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3D local qupit quantum code without string logical operator
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of my name is here from Caltech and today I'm going to talk about that in much the same kind of people with a similar property about OK so
00:31
here's a general picture of me entry barriers of many local it fun errorcorrecting codes so as we already probably note that in 2 D is a test for a cold and color codes we have part when citations so we can have is that at finite temperature but there will be a nonzero probability that the particles and antiparticles can be created and they'll just diffuse awaited out extra energy costs and if they make making non contractible who and somewhere but we do know at the same time of year the fourdimensional system which is the oratorical a which is a finite energy barrier and the way you can like visualize this is basically of the elementary particle it's at the elementary excitation looks like a cool stringlike excitation with a finite string tension so to cause the logical where you actually need to stretch out the course string all the way up until you of fill out the whole surface and that's where we get the the linear in respect and in this Treaty of the natural generalization of the lattice gage models success story codes can be basically thought in that pretty much the same way as in the same ways as into the but here it instead of having a particle and particle we have particles and string with since we do have a particle anyway but it was still cause a logical error and this is believed to be a general picture for a very long time until the last year of our John holiday introduced a new highend cold and as in the previous in 2011 brought in hot show that there is actually lot within the inner barrier for a trading loss where so just to have a brief recap of the previous talk about I just want to ask the following question which is enhanced code word the to lower the energy barrier for the logic where it comes from and the answer turns
02:20
out to be fairly simple which is the existence of constant aspect ratio so as you see in this picture on the top on the top the this guy and this guy's pretty happy with this was not happy and that is the reason why this guy's not so happy is because compared to the size of the anchor of the distance between 2 actors are too far away so which is not allowed like model and this is 1 of the defining properties of Aloha's model and we're going to look through models that the more general cause that's the same kind of property so the real question that I want to
02:54
ask was obviously are they're similar kind of colds and to think about this question we revisit how hard I found this cult which is to basically he found numerically after exhaustive search over binary stabilizer called with certain possible assumptions such as they have to commute and a translation invariant and they have to stabilize generators for each cube etc. and our approach is gonna be slightly different because if you take the samples were going write the same code anyway and the primary differences that we're going to search to reach you did stabilizer called instead she bits the and it turns out that the stabilizer formalism carries out pretty much the same way when the p is a prime number so we study you quantum called hence the explanation for the exotic title and here main differences between the cost code and our call that I'm going to present at the 1st difference is that the local particle dimension in our is to whereas in my code it's a prime number in general and hostile that he has 2 particles of her resin Oracle will have just 1 per cent and the 1st all we would open and it'll be the same kind of light as a cubic lattice but instead of having to on generators for hospitals which resulted a CSS call what does have 1 generator for each Q. so this would not give users
04:15
and basically forward and did there's a generic tool that is applicable and these had situations and the the firstinthenation notices that instead of a x operator so we should not pass through generalized shift operator and also a generalized phase operator z where omega responses these food of the unity and the date and satisfy the following quote unquote a commutation relation which is encoded by the and and the 1 thing they use remember is that the commutation relation of the generalized power operator not completely encapsulated by the socalled symplectic of between the symplectic pairs alpha and beta and alpha and beta here is actually a combination of 2 alpha 1 alpha 2 and beta 1 and beta 2 so you can see from this relation dead to generalize how the operators community Charter if and or if and only if that the simpler a part of it is equal to 0 model that the because then go on it to the indeed becomes exactly why and also for the multiples of the so our stabilizer generator will look pretty much the same as in a hospital as you can see we have a cubic generator and we have 8 on it's simply the pairs and as I mentioned in the previous slide after will be a general simplex pair which represents a generalized Haley operators and as in hospital we'll consider a translation of You in 3 different directions and also we assume a periodic boundary conditions and you can see from the previous slide that you is a general unitary operator and and when these equal to 2 the nice thing about the poly operators are there's a novel unitary but they're also permission but we don't have the luxury anymore in general be so to actually write down the 1 made by the Hamiltonian you actually have to permit ties unitary operator and hence the Hamiltonian becomes like following which is summed over all the cute and if you think about it and
06:21
this generic model will not result in any kind of meaningful cold because unless we unless we supply some kind of constraints on it for 1 thing since studying the
06:33
stabilizer cold all the stabilizer generated have to commute with right so that's the 1st condition that we're going to impose on the cold and the 2nd condition is pretty much all the property the defining property of harsco which is the absence of string logical and that basically there are many ways to interpret it but the way I of understood half paper willing to steps there there really 2 steps in showing the absence of string logical operator the 1st step is the socalled deformability conditions and those in the house language it was an eraser and the basic idea is that if we have a shot boundary and logical operator we can somehow smooth and out in 2 ways of in twin words smooth and surface and the next 1 is the constant aspect ratio which is that if we have the financing segment with the logical string operator cannot get too long so once we take it for granted for these 2 facts about a cold than the basic logic behind how you show that there is no string logical operators something like this so what suppose we have a string logical operator there we take a look into a particular segment of that and if it is too long because of the 2nd condition here we conclude that it becomes a does a trivial part and then we are ending up with a string with a opening but then we when the but open string and then we can apply the 1st condition with is disability conditions and the open end of the string will be very sharp so we'll just deform it away until we arrive at the trivial so the amazing thing about this is that 1st of all I alpha bar is the minus alpha model with the general it turns out that the commutation relation the deformability condition which was the 1st 2 conditions that I introduced in the previous slide on constraints in the stabilized generator in a very peculiar way and I have to make sure I have to let you know that that the data generated here and the generator here response to different cold out there in the the their continue different color and as you can see but there either symmetric were and the symmetry of against the under the inversion operation and there's also this extra constraint that the symplectic product between the the parameters of the on coal which is alpha beta and gamma delta should not be good which is a fairly on simple constraint that you can check by check with your hands so far from now on I will denote the cold that is represented by this kind of stabilizer generator as a symmetrical and the 2nd 1 is asymmetrical and as you can see it's basically represent life or parameters which is a science which are simply pairs so those are a number of altogether and before you get into the detail I just want to point out there is a problem in a very nice equivalence relation relation with the 1st of which is the lattice symmetry so if we rotate the lattice and if we end up with the same 1 called we should probably count them as the same kind of cold and it turns out that under those lattice symmetry actually of to hold 2nd candidate then applied if I if they can then be mapped into each other under the permutations of the parameters of the cold and also you it's to calls can be mapped into each other under the local unitary operation especially local FIFA transformation then again you can identify those 2 about colds as a same 1 was is again represented by special when your group of to like to special in your group right for a finite field the and there is 1 final rather exotic like full installation which is based in equivalence relation between the symmetric cold and the antisymmetrical and this is what was subtle because if you look at this symmetrical cheer in the antisymmetrical here you can imagine applying a Clifford operation on the 2nd 1 on the event's layer so in that case you will not delta bore into the delta and gamma pour into gamma and beta gamma bia ba into beta and help of war into alpha and then the next and and for the stabilize Genet behind this kind of rulebased map out fine to alpha barbarians better but since they're basically the same stabilizer generators because if you basically multiple because of this group they only differ by sign and modern matters is a connotation which so you can see that in the ball of there's a there's a correspondence between the symmetrical and antisymmetrical but it turns out that in general if we have a periodic boundary conditions especially the length time in the x direction here is on in that case we cannot support we can apply that argument anymore so it turns out that in that case those 2 culture not accept that actually identify
11:32
so to summarize a little bit under the lattice symmetry we can permit the can basically permute the parameters of the cold so from now on instead of writing down all the 4 elements of this denoted as a set which is a set of alphabet this set of 4 parameters alpha beta gamma delta and since thy there's a local FIFA transmission problems and I will get identified to search can be identified Tatars through a special linear group of element of a special when you the last 1 is a really important 1 because of basically when proving the absence of string logical operator remember there were 2 conditions which 1st of which was deformability condition and the 2nd 1 was the the finite aspect ratio as these 2 conditions we do not assume anything about the boundary conditions right which means basically to prove that absence of strain a lot we only need to study the ball property of the call so you can show that certain symmetry cold does not have string logical operator like we can like being the 1st 3 that there isn't corresponding antisymmetrical string logical operator so
12:42
this is the main result I sold the the basically says that the following 3 conditions on the parameters of the coal of guarantees an aspect ratio by word of forties 2 poles but important thing is that the ratio is finite because that's where we get the lover and again as you get there and you can see the 1st condition was a is the deformability condition which is expressed great nicely and the 2nd condition is something that we expect to happen CIO we're looking for a 1 colors that does not have any string lots of operator right and so it's only natural to assume that there is not any string logical operator would of woods particularly equal to 1 and what's amazing about this condition is that there's this 1 extra simple condition and that really combines all the all of them together and guarantees a finite aspect ratio and so basically given at 1 called that is represented by these 4 parameters you can easily check if it they have it their absent of string much about the couple observations that you can make hard turns out that the previous and thus the equation setting introduced in the previous slide does not have a solution when D is equal to 2 word PC but it does have a solution when these equal to 5 which is the following and it turns out that for sufficiently large number of the there's always a cold that satisfies the condition and also most importantly as constantly advertise these calls have evolved in an energy barrier for logic where basically even though are brought in hot result related was was about I binary stabilizer called the idea behind proving the lover and theory can be the exactly extended in the same way but that they
14:30
might be potential objections which is 10 maybe there's no introverted you to that all maybe that's why we didn't have any string logical and 4 there is a nice response that which to stand at least for the antisymmetrical there is a 1 encode included and the way you can see this is given an tubes there are m physical qubits and be and generators and there's at least 1 untrue constraint between the generators which is to basically multiple everything so take a look at the 2nd generator here and if we multiply everything for each vertex we have now for each symplectic pair alpha we have the exact minus of that which is half a block so alpha times and the generalized poly operator represented represented by alpha and the jealous poly operator represented by alpha bark when multiplied together becomes an identity and same thing for all of the rest of the 3 of the so that corresponds to 1 untruth constraint on the cold so we always end up having at least 1 encode acute in this error correcting so how does a logical for look like it's conjectured that there is a fact logical operated I have to admit this is something that I'm not really sure about at this point but when we change the system size it it turns out that the Bronsted generous changes as the system size to and appellate commutation relation between spectral operators seems to be the very heart of compute but they're very nice of logical operators which are not contractible services that have not to be commutation relations with each other and they have a commutation relation with each other this the intersection length of those services have it as an introspection length is not equal to 0 model at the word these 2 local particle dimension then these early examples and so the normal vector is up the logical operators are and an X or Y or Z direction so for instance if we try to compute the comic adulation between this guy and this guy it'll it'll depend on the intersection length between those 2 which is a single line so to conclude there is the large family of threedimensional local coals resembling the properties of possible so this tells us that the Harsco this matters of look this is a very general property of those with those kind of systems which have long been of the energy barrier which of grousing degenerative chains with system size and logical operators are our might affect all members but most importantly did not have anything and I want mentioned that there is something that I didn't really introduce through the paper which is that even though the the the smell apply to the course recalled they're out there are actually some numerical evidence suggesting that there are some people see prose with finite aspect ratio to be more particular fired aspect ratio with street but I couldn't prove it so would be desirable to have a proof for that particular and it would be also interesting see other chronicles with me on a different lattice with similar properties thank you Christians so you look take you picked up the vertices of course as young 1 house has to QB bits so have you looked at like combinations like that with tensor product is a technical you look at like all his new trick combination for example to cue tree is there was this stuff possible but I just wanted to start from the simple example after this initial motivation was this decent equals recall that I mentioned last and then I tried to generalize it to see if there are any calls that can improve the same thing but the general ideas yeah like the the idea for proving the absence of string logical operator is quite general it's just that but it depends on but what I can prove it or not depends on the problem of the cold so yeah I'll have to you actually see the cold to see if I can call that something similar were not part of the the method is quite general to be erected shouldn't easily or not easily them look at Hubert's competition so basically the way I prove this is to map the problem into a linear algebra problem which is basically kind of certain matrix is a matrix and reducing to that problem is trivial it's the same but given the matrix of proving counting the rank might be an easy problem or may not be a problem so that's where the difficulties coming as other questions and stick with
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Formale Metadaten
Titel  3D local qupit quantum code without string logical operator 
Serientitel  Second International Conference on Quantum Error Correction (QEC11) 
Autor 
Kim, Isaac

Mitwirkende 
University of Southern California (USC)

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  10.5446/35196 
Herausgeber  University of Southern California (USC) 
Erscheinungsjahr  2011 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Informatik, Mathematik, Physik 
Abstract  Recently Haah introduced a new class of local quantum error correcting code embedded on a cubic lattice without any string logical operator. We present new codes with same property by relaxing the condition on the local particle dimension. The resulting code is welldefined when the local Hilbert space dimension is prime. These codes can be divided into two different classes: the local stabilizer generators are either symmetric or antisymmetric with respect to the inversion operation. We lower bound the number of encoded qudits by computing the commutation relation between the logical operators confined on a plane. 