive here the morphisms from the channel this is a nice way of copper and which you can also have the figures so which 1 wins I call but you can also make a distance with meticulous wrote of 1 but it depends state but which existed so this this is a
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filtered version of the prior history might give you so what is this this is trying to find a child isotopes any the school possible to sweet and was yet and again to be supportive of finding the ultimate for approximate error correction and this is a deal so it implies the inner abnormalities defend and lose the court room very general enzymatic editions of his you the other side so as a proof that the proof Lantos's graphical language before so this year that's a work which in which that we debates we use a certain religion to wars the element in this the same the so attitude and you get the lefthand side and if you read it in the other direction you get the right answer this is the end user staffing Division a for an interview users that for the size a division of roles and the tree is that if you really it's a shame there with this as the lefthand side then as a message of you quest for ultimate amusing over the provision of and this you prime comes from the definition of the futility where did you really enjoy the direction to get the right hand side then it is you Prime which is the provision of our prime this you which we go from the definition of which come from the definition of the day so I what and you've intermissionless that can lead me for now so now I'm going to get a better
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energy which is the whiskey so I mean I as the when and for all possible input status this is the worst case is better because it doesn't depend on in state because my thoughts and attribute
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it written up so that there is a that is the same as before and the proof is the same except that as a norm minimization in between the 2 maximization so we have to use the minimax theorem to swap things around and that is that it's inequality which is quite remarkable so again
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what's less is this all of the is that we can do the same thing before you and that is a
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Pareto we can solve the dual problem so that was thing on the right hand side of the equation in that there and it's easy to prove that it is bonded so this is bonus trivial this weighted to prove using the triangle inequality Of course the metric that cost ability and of course you the entropy is this by the by the pretend approximates channel simulation and we did a good bond because this node motion affect all his nesting there and when it goes to 1 then you good this goes to 1 as well so this mean that this quantity here is a good guess for the optimal value here so for instance when the identities approximate column errorcorrection then we get this explicit form for this I guess for his good guess so here you see it doubt though is that now traced it you can put whatever that you want here and this in a cube use among the reason for this is that this is the here we have the channel which is complementarity identity but as a whole family of the Chern and in fact any challenges constant this commentators added so it always has the value of that cost function can choose whatever which 1 you want saddam what's in knowledge which element of the lake you stress of commendation the truth we soon emission of a role because this is the worst case the that this is a call except innovations which set of American so for instance if project under fact on derive and this this gives us 0 so of a good bond for of good estimation of the optimal affinity for approximate system called immigration and of course it just generalizes toward you as a person you what MIT might to force spaghetti height the families of trace or consultant and that this was some form of approximate prevent privacy very cold so we can even build this
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Nyrup dimensional so if we find the state role which minimizes expression which is called the and include sigma then you can build such correction channel which perform as well as about the which work in this way so what is this and hat and digress Sigmar it is additionally it's it's almost a duality that you channel but with see on both sides but we always choose a course but also an issue that I G it is almost on the jejunum and this overlay his body ages anomaly vision because it is not this is sitting there but it's not trace preserving this is a way to take any another make interest presenting and is only defined as genome was so this this course you should just comes out from a particle position of some of which this is why we get this universal let me just give you this 1
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application of this of this result so suppose we're better but the noise is operative edition so it's sort of a small parameter T so the 1st order of the sorry there was a while we assume that the 1st of the year that additional and then if we're always has been but for and if we just that they called good guess of what you killed correction to the the is an expanded as part of the book and then we get the necessary and sufficient condition
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for correcting its to 1st order and this condition is just as in the last Commission applied to the Linde AG workers and I and this is the necessary and sufficient well the fact that this situation is prove that affect necessary to be taught as the science equation which was posing newspaper an approximate error correction Our
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study different situations awful so that for America action is the full system interacts with reason environment or by according to reach Hamming thing and so that is the energy as a unit of energy and differences the strength of of Lamitan and this will is the net and the imagine that these Apatow's Jay's acts on the system and the case and the environment on the back so here's another particles have this form in a better at expression in excellent so Eckart was to them so that only leaves and zeros 1 has the identity complement and just start to order etc. so these guys in the eyes are the fortunate knows about of G. J and J. J where this this assume that we know where the initial state of the environment is and call it 0 and this is the environment the basis which includes this is there and so
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this gives us affairs that this sentence tractable to our actions there so it's always it turned out it was going to at and need to do anything but it's order to this greater direction where if and only if others in the left and was not despite forces guys these are you know as a Petros but we don't need to apply them to the next to the 1 which has the identity component and discussions on that only necessary but a sufficient but unnecessary so are slightly weaker than the so is better addressed proven that discussions are sufficient joint discussion including Zico there are sufficient in because we don't even know the student and firm additional acidity environment then this second week equations are not only the surrealist and that's
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all if at the questions because all this question is is about computational aspects of this so if if you if somebody gives you 2 channels can is even decidable whether 1 is dominating the other in that postprocessing water well so yeah I mean the that well it depends on the cost so miracle yes words that of social sciences so fine dimension right Sutherland in the special case where M is a projector and the Pareto then it's also it's easy to compute lazy depend I know it's clearly and yes to the general case general and it doesn't lead to a convex optimization problem why is it so in the case or whether it is there is no admission to do so it's just check inequality is evident case it just orbits of optimism for the socially is decidable them yes further questions no it's not that let's thank basically at
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Titel  Approximate simulation of quantum channels 
Serientitel  Second International Conference on Quantum Error Correction (QEC11) 
Autor 
Beny, Cedric

Lizenz 
CCNamensnennung  keine kommerzielle Nutzung  keine Bearbeitung 3.0 Deutschland: Sie dürfen das Werk bzw. den Inhalt in unveränderter Form zu jedem legalen und nichtkommerziellen Zweck nutzen, vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. 
DOI  10.5446/35289 
Herausgeber  University of Southern California (USC) 
Erscheinungsjahr  2011 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Informatik, Mathematik, Physik 
Abstract  We generalize the KnillLaflamme condition to the problem of using a given noisy channel N to approximately simulate another given channel M. In particular, this yields a way to compute a nearoptimal error for approximate subsystem quantum error correction. For approximate QEC, this also yield a nearoptimal recovery channel. We use this result to derive a simple necessary and sufficient condition for lowest order perturbative QEC. That is, given a noise model function of a small parameter, to determine whether a given code can correct it up to lowest order in the fidelity. 