Universal topological phase of 2D stabilizer codes
Bombin, Hector
We show that 2D topological stabilizer codes can all be characterized in terms of a finite number of charges and string operators. This is true either for subspace or subsystem codes, and it has direct applications for error correction. Subspace codes are directly connected to topologically ordered systems, and we show that all 2D topological stabilizer codes are locally equivalent to several copies of one universal phase: Kitaev's topological code.
 polygical phasing to destabilizer codes. So thanks, Daniel, for inviting me to this nice conference. This is a word that is a collaboration with David Poulin who couldn't make it to this conference in the last minute. And one of his students, Guillem de Closien, said he will be giving a related talk on Friday. So this talk is about topological codes. And well, since although we have already heard about them, let me remind you that the idea that in a topological code, qubits are in some sort of manifold. And it's at cost of the check operators that we need to measure to recover the error syndrome at low cost so that they just involve a few neighboring qubits. And at the same time, technical operators have a global nature so that any operator that has a support on a region that doesn't see the global structure of the manifold won't be able to recover any information about the encoded states. So in this talk, I want to take a systematic approach towards these codes and try to understand on their certain constraints, what is the general structure. In particular, I will be focusing on two dimensional systems and which are traditionally valid in a certain sense and defined by qubit systems with codes, the stabilizer codes. So topological codes are given as families of codes and in all the examples that we have, the code will have some bulk and then maybe there will be boundaries and other defects but it's always the case that this bulk is a translationary invariant. So our approach is to classify codes according to this bulk so that we can forget about all these different effects and so on. Now of course, such codes are closely related to topological order models so in particular I can consider such a Hamiltonian which is a sum of the stabilizer generators. And the idea is that the ground state correspond to all these guys having a game value one so that the ground state, it's our ground state, the space of ground states, it's correspond to the code subspace and the excitations correspond to error syndrome. So whenever one of these negative game value, I will have an excitations with two units of energy. The ground state is of course, the generate just as a code and from this property of topological codes, we get that the ground states have a locally indistinguishable. So this is basically, this is a property that defines topological order and there is a nice perspective on topological order introduced by Cheng-Wen Wen. So what they say is that topological order describes the equivalent classes defined by local unitary evolution. Let's see what they mean by this. So what they argue is that if we have two gap ground states, they will be in the same phase, so only if they can be connected the adiabatically without closing this gap. And then this amount to them be connected by a local unitary transformation or for us to the existence of some quantum circuit of constant depth, constant in the size, with respect to the size of the system. Okay, so more pictorically what they are saying is that in order to classify topological codes or topological order, we are doing this classifying long range entanglement patterns. So for them, a short range entanglement pattern is one, so it's something, some state that we can transform for local operations into a product state. And other states will have long range entanglement and will be, so these long range entanglement will be defined by this equivalence classes up to local transformations. So we have two main goals. The first is to understand the structure of two dimensional topological stabilizer codes, including subsystem codes that are not so closely related to this topological order that I was discussing. And then in addition for sub space codes, which have the Hamiltonian analog, we would like to see that whenever they have the same same structure in a certain sense, we can transform one into another through local transformations. As we should according to this prescription by Wien and collaborators. Okay, so here's the outline. I will start explaining the role of onions in the topical very briefly. Then I will go to the main result, which is that the third code is somehow universal for two dimensional codes. And then I will jump to subsystem codes which have quite a nice algebraic structure. Okay, so what is special? Well, there are several things special about two dimensions, but one of them is that there are two physically distinct ways to exchange two particles, either identical or not. And so the idea is that if I exchange them in this way, in this way I cannot, the two processes cannot be the form one into the other without the word lines crossing. So this gives rise to the possibility that there could be some topological interaction that distinguishes between these two paths. And the particles that suffer such or are subject to such topological interactions are called anions. So the simplest case is that of abelian anions which are all we need. And such models are described by three elements. So we have first of all a set of topological charges. These charges are the equivalence of sort of excitations. So if I take a given region and I have a given configuration of excitations, I could apply some local transformations and get another set of excitations. And as I move, as I do this transformation, I will move in a certain equivalence class. And these are the labels for these equivalence classes. So that charge is something that is conserved in a region unless it's exchanged with some other outside region. So the second, oh sorry, whoops. So the second element will be fusion rules that tell us what is the total charge given of two regions given the individual charge of each of them. Okay, so in the abelian case, this amounts to basically a product. Okay, so the diffusion rules will tell me what is the total charge given these two charges. Okay, and finally, there are some rating rules that tell us what happens as long as topological interactions are involved when I sense two annuals in one of the two possible ways. Okay, so let's see this in this very relevant example, which is the Kittar story code. So I would like to emphasize that in the story code, we can visualize poly operators, which are products of X and Zs, graphically, so that an X will correspond to a direct link and a Z to a dual link, link in the dual lattice. And this way, both the kind of terms in the Hamiltonian or the check operators can be regarded as plaquettes. Okay. So two kinds of excitations. Yes, excitations in this model, we have excitations in dual plaquettes that can be created by such syn operators, which are products of X operators here. Okay, so the idea is that whenever I apply such a string, I will not sense the even or odd number of, whether it's an odd or an even number of such excitations. And similarly for the direct plaquettes, sorry, this would be a direct plaquette, it's neither a dual or a direct plaquette. Okay, and the idea is that the total there is, there are four charges, so that I could have an even number of both dual and direct excitations, then I'm in the trivial sector. If I have another number of the dual ones, I say that I have an electric charge. If the odd number, if I have another number of direct ones and even number of these, I have magnetic charge, and then finally there is a cut, which corresponds to the case where I have another number of both. So basically this already includes the fusion rules because any charge composed with itself will give me the trivial sector. And I only need the topological interactions. So to recover them, we can use the fact that these charges are created at the end points of these strings, and recover the topological interactions from the commutation properties of the string operator. So if this is a sort of, let's see, I guess magnetic string, and this is an electric one, they will anticommute, and then I can consider two processes that are products of these string operators. And when seen in time, developing in time, one of the processes describes, so in both cases, the two electric and the magnetic charges are in the same place, but in one case the pads are, the war lines are linked, in the other case they are not, and we can see that a minus and appears differentiating the two. So this, the last that these two charges have are simulations. Okay. And then we similarly can recover the, the self statistics or topological spins with tell us what happens when they exchange to two charges of the same type. So in this case, we will need three strings with a common endpoint. So in the case of electric and magnetic charges, they are both, so it's not very, a lot of fun, but the composite tends out to be a fermion, and this can be checked this way. So notice that the composite will be moved from one place to another through such machine operators, which is the product of an electric and a magnetic strings. And I take three of these, and then by the ideas that I have, and in here, one in here, and then by ordering these operators in two different ways, I can either exchange the two excitations or leave them where they are. And the difference again is a minus sign. Then of course we have the, they are fermions because when you exchange fermions, you get a minus sign. Okay. So the main message is that we have some Hamiltonian or some code, and we have extracted an abstract new model from it. In this case with four charges, there's confusion rules that correspond to the C2 times C2 group and then some braiding rules. And everything can be recovered from the string operators. This is the main message. Okay. So now I can jump to the main results. So first thing that we need is to abstract ourselves from a family of topological codes to a single stabilizer group that's defined in an infinite lattice. So, you know, for this family of codes, there will be some recipe to construct the bulk and we just apply it to the infinite lattice. Now, of course, this, this resulting stabilizer will have some properties because it comes from a family of topological codes. It cannot be just any stabilizer. And we must remember that in a topological code, local and detectable errors should not affect encoded states. It's not a case of putting bit of a political and let's see what that translates into. So if I have some polar operator here in the infinite lattice and it commutes with all the stabilizer generators, we can find an element of this family, infinite family of codes that is big enough so that this guy fits in the bulk. And then it will also belong to the centralizer of the stabilizer from which it follows that from this code being topological that p itself belongs to the stabilizer. And then, well, without going into details, it's possible to show that either p belongs to the stabilizer here or we can assume that it does actually. Okay, so the message is that we have this condition that the centralizer of the stabilizer is up to face of the stabilizer itself and this is the topological condition that we impose. So these are the two ingredients, translation and invariance and this topological condition. And then we want to classify all these guys. So we seek to be able to relate codes through local transformations and indeed it turns out that it's enough to consider the following ones. First we have a cross-graining of the lattice which basically amounts to doing nothing, it's just looking at the lattice from a different perspective. We have local and translation invariant key for mappings so that we map the polygroup to itself. In fact, locality in this case in the evening lattice follows from the translation invariance. Given that I am only considering the polygroup to consist of operators with a final support. And then finally I would like to be able to add and remove this entangled qubit. So qubits that have their own single qubit stabilizer so that from the perspective of the topological code they are not interesting. Okay, so it's easy to see that given these three transformations by commanding them, one will, so if my code has a certain annual model, suppose then this annual model will not change through these transformations simply because it's described with the stenopedors that can be arbitrary large and so on. But the interesting thing is that the contract is also true and whenever two codes have the same annual model, they can be transformed one into the other through these basic operations. Then if we rule out the possibility of so-called kaianonions which we can do according to the guitar. Then when I rise to the conclusion that, but I will give more details that every two dimensional topological stabilizer group in this sense is locally equivalent to a finite number of copies of the topical which is our main result. So let me give some clues about the proof of this. So well first of all there are two technical boring results but very important. So first that in 2D for these stabilizer groups, we can find a set of independent generators. Okay, so that if you translate it into the symplectic vector space formalism, I'm just saying that there are no linear constraints between them. So the thing is that in 2D we can find such a local and translational invariant independent generators and this is actually not true in 3D. In the proof it also will use the fact that these stabilizers are the centralizer of something actually of themselves. So if that was not the case maybe it wouldn't be true. This property, it can be not true. So this is the first technical thing. The second technical thing is to show that the number of charges is finite. And once you have this it's very easy. You just obtain the annual model by constructing the stream operators and checking what are the commutation properties and so on. And then once you have these stream operators you construct a framework, a series of lattices made of stream segments, we'll see how. And the framework of stream segments, the resulting the placate stabilizers are, give all the, give generators for the stabilizer, give all the generators for the stabilizer that have charge and then the rest of generators have no charge. And then the mapping between two codes where we have done all this, good amount to map these stream segments to the corresponding stream segments with the same charge and all those extra charge stabilizers, we map them to single qubit stabilizers. So they are the capital from the code. So some small details. So first of all we need to cross grain the lattice till at any site we can find any possible charge until whenever we have two sites with excitations here in this site and this site with a given charge that is equal and we can path it to them. If we cross grain enough we are going to be able to construct an operator with support in this gray region here so just a bit thicker than the path such that it creates these two excitations. So this is what we mean for a stream operator here. We can define them for arbitrary paths. And once we have stream operators we can recover motorized statistics and the vertical spin with a trick that I already explained for the third code. And to see that this actually makes sense, consider for example that we have these two pairs of strings so they should have the same commutator. And to check it we can extend the picture of it. And now I have these closest strings since they have no endpoints they create no excitations so they belong to the stabilizer up to a phase. And therefore the two of them commute the two commutators are the same. And thus this guy here is well defined. It's independent of the choice of the strings. And similarly for this other guy here. So once we have our new model and the two little interactions we can find a set of canonical generators for the group of charges. And this is what one finds. So there will be a certain number of pairs of generators. Each of them. So this charge here will have some unique mutual statistics with this other guy. And both of them interact with any other generator. And then all of them are bosons as that maybe the two last ones which could be both thermions. Now then each of these pairs corresponds to the anions in the third code. Okay so we have several copies of the third code and then maybe this kind of part which. So then we. In any case we can construct several largeses of operators. Okay so this will be large operators after all the course grading and everything. We'll have some string operators here forming a square larges for its charge and each of these largeses. So they are displaced a bit with respect to each other. And then the reason to do so is that we want the commutators to depend between all these string segments. We want them these commutators to depend only on the annual model, not on details of the strings. Okay so that's the reason to do it that way. So we end up with this collection of string segments that have commutators that are independent of the original model. Only the model. And then in order to do the mapping we will just, so here we have two codes. In each of them we need all these processors so we arrive to these largeses and then for each larges. So for example I will have this, first say electric charge and the corresponding one in the other code and I will map string segments to string segments and then it's possible. So there will be, apart from the stabilizers coming from from plaquettes I will have other stabilizers but they have no charge and they can be mapped to single qubit stabilizers. I will not go into any more details. So let me move on to the subsystem case. So well we have heard of this but just very fast. In subsystem codes rather than encoding in a subspace of the Hilversys we will just take a subsystem. These are our logical qubits and then each qubit about which we don't care. And in the case of, in the stabilizer formalism as described by Poulin we are still described our subspace through a stabilizer but then in addition we have a gauge group whose action on that doesn't affect in code states. And the important thing is that the stabilizer and the gauge group are related through these conditions so that up to phase as the stabilizer is the center of the gauge group. Now in the topological case what we mean for a topological stabilizer subsystem code will be a family of codes it's with a stabilizer and a gauge group and both of them have local generators. Now again we want to extract the recipe for the bulk and put it in an infinite lattice and obtain some stabilizer and some gauge group. As before we want to impose a topological condition on these guys and in this case, whoops, in this case if I have Poulin operator here that commutes with all the stabilizer generators. When I carry it to a large enough code it will also commute with all of them therefore it will belong to the gauge group now. And one can see that again this in place that we can assume that the gauge group here contains this Poulin operator. So we are left with this condition that the gauge group is the centralizer of the stabilizer so it contains all the Poulin operators that commute with the stabilizer generators. But once we have this condition we could just as well define the gauge group through it and then we get a condition on the stabilizer that guarantees that our, well not guarantees but that we need to know that becomes our topological condition. So it doesn't get grouped anymore and it's that the centralizer of the stabilizer is in the sense of stabilizer up to phases. So of course this would be trivial for a finite number of qubits but it's not necessarily true in the case of an infinite lattice. Okay. So the main obstacle to move to such a gauge goals from such space goals is that we need to generalize the notion of charts. So the trick is to make it a bit abstract so remember that charts classifies equivalent classes of excitations up to trivial sets of excitations that can be created by applying some Poulin operator. And this we can extract a bit by considering that our configuration of excitations are actually or if you want the syndrome, possible syndromes with morphisms from the stabilizer group to C2. And the trivial morphisms will correspond to those that can be obtained as a commutator with some Poulin operator. So this quotient will give the group of charges for the stabilizer but not that we have this sort of slightly more abstract definition. We can generalize it immediately to the gauge group. We just substitute S with G and we have a new group of charges that don't have the same sort of obvious meaning in terms of states or anything like that but are very useful. And we still have the old charges for the stabilizer group. So we have two sets of two groups of charges and we could as before try to define topological interactions between them. So consider for example, so in order for these interactions to make sense, remember that we have this trick, these two closest in operators that have to commute so that it didn't matter which pair of strings I took to define the sort of interactions. So consider for example, that they have, that the charge of these strings came from the, from the, from the, from gauge charges, right? So it's a gauge charge. So in that case, having a closest string means that this guy will commute with all the gauge operators and therefore it will belong, this closest thing will belong to the stabilizer. Similarly, if I have a string with a stabilizer charge, this means that the closest thing will commute with all the stabilizers and therefore it belongs to the gauge group. So in this case, the two strings commute and the two commutators are the same. This would also be true if I have taken two gauge charges because then I would have two stabilizers, they commute. But if I, if I laser charge another stabilizer charge, I will have two gauge elements but then they don't necessarily commute. So I cannot define such topological interactions in that case. So what we are left with is that for two gauge charges, we have the, we'll define a mutual statistics, we have topological spin. And we also have, whenever I have a gauge charge and a stabilizer charge, I also have this, this mutual statistics. But nothing else. So I don't have a topological spin for stabilizer charges nor interactions between them. So really I have a nanomobile for the gauge charges and then I could regard this as some sort of board basis for these anions. Good. So of course, I come up gauge charges to stabilizer charges simply because amorphism from G to C2 can be restricted to amorphism as a to C2. I will graphically depict this this way so that a gauge charge is a circle with some color and then the image will have the same color and square shape. And it's, well, this is a natural morphism. It doesn't depend on any choice of generators or anything like that. And it will preserve this, this commutator so that if I just pick up two gauge charges and I map one of them, I will get the same commutator. Of course, I cannot map the two of them because then the thing on the other side is not well defined. And as in the case of SASPACE codes, again I can find a canonical set of generators for the gauge charges and for the stabilizer charges. Let me first discuss the gauge charges. So we have first sector that looks very much like what we've had in SASPACE codes, several bosons, and then maybe a chiral pair that now can actually appear. And in addition, we have something new. We have several bosons and maybe one fermion. All of them non-interactive, so they are not, they are not annuals. And then for the stabilizer charges, we can find the same number of generators. And this is because actually there is a duality between the two. So well, this would be ordered in a different way. So the thing is that this stabilizer charge interacts only with this guy but not with any other one. Similarly, the black one here with the red one here, so one for these guys. And then I have these stabilizer charges that are not the image of anybody here. And they are dual respectively to the charges with the same number. Through the morphism, if you want this map to the trivial charge in the stabilizer. So there are many annual models that we could have, but all of them can be obtained by combining four basic ones. So the first of them correspond to what we find in the TORI codes with these two bosons with unique mutual statistics. Then the second one is the same but with two fermions. So that, well, the three non-trivial charges are fermionic. And this can be found in topolecule subsystem color code that we discussed by Robert yesterday. We could also have a single boson. This can be recovered from what I call the subsystem TORI code, which would be a TORI code where to the stabilizer, to get the gauge group, we add all single qubit X or C operators, whatever you want. And we can also have a single fermion. And this is what we get if we look at the Kittives HONICOM model as a subsystem code. So in these two cases, actually, we have no encoded qubits. But this fits into the classification. And actually, it's important to keep it, even though it's not useful from the encoding point of view. So once again, we can construct a framework of single operators. And we would like to recover the stabilizer and gauge generators as plaquettes of these largeses. And we can do so. So in the case of the gauge group, it's possible to find a set of generators for the gauge that include these plaquettes. So this would be coming from the largeses that where strings have this chart. And this is the dual chart. Well, I should better say it. Can't you get this case? So the thing is that this is the chart of this plaquette in this basis and this set of generators, similarly for this one here. And then we also have generators that have these charges which are not unique. And they come from plaquettes made up with strings that have a chart that corresponds to these guys that were not dual to this other gauge. And similarly for the stabilizer. So the idea is that the rest of generators of either the gauge or the stabilizer will have a trivial chart. OK, so the idea is that with this, we put the codes in a standard form. And therefore, we can see that all codes can be understood in terms of these anions. In addition, we could put these largeses back into a finance system like a torus. And then once is, well, what do you expect that you have again, that the homology gives you the logical operators and so on. So each of these pairs will contribute two qubits in the code in the torus. And these guys contribute nothing. And in addition, they have a problem, is that when put in a finance largeses, we need to add either a gauge or a stabilizer generator that are global. And you need to add one of these generators and per generator of the first homology group. Yeah, so the lesson is basically this that we can understand two-dimensional topological subsystem codes in terms of anions. And yeah, so with that, let me just leave some questions. So first of all, we have been considering just the bulk. But we could also look at the defects in the codes. So we could have a code with several different bugs put together. Either of these could be just a vacuum at the end of the code. So we have boundaries and we have point defects. But we can also classify this potentially using this tool. So we could go to a setting where we have two semi-infinite planes each with a different representing different bulk. So this is translationally invariant. Then we have a boundary that is translationally invariant in this direction. And then we have some point defect. And this way, we could classify all such possible things up to local transformations and so on. Of course, we could consider more general two-dimensional codes. And finally, maybe the most exciting is going to 3D, because as we know, as we learned this morning, I think, there are much more exotic things waiting for us there. So the classification is more than just homology. OK, so that's it. Thank you. So questions? I have a question. So what's the defining property of the 2D topological stabilizing code? So do you assume the locality of the operator and find a distance, because you didn't specify it at the beginning? So all you impose is the locality of the stabilizer generators. And then that, so you will have a family. And then you impose that any operator. So you could do it in different ways. But the idea is that there will be always a large code in your family, so when you go to large enough number, where any operator will be local in the sense that it doesn't affect that. And so if your coarse-grained operators are limited in width by their distance? So the coarse-graining we only do in the infinite lattice and so on? Yes, but they are limited in size by the locality of the stabilizer generators. Do you mean whether? What's the limit on the required coarse-graining? Oh, OK. Let's see. So there is a limit, but yeah, I cannot tell you. There are many exceptions to proof that are just like, OK, there is some coarse-graining that will give you this. But it's a bit involved. So there are always, I mean, you can put a bound on what coarse-graining you are going to need depending on the size of the generators, but I cannot tell you like, expression. It is possible to give it. For the standard stabilizer case, was the condition that the centralizer of the stabilizer is modulo phase is the stabilizer? In the infinite lattice. Yeah, so yeah, that's exactly what I'm asking about, sort of the encoded operators. I mean, what about this? I mean, do you put a sort of a size? I mean, is it on the entire group or elements of bounded size? So this appear, encoded operators will appear when you go back to some finite code. Yes. But not, of course, in the infinite lattice. So the infinite lattice is just a picture. The idea of going through the infinite lattice is that everything becomes local. So everything is automatically local, so you don't have to care about. That's why I wasn't careful in defining locality or anything. So the thing is that whatever definition you take, it will basically work. And then once you go to the infinite lattice, everything is local. And then it's easier. OK, thank you. Other questions? I think the speaker. Thank you.
 polygical phasing to destabilizer codes.  So thanks, Daniel, for inviting me to this nice conference.  This is a word that is a collaboration with David Poulin  who couldn't make it to this conference in the last minute.  And one of his students, Guillem de Closien,  said he will be giving a related talk on Friday.  So this talk is about topological codes.  And well, since although we have already heard about them,  let me remind you that the idea that in a topological code,  qubits are in some sort of manifold.  And it's at cost of the check operators  that we need to measure to recover the error syndrome  at low cost so that they just involve  a few neighboring qubits.  And at the same time,  technical operators have a global nature  so that any operator that has a support on a region  that doesn't see the global structure of the manifold  won't be able to recover any information  about the encoded states.  So in this talk, I want to take a systematic approach  towards these codes and try to understand  on their certain constraints,  what is the general structure.  In particular, I will be focusing on two dimensional systems  and which are traditionally valid in a certain sense  and defined by qubit systems with codes,  the stabilizer codes.  So topological codes are given as families of codes  and in all the examples that we have,  the code will have some bulk  and then maybe there will be boundaries and other defects  but it's always the case that this bulk  is a translationary invariant.  So our approach is to classify codes according  to this bulk so that we can forget  about all these different effects and so on.  Now of course, such codes are closely related  to topological order models  so in particular I can consider such a Hamiltonian  which is a sum of the stabilizer generators.  And the idea is that the ground state  correspond to all these guys having a game value one  so that the ground state,  it's our ground state, the space of ground states,  it's correspond to the code subspace  and the excitations correspond to error syndrome.  So whenever one of these negative game value,  I will have an excitations with two units of energy.  The ground state is of course,  the generate just as a code  and from this property of topological codes,  we get that the ground states  have a locally indistinguishable.  So this is basically,  this is a property that defines topological order  and there is a nice perspective on topological order  introduced by Cheng-Wen Wen.  So what they say is that topological order  describes the equivalent classes  defined by local unitary evolution.  Let's see what they mean by this.  So what they argue is that  if we have two gap ground states,  they will be in the same phase,  so only if they can be connected  the adiabatically without closing this gap.  And then this amount to them  be connected by a local unitary transformation  or for us to the existence of some quantum circuit  of constant depth,  constant in the size,  with respect to the size of the system.  Okay, so more pictorically what they are saying  is that in order to classify topological codes  or topological order,  we are doing this classifying long range  entanglement patterns.  So for them,  a short range entanglement pattern is one,  so it's something,  some state that we can transform  for local operations into a product state.  And other states will have long range entanglement  and will be,  so these long range entanglement will be defined  by this equivalence classes up to local transformations.  So we have two main goals.  The first is to understand the structure  of two dimensional topological stabilizer codes,  including subsystem codes  that are not so closely related  to this topological order that I was discussing.  And then in addition for sub space codes,  which have the Hamiltonian analog,  we would like to see that  whenever they have the same same structure  in a certain sense,  we can transform one into another  through local transformations.  As we should according to this prescription  by Wien and collaborators.  Okay, so here's the outline.  I will start explaining the role of onions  in the topical very briefly.  Then I will go to the main result,  which is that the third code is somehow universal  for two dimensional codes.  And then I will jump to subsystem codes  which have quite a nice algebraic structure.  Okay, so what is special?  Well, there are several things special about two dimensions,  but one of them is that there are two physically  distinct ways to exchange two particles,  either identical or not.  And so the idea is that if I exchange them  in this way,  in this way I cannot,  the two processes cannot be the form one into the other  without the word lines crossing.  So this gives rise to the possibility that  there could be some topological interaction  that distinguishes between these two paths.  And the particles that suffer such  or are subject to such topological interactions  are called anions.  So the simplest case is that of abelian anions  which are all we need.  And such models are described by three elements.  So we have first of all a set of topological charges.  These charges are  the equivalence of sort of excitations.  So if I take a given region  and I have a given configuration of excitations,  I could apply some local transformations  and get another set of excitations.  And as I move, as I do this transformation,  I will move in a certain equivalence class.  And these are the labels for these equivalence classes.  So that charge is something that is conserved in a region  unless it's exchanged with some other outside region.  So the second, oh sorry, whoops.  So the second element will be fusion rules  that tell us what is the total charge  given of two regions given the individual charge  of each of them.  Okay, so in the abelian case,  this amounts to basically a product.  Okay, so the diffusion rules will tell me  what is the total charge given these two charges.  Okay, and finally, there are some rating rules  that tell us what happens  as long as topological interactions are involved  when I sense two annuals in one of the two possible ways.  Okay, so let's see this in this very relevant example,  which is the Kittar story code.  So I would like to emphasize that in the story code,  we can visualize poly operators,  which are products of X and Zs, graphically,  so that an X will correspond to a direct link  and a Z to a dual link, link in the dual lattice.  And this way, both the kind of terms in the Hamiltonian  or the check operators can be regarded as plaquettes.  Okay.  So two kinds of excitations.  Yes, excitations in this model,  we have excitations in dual plaquettes  that can be created by such syn operators,  which are products of X operators here.  Okay, so the idea is that whenever I apply such a string,  I will not sense the even or odd number of,  whether it's an odd or an even number of such excitations.  And similarly for the direct plaquettes,  sorry, this would be a direct plaquette,  it's neither a dual or a direct plaquette.  Okay, and the idea is that the total there is,  there are four charges,  so that I could have an even number  of both dual and direct excitations,  then I'm in the trivial sector.  If I have another number of the dual ones,  I say that I have an electric charge.  If the odd number,  if I have another number of direct ones  and even number of these,  I have magnetic charge,  and then finally there is a cut,  which corresponds to the case  where I have another number of both.  So basically this already includes the fusion rules  because any charge composed with itself  will give me the trivial sector.  And I only need the topological interactions.  So to recover them,  we can use the fact that these charges are created  at the end points of these strings,  and recover the topological interactions  from the commutation properties of the string operator.  So if this is a sort of,  let's see, I guess magnetic string,  and this is an electric one,  they will anticommute,  and then I can consider two processes  that are products of these string operators.  And when seen in time, developing in time,  one of the processes describes,  so in both cases,  the two electric and the magnetic charges  are in the same place,  but in one case the pads are,  the war lines are linked,  in the other case they are not,  and we can see that a minus and appears  differentiating the two.  So this, the last that these two charges  have are simulations.  Okay.  And then we similarly can recover the,  the self statistics or topological spins  with tell us what happens when they exchange  to two charges of the same type.  So in this case,  we will need three strings with a common endpoint.  So in the case of electric and magnetic charges,  they are both,  so it's not very, a lot of fun,  but the composite tends out to be a fermion,  and this can be checked this way.  So notice that the composite will be  moved from one place to another through such  machine operators,  which is the product of an electric and a magnetic strings.  And I take three of these,  and then by the ideas that I have,  and in here, one in here,  and then by ordering these operators in two different ways,  I can either exchange the two excitations  or leave them where they are.  And the difference again is a minus sign.  Then of course we have the,  they are fermions because when you exchange fermions,  you get a minus sign.  Okay.  So the main message is that we have some Hamiltonian  or some code,  and we have extracted an abstract new model from it.  In this case with four charges,  there's confusion rules that correspond to the  C2 times C2 group and then some braiding rules.  And everything can be recovered from the string operators.  This is the main message.  Okay. So now I can jump to the main results.  So  first thing that we need is to abstract ourselves  from a family of topological codes to a  single stabilizer group that's defined in an infinite lattice.  So, you know, for this family of codes,  there will be some recipe to construct the bulk  and we just apply it to the infinite lattice.  Now, of course, this,  this resulting stabilizer will have some properties  because it comes from a family of topological codes.  It cannot be just any stabilizer.  And we must remember that in a topological code,  local and detectable errors should not affect  encoded states.  It's not a case of putting bit of a political  and let's see what that translates into.  So if I have some polar operator here in the infinite lattice  and it commutes with all the stabilizer generators,  we can find an element of this family,  infinite family of codes that is big enough  so that this guy fits in the bulk.  And then it will also belong to the  centralizer of the stabilizer from which it follows  that from this code being topological that  p itself belongs to the stabilizer.  And then, well, without going into details,  it's possible to show that either p belongs  to the stabilizer here or we can assume that it does actually.  Okay, so the message is that we have this condition  that the centralizer of the stabilizer is up to face  of the stabilizer itself and this is the topological  condition that we impose.  So these are the two ingredients,  translation and invariance and this topological condition.  And then we want to classify all these guys.  So we seek to be able to relate codes  through local transformations and indeed it turns out  that it's enough to consider the following ones.  First we have a cross-graining of the lattice  which basically amounts to doing nothing,  it's just looking at the lattice  from a different perspective.  We have local and translation invariant  key for mappings so that we map the polygroup to itself.  In fact, locality in this case in the evening lattice  follows from the translation invariance.  Given that I am only considering the polygroup  to consist of operators with a final support.  And then finally I would like to be able to add  and remove this entangled qubit.  So qubits that have their own single qubit stabilizer  so that from the perspective of the topological code  they are not interesting.  Okay, so it's easy to see that given these three  transformations by commanding them,  one will, so if my code has a certain annual model,  suppose then this annual model will not change  through these transformations simply because  it's described with the stenopedors  that can be arbitrary large and so on.  But the interesting thing is that the contract is also true  and whenever two codes have the same annual model,  they can be transformed one into the other  through these basic operations.  Then if we rule out the possibility of so-called  kaianonions which we can do according to the guitar.  Then when I rise to the conclusion that,  but I will give more details that every two dimensional  topological stabilizer group in this sense  is locally equivalent to a finite number of copies  of the topical which is our main result.  So let me give some clues about the proof of this.  So well first of all there are two technical boring results  but very important.  So first that in 2D for these stabilizer groups,  we can find a set of independent generators.  Okay, so that if you translate it into the  symplectic vector space formalism,  I'm just saying that there are no linear constraints  between them.  So the thing is that in 2D we can find such a local  and translational invariant independent generators  and this is actually not true in 3D.  In the proof it also will use the fact that  these stabilizers are the centralizer of something  actually of themselves.  So if that was not the case maybe it wouldn't be true.  This property, it can be not true.  So this is the first technical thing.  The second technical thing is to show that  the number of charges is finite.  And once you have this it's very easy.  You just obtain the annual model by constructing  the stream operators and checking what are the  commutation properties and so on.  And then once you have these stream operators  you construct a framework, a series of lattices  made of stream segments, we'll see how.  And the framework of stream segments,  the resulting the placate stabilizers are,  give all the, give generators for the stabilizer,  give all the generators for the stabilizer that have  charge and then the rest of generators have no charge.  And then the mapping between two codes where we have  done all this, good amount to map these stream segments  to the corresponding stream segments with the same charge  and all those extra charge stabilizers,  we map them to single qubit stabilizers.  So they are the capital from the code.  So some small details.  So first of all we need to cross grain the lattice  till at any site we can find any possible charge  until whenever we have two sites with excitations here  in this site and this site with a given charge  that is equal and we can path it to them.  If we cross grain enough we are going to be able  to construct an operator with support in this gray region  here so just a bit thicker than the path  such that it creates these two excitations.  So this is what we mean for a stream operator here.  We can define them for arbitrary paths.  And once we have stream operators we can recover  motorized statistics and the vertical spin  with a trick that I already explained for the third code.  And to see that this actually makes sense,  consider for example that we have these two pairs of strings  so they should have the same commutator.  And to check it we can extend the picture of it.  And now I have these closest strings  since they have no endpoints they create no excitations  so they belong to the stabilizer up to a phase.  And therefore the two of them commute  the two commutators are the same.  And thus this guy here is well defined.  It's independent of the choice of the strings.  And similarly for this other guy here.  So once we have our new model and the two little interactions  we can find a set of canonical generators  for the group of charges.  And this is what one finds.  So there will be a certain number of pairs of generators.  Each of them.  So this charge here will have some unique  mutual statistics with this other guy.  And both of them interact with any other generator.  And then all of them are bosons  as that maybe the two last ones  which could be both thermions.  Now then each of these pairs  corresponds to the anions in the third code.  Okay so we have several copies of the third code  and then maybe this kind of part which.  So then we.  In any case we can construct several  largeses of operators.  Okay so this will be large operators  after all the course grading and everything.  We'll have some string operators here  forming a square larges for its charge  and each of these largeses.  So they are displaced a bit with respect to each other.  And then the reason to do so  is that we want the commutators to depend  between all these string segments.  We want them these commutators to depend  only on the annual model, not on details of the strings.  Okay so that's the reason to do it that way.  So we end up with this collection of string segments  that have commutators that are independent  of the original model.  Only the model.  And then in order to do the mapping we will just,  so here we have two codes.  In each of them we need all these processors  so we arrive to these largeses and then for each larges.  So for example I will have this,  first say electric charge and the corresponding one  in the other code and I will map  string segments to string segments  and then it's possible.  So there will be, apart from the stabilizers coming  from from plaquettes I will have other stabilizers  but they have no charge and they can be mapped  to single qubit stabilizers.  I will not go into any more details.  So let me move on to the subsystem case.  So well we have heard of this but just very fast.  In subsystem codes rather than encoding in a subspace  of the Hilversys we will just take a subsystem.  These are our logical qubits and then each qubit  about which we don't care.  And in the case of, in the stabilizer formalism  as described by Poulin we are still described  our subspace through a stabilizer  but then in addition we have a gauge group  whose action on that doesn't affect in code states.  And the important thing is that the stabilizer  and the gauge group are related through these conditions  so that up to phase as the stabilizer  is the center of the gauge group.  Now in the topological case what we mean for a topological  stabilizer subsystem code will be a family of codes  it's with a stabilizer and a gauge group  and both of them have local generators.  Now again we want to extract the recipe for the bulk  and put it in an infinite lattice  and obtain some stabilizer and some gauge group.  As before we want to impose a topological condition  on these guys and in this case, whoops,  in this case if I have Poulin operator here  that commutes with all the stabilizer generators.  When I carry it to a large enough code  it will also commute with all of them  therefore it will belong to the gauge group now.  And one can see that again this in place  that we can assume that the gauge group here  contains this Poulin operator.  So we are left with this condition  that the gauge group is the centralizer of the stabilizer  so it contains all the Poulin operators  that commute with the stabilizer generators.  But once we have this condition we could just as well  define the gauge group through it  and then we get a condition on the stabilizer  that guarantees that our, well not guarantees  but that we need to know that becomes our topological condition.  So it doesn't get grouped anymore  and it's that the centralizer of the stabilizer  is in the sense of stabilizer up to phases.  So of course this would be trivial for a finite number  of qubits but it's not necessarily true  in the case of an infinite lattice.  Okay.  So the main obstacle to move to such a gauge goals  from such space goals is that we need to generalize  the notion of charts.  So the trick is to make it a bit abstract  so remember that charts classifies equivalent classes  of excitations up to trivial sets of excitations  that can be created by applying some Poulin operator.  And this we can extract a bit by considering  that our configuration of excitations are actually  or if you want the syndrome, possible syndromes  with morphisms from the stabilizer group to C2.  And the trivial morphisms will correspond to those  that can be obtained as a commutator  with some Poulin operator.  So this quotient will give the group of charges  for the stabilizer but not that we have this sort of  slightly more abstract definition.  We can generalize it immediately to the gauge group.  We just substitute S with G and we have a new group  of charges that don't have the same sort of  obvious meaning in terms of states or anything like that  but are very useful.  And we still have the old charges for the stabilizer group.  So we have two sets of two groups of charges  and we could as before try to define  topological interactions between them.  So consider for example, so in order for these interactions  to make sense, remember that we have this trick,  these two closest in operators that have to commute  so that it didn't matter which pair of strings I took  to define the sort of interactions.  So consider for example, that they have,  that the charge of these strings came from the,  from the, from the, from gauge charges, right?  So it's a gauge charge.  So in that case, having a closest string means that  this guy will commute with all the gauge operators  and therefore it will belong,  this closest thing will belong to the stabilizer.  Similarly, if I have a string with a stabilizer charge,  this means that the closest thing will commute  with all the stabilizers and therefore  it belongs to the gauge group.  So in this case, the two strings commute  and the two commutators are the same.  This would also be true if I have taken two gauge charges  because then I would have two stabilizers, they commute.  But if I, if I laser charge another stabilizer charge,  I will have two gauge elements  but then they don't necessarily commute.  So I cannot define such topological interactions  in that case.  So what we are left with is that for two gauge charges,  we have the, we'll define a mutual statistics,  we have topological spin.  And we also have, whenever I have a gauge charge  and a stabilizer charge, I also have this,  this mutual statistics.  But nothing else.  So I don't have a topological spin  for stabilizer charges nor interactions between them.  So really I have a nanomobile for the gauge charges  and then I could regard this as some sort of  board basis for these anions.  Good.  So of course, I come up gauge charges  to stabilizer charges simply because amorphism  from G to C2 can be restricted to amorphism  as a to C2.  I will graphically depict this this way  so that a gauge charge is a circle with some color  and then the image will have the same color  and square shape.  And it's, well, this is a natural morphism.  It doesn't depend on any choice of generators  or anything like that.  And it will preserve this, this commutator  so that if I just pick up two gauge charges  and I map one of them, I will get the same commutator.  Of course, I cannot map the two of them  because then the thing on the other side  is not well defined.  And as in the case of SASPACE codes,  again I can find a canonical set of generators  for the gauge charges and for the stabilizer charges.  Let me first discuss the gauge charges.  So we have first sector that looks very much  like what we've had in SASPACE codes, several bosons,  and then maybe a chiral pair that now can actually appear.  And in addition, we have something new.  We have several bosons and maybe one fermion.  All of them non-interactive, so they are not,  they are not annuals.  And then for the stabilizer charges,  we can find the same number of generators.  And this is because actually there is a duality  between the two.  So well, this would be ordered in a different way.  So the thing is that this stabilizer charge  interacts only with this guy but not with any other one.  Similarly, the black one here with the red one here,  so one for these guys.  And then I have these stabilizer charges  that are not the image of anybody here.  And they are dual respectively to the charges  with the same number.  Through the morphism, if you want this map  to the trivial charge in the stabilizer.  So there are many annual models that we could have,  but all of them can be obtained by combining four basic ones.  So the first of them correspond to what  we find in the TORI codes with these two bosons  with unique mutual statistics.  Then the second one is the same but with two fermions.  So that, well, the three non-trivial charges are fermionic.  And this can be found in topolecule subsystem color  code that we discussed by Robert yesterday.  We could also have a single boson.  This can be recovered from what I  call the subsystem TORI code, which would be a TORI code where  to the stabilizer, to get the gauge group,  we add all single qubit X or C operators, whatever you want.  And we can also have a single fermion.  And this is what we get if we look at the Kittives  HONICOM model as a subsystem code.  So in these two cases, actually, we  have no encoded qubits.  But this fits into the classification.  And actually, it's important to keep it,  even though it's not useful from the encoding point of view.  So once again, we can construct a framework of single operators.  And we would like to recover the stabilizer and gauge  generators as plaquettes of these largeses.  And we can do so.  So in the case of the gauge group,  it's possible to find a set of generators for the gauge  that include these plaquettes.  So this would be coming from the largeses  that where strings have this chart.  And this is the dual chart.  Well, I should better say it.  Can't you get this case?  So the thing is that this is the chart of this plaquette  in this basis and this set of generators,  similarly for this one here.  And then we also have generators that  have these charges which are not unique.  And they come from plaquettes made up  with strings that have a chart that corresponds to these guys  that were not dual to this other gauge.  And similarly for the stabilizer.  So the idea is that the rest of generators  of either the gauge or the stabilizer  will have a trivial chart.  OK, so the idea is that with this,  we put the codes in a standard form.  And therefore, we can see that all codes can be understood  in terms of these anions.  In addition, we could put these largeses back  into a finance system like a torus.  And then once is, well, what do you  expect that you have again, that the homology gives you  the logical operators and so on.  So each of these pairs will contribute  two qubits in the code in the torus.  And these guys contribute nothing.  And in addition, they have a problem,  is that when put in a finance largeses,  we need to add either a gauge or a stabilizer generator that  are global.  And you need to add one of these generators  and per generator of the first homology group.  Yeah, so the lesson is basically this  that we can understand two-dimensional  topological subsystem codes in terms of anions.  And yeah, so with that, let me just leave some questions.  So first of all, we have been considering just the bulk.  But we could also look at the defects in the codes.  So we could have a code with several different bugs put  together.  Either of these could be just a vacuum at the end of the code.  So we have boundaries and we have point defects.  But we can also classify this potentially  using this tool.  So we could go to a setting where  we have two semi-infinite planes  each with a different representing different bulk.  So this is translationally invariant.  Then we have a boundary that is translationally  invariant in this direction.  And then we have some point defect.  And this way, we could classify all such possible things  up to local transformations and so on.  Of course, we could consider more general two-dimensional codes.  And finally, maybe the most exciting is going to 3D,  because as we know, as we learned this morning, I think,  there are much more exotic things waiting for us there.  So the classification is more than just homology.  OK, so that's it.  Thank you.  So questions?  I have a question.  So what's the defining property of the 2D topological  stabilizing code?  So do you assume the locality of the operator  and find a distance, because you didn't specify it  at the beginning?  So all you impose is the locality of the stabilizer  generators.  And then that, so you will have a family.  And then you impose that any operator.  So you could do it in different ways.  But the idea is that there will be always  a large code in your family, so when  you go to large enough number, where any operator will be  local in the sense that it doesn't affect that.  And so if your coarse-grained operators are limited in width  by their distance?  So the coarse-graining we only do in the infinite lattice  and so on?  Yes, but they are limited in size  by the locality of the stabilizer generators.  Do you mean whether?  What's the limit on the required coarse-graining?  Oh, OK.  Let's see.  So there is a limit, but yeah, I cannot tell you.  There are many exceptions to proof that are just like, OK,  there is some coarse-graining that will give you this.  But it's a bit involved.  So there are always, I mean, you  can put a bound on what coarse-graining you are going  to need depending on the size of the generators,  but I cannot tell you like, expression.  It is possible to give it.  For the standard stabilizer case,  was the condition that the centralizer of the stabilizer  is modulo phase is the stabilizer?  In the infinite lattice.  Yeah, so yeah, that's exactly what I'm  asking about, sort of the encoded operators.  I mean, what about this?  I mean, do you put a sort of a size?  I mean, is it on the entire group  or elements of bounded size?  So this appear, encoded operators  will appear when you go back to some finite code.  Yes.  But not, of course, in the infinite lattice.  So the infinite lattice is just a picture.  The idea of going through the infinite lattice  is that everything becomes local.  So everything is automatically local,  so you don't have to care about.  That's why I wasn't careful in defining locality or anything.  So the thing is that whatever definition you take,  it will basically work.  And then once you go to the infinite lattice,  everything is local.  And then it's easier.  OK, thank you.  Other questions?  I think the speaker.  Thank you.
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polygical phasing to destabilizer codes. So thanks, Daniel, for inviting me to this nice conference. This is a word that is a collaboration with David Poulin who couldn't make it to this conference in the last minute. And one of his students, Guillem de Closien, said he will be giving a related talk on Friday. So this talk is about topological codes. And well, since although we have already heard about them, let me remind you that the idea that in a topological code, qubits are in some sort of manifold. And it's at cost of the check operators that we need to measure to recover the error syndrome at low cost so that they just involve a few neighboring qubits. And at the same time, technical operators have a global nature so that any operator that has a support on a region that doesn't see the global structure of the manifold won't be able to recover any information about the encoded states. So in this talk, I want to take a systematic approach towards these codes and try to understand on their certain constraints, what is the general structure. In particular, I will be focusing on two dimensional systems and which are traditionally valid in a certain sense and defined by qubit systems with codes, the stabilizer codes. So topological codes are given as families of codes and in all the examples that we have, the code will have some bulk and then maybe there will be boundaries and other defects but it's always the case that this bulk is a translationary invariant. So our approach is to classify codes according to this bulk so that we can forget about all these different effects and so on. Now of course, such codes are closely related to topological order models so in particular I can consider such a Hamiltonian which is a sum of the stabilizer generators. And the idea is that the ground state correspond to all these guys having a game value one so that the ground state, it's our ground state, the space of ground states, it's correspond to the code subspace and the excitations correspond to error syndrome. So whenever one of these negative game value, I will have an excitations with two units of energy. The ground state is of course, the generate just as a code and from this property of topological codes, we get that the ground states have a locally indistinguishable. So this is basically, this is a property that defines topological order and there is a nice perspective on topological order introduced by Cheng-Wen Wen. So what they say is that topological order describes the equivalent classes defined by local unitary evolution. Let's see what they mean by this. So what they argue is that if we have two gap ground states, they will be in the same phase, so only if they can be connected the adiabatically without closing this gap. And then this amount to them be connected by a local unitary transformation or for us to the existence of some quantum circuit of constant depth, constant in the size, with respect to the size of the system. Okay, so more pictorically what they are saying is that in order to classify topological codes or topological order, we are doing this classifying long range entanglement patterns. So for them, a short range entanglement pattern is one, so it's something, some state that we can transform for local operations into a product state. And other states will have long range entanglement and will be, so these long range entanglement will be defined by this equivalence classes up to local transformations. So we have two main goals. The first is to understand the structure of two dimensional topological stabilizer codes, including subsystem codes that are not so closely related to this topological order that I was discussing. And then in addition for sub space codes, which have the Hamiltonian analog, we would like to see that whenever they have the same same structure in a certain sense, we can transform one into another through local transformations. As we should according to this prescription by Wien and collaborators. Okay, so here's the outline. I will start explaining the role of onions in the topical very briefly. Then I will go to the main result, which is that the third code is somehow universal for two dimensional codes. And then I will jump to subsystem codes which have quite a nice algebraic structure. Okay, so what is special? Well, there are several things special about two dimensions, but one of them is that there are two physically distinct ways to exchange two particles, either identical or not. And so the idea is that if I exchange them in this way, in this way I cannot, the two processes cannot be the form one into the other without the word lines crossing. So this gives rise to the possibility that there could be some topological interaction that distinguishes between these two paths. And the particles that suffer such or are subject to such topological interactions are called anions. So the simplest case is that of abelian anions which are all we need. And such models are described by three elements. So we have first of all a set of topological charges. These charges are the equivalence of sort of excitations. So if I take a given region and I have a given configuration of excitations, I could apply some local transformations and get another set of excitations. And as I move, as I do this transformation, I will move in a certain equivalence class. And these are the labels for these equivalence classes. So that charge is something that is conserved in a region unless it's exchanged with some other outside region. So the second, oh sorry, whoops. So the second element will be fusion rules that tell us what is the total charge given of two regions given the individual charge of each of them. Okay, so in the abelian case, this amounts to basically a product. Okay, so the diffusion rules will tell me what is the total charge given these two charges. Okay, and finally, there are some rating rules that tell us what happens as long as topological interactions are involved when I sense two annuals in one of the two possible ways. Okay, so let's see this in this very relevant example, which is the Kittar story code. So I would like to emphasize that in the story code, we can visualize poly operators, which are products of X and Zs, graphically, so that an X will correspond to a direct link and a Z to a dual link, link in the dual lattice. And this way, both the kind of terms in the Hamiltonian or the check operators can be regarded as plaquettes. Okay. So two kinds of excitations. Yes, excitations in this model, we have excitations in dual plaquettes that can be created by such syn operators, which are products of X operators here. Okay, so the idea is that whenever I apply such a string, I will not sense the even or odd number of, whether it's an odd or an even number of such excitations. And similarly for the direct plaquettes, sorry, this would be a direct plaquette, it's neither a dual or a direct plaquette. Okay, and the idea is that the total there is, there are four charges, so that I could have an even number of both dual and direct excitations, then I'm in the trivial sector. If I have another number of the dual ones, I say that I have an electric charge. If the odd number, if I have another number of direct ones and even number of these, I have magnetic charge, and then finally there is a cut, which corresponds to the case where I have another number of both. So basically this already includes the fusion rules because any charge composed with itself will give me the trivial sector. And I only need the topological interactions. So to recover them, we can use the fact that these charges are created at the end points of these strings, and recover the topological interactions from the commutation properties of the string operator. So if this is a sort of, let's see, I guess magnetic string, and this is an electric one, they will anticommute, and then I can consider two processes that are products of these string operators. And when seen in time, developing in time, one of the processes describes, so in both cases, the two electric and the magnetic charges are in the same place, but in one case the pads are, the war lines are linked, in the other case they are not, and we can see that a minus and appears differentiating the two. So this, the last that these two charges have are simulations. Okay. And then we similarly can recover the, the self statistics or topological spins with tell us what happens when they exchange to two charges of the same type. So in this case, we will need three strings with a common endpoint. So in the case of electric and magnetic charges, they are both, so it's not very, a lot of fun, but the composite tends out to be a fermion, and this can be checked this way. So notice that the composite will be moved from one place to another through such machine operators, which is the product of an electric and a magnetic strings. And I take three of these, and then by the ideas that I have, and in here, one in here, and then by ordering these operators in two different ways, I can either exchange the two excitations or leave them where they are. And the difference again is a minus sign. Then of course we have the, they are fermions because when you exchange fermions, you get a minus sign. Okay. So the main message is that we have some Hamiltonian or some code, and we have extracted an abstract new model from it. In this case with four charges, there's confusion rules that correspond to the C2 times C2 group and then some braiding rules. And everything can be recovered from the string operators. This is the main message. Okay. So now I can jump to the main results. So first thing that we need is to abstract ourselves from a family of topological codes to a single stabilizer group that's defined in an infinite lattice. So, you know, for this family of codes, there will be some recipe to construct the bulk and we just apply it to the infinite lattice. Now, of course, this, this resulting stabilizer will have some properties because it comes from a family of topological codes. It cannot be just any stabilizer. And we must remember that in a topological code, local and detectable errors should not affect encoded states. It's not a case of putting bit of a political and let's see what that translates into. So if I have some polar operator here in the infinite lattice and it commutes with all the stabilizer generators, we can find an element of this family, infinite family of codes that is big enough so that this guy fits in the bulk. And then it will also belong to the centralizer of the stabilizer from which it follows that from this code being topological that p itself belongs to the stabilizer. And then, well, without going into details, it's possible to show that either p belongs to the stabilizer here or we can assume that it does actually. Okay, so the message is that we have this condition that the centralizer of the stabilizer is up to face of the stabilizer itself and this is the topological condition that we impose. So these are the two ingredients, translation and invariance and this topological condition. And then we want to classify all these guys. So we seek to be able to relate codes through local transformations and indeed it turns out that it's enough to consider the following ones. First we have a cross-graining of the lattice which basically amounts to doing nothing, it's just looking at the lattice from a different perspective. We have local and translation invariant key for mappings so that we map the polygroup to itself. In fact, locality in this case in the evening lattice follows from the translation invariance. Given that I am only considering the polygroup to consist of operators with a final support. And then finally I would like to be able to add and remove this entangled qubit. So qubits that have their own single qubit stabilizer so that from the perspective of the topological code they are not interesting. Okay, so it's easy to see that given these three transformations by commanding them, one will, so if my code has a certain annual model, suppose then this annual model will not change through these transformations simply because it's described with the stenopedors that can be arbitrary large and so on. But the interesting thing is that the contract is also true and whenever two codes have the same annual model, they can be transformed one into the other through these basic operations. Then if we rule out the possibility of so-called kaianonions which we can do according to the guitar. Then when I rise to the conclusion that, but I will give more details that every two dimensional topological stabilizer group in this sense is locally equivalent to a finite number of copies of the topical which is our main result. So let me give some clues about the proof of this. So well first of all there are two technical boring results but very important. So first that in 2D for these stabilizer groups, we can find a set of independent generators. Okay, so that if you translate it into the symplectic vector space formalism, I'm just saying that there are no linear constraints between them. So the thing is that in 2D we can find such a local and translational invariant independent generators and this is actually not true in 3D. In the proof it also will use the fact that these stabilizers are the centralizer of something actually of themselves. So if that was not the case maybe it wouldn't be true. This property, it can be not true. So this is the first technical thing. The second technical thing is to show that the number of charges is finite. And once you have this it's very easy. You just obtain the annual model by constructing the stream operators and checking what are the commutation properties and so on. And then once you have these stream operators you construct a framework, a series of lattices made of stream segments, we'll see how. And the framework of stream segments, the resulting the placate stabilizers are, give all the, give generators for the stabilizer, give all the generators for the stabilizer that have charge and then the rest of generators have no charge. And then the mapping between two codes where we have done all this, good amount to map these stream segments to the corresponding stream segments with the same charge and all those extra charge stabilizers, we map them to single qubit stabilizers. So they are the capital from the code. So some small details. So first of all we need to cross grain the lattice till at any site we can find any possible charge until whenever we have two sites with excitations here in this site and this site with a given charge that is equal and we can path it to them. If we cross grain enough we are going to be able to construct an operator with support in this gray region here so just a bit thicker than the path such that it creates these two excitations. So this is what we mean for a stream operator here. We can define them for arbitrary paths. And once we have stream operators we can recover motorized statistics and the vertical spin with a trick that I already explained for the third code. And to see that this actually makes sense, consider for example that we have these two pairs of strings so they should have the same commutator. And to check it we can extend the picture of it. And now I have these closest strings since they have no endpoints they create no excitations so they belong to the stabilizer up to a phase. And therefore the two of them commute the two commutators are the same. And thus this guy here is well defined. It's independent of the choice of the strings. And similarly for this other guy here. So once we have our new model and the two little interactions we can find a set of canonical generators for the group of charges. And this is what one finds. So there will be a certain number of pairs of generators. Each of them. So this charge here will have some unique mutual statistics with this other guy. And both of them interact with any other generator. And then all of them are bosons as that maybe the two last ones which could be both thermions. Now then each of these pairs corresponds to the anions in the third code. Okay so we have several copies of the third code and then maybe this kind of part which. So then we. In any case we can construct several largeses of operators. Okay so this will be large operators after all the course grading and everything. We'll have some string operators here forming a square larges for its charge and each of these largeses. So they are displaced a bit with respect to each other. And then the reason to do so is that we want the commutators to depend between all these string segments. We want them these commutators to depend only on the annual model, not on details of the strings. Okay so that's the reason to do it that way. So we end up with this collection of string segments that have commutators that are independent of the original model. Only the model. And then in order to do the mapping we will just, so here we have two codes. In each of them we need all these processors so we arrive to these largeses and then for each larges. So for example I will have this, first say electric charge and the corresponding one in the other code and I will map string segments to string segments and then it's possible. So there will be, apart from the stabilizers coming from from plaquettes I will have other stabilizers but they have no charge and they can be mapped to single qubit stabilizers. I will not go into any more details. So let me move on to the subsystem case. So well we have heard of this but just very fast. In subsystem codes rather than encoding in a subspace of the Hilversys we will just take a subsystem. These are our logical qubits and then each qubit about which we don't care. And in the case of, in the stabilizer formalism as described by Poulin we are still described our subspace through a stabilizer but then in addition we have a gauge group whose action on that doesn't affect in code states. And the important thing is that the stabilizer and the gauge group are related through these conditions so that up to phase as the stabilizer is the center of the gauge group. Now in the topological case what we mean for a topological stabilizer subsystem code will be a family of codes it's with a stabilizer and a gauge group and both of them have local generators. Now again we want to extract the recipe for the bulk and put it in an infinite lattice and obtain some stabilizer and some gauge group. As before we want to impose a topological condition on these guys and in this case, whoops, in this case if I have Poulin operator here that commutes with all the stabilizer generators. When I carry it to a large enough code it will also commute with all of them therefore it will belong to the gauge group now. And one can see that again this in place that we can assume that the gauge group here contains this Poulin operator. So we are left with this condition that the gauge group is the centralizer of the stabilizer so it contains all the Poulin operators that commute with the stabilizer generators. But once we have this condition we could just as well define the gauge group through it and then we get a condition on the stabilizer that guarantees that our, well not guarantees but that we need to know that becomes our topological condition. So it doesn't get grouped anymore and it's that the centralizer of the stabilizer is in the sense of stabilizer up to phases. So of course this would be trivial for a finite number of qubits but it's not necessarily true in the case of an infinite lattice. Okay. So the main obstacle to move to such a gauge goals from such space goals is that we need to generalize the notion of charts. So the trick is to make it a bit abstract so remember that charts classifies equivalent classes of excitations up to trivial sets of excitations that can be created by applying some Poulin operator. And this we can extract a bit by considering that our configuration of excitations are actually or if you want the syndrome, possible syndromes with morphisms from the stabilizer group to C2. And the trivial morphisms will correspond to those that can be obtained as a commutator with some Poulin operator. So this quotient will give the group of charges for the stabilizer but not that we have this sort of slightly more abstract definition. We can generalize it immediately to the gauge group. We just substitute S with G and we have a new group of charges that don't have the same sort of obvious meaning in terms of states or anything like that but are very useful. And we still have the old charges for the stabilizer group. So we have two sets of two groups of charges and we could as before try to define topological interactions between them. So consider for example, so in order for these interactions to make sense, remember that we have this trick, these two closest in operators that have to commute so that it didn't matter which pair of strings I took to define the sort of interactions. So consider for example, that they have, that the charge of these strings came from the, from the, from the, from gauge charges, right? So it's a gauge charge. So in that case, having a closest string means that this guy will commute with all the gauge operators and therefore it will belong, this closest thing will belong to the stabilizer. Similarly, if I have a string with a stabilizer charge, this means that the closest thing will commute with all the stabilizers and therefore it belongs to the gauge group. So in this case, the two strings commute and the two commutators are the same. This would also be true if I have taken two gauge charges because then I would have two stabilizers, they commute. But if I, if I laser charge another stabilizer charge, I will have two gauge elements but then they don't necessarily commute. So I cannot define such topological interactions in that case. So what we are left with is that for two gauge charges, we have the, we'll define a mutual statistics, we have topological spin. And we also have, whenever I have a gauge charge and a stabilizer charge, I also have this, this mutual statistics. But nothing else. So I don't have a topological spin for stabilizer charges nor interactions between them. So really I have a nanomobile for the gauge charges and then I could regard this as some sort of board basis for these anions. Good. So of course, I come up gauge charges to stabilizer charges simply because amorphism from G to C2 can be restricted to amorphism as a to C2. I will graphically depict this this way so that a gauge charge is a circle with some color and then the image will have the same color and square shape. And it's, well, this is a natural morphism. It doesn't depend on any choice of generators or anything like that. And it will preserve this, this commutator so that if I just pick up two gauge charges and I map one of them, I will get the same commutator. Of course, I cannot map the two of them because then the thing on the other side is not well defined. And as in the case of SASPACE codes, again I can find a canonical set of generators for the gauge charges and for the stabilizer charges. Let me first discuss the gauge charges. So we have first sector that looks very much like what we've had in SASPACE codes, several bosons, and then maybe a chiral pair that now can actually appear. And in addition, we have something new. We have several bosons and maybe one fermion. All of them non-interactive, so they are not, they are not annuals. And then for the stabilizer charges, we can find the same number of generators. And this is because actually there is a duality between the two. So well, this would be ordered in a different way. So the thing is that this stabilizer charge interacts only with this guy but not with any other one. Similarly, the black one here with the red one here, so one for these guys. And then I have these stabilizer charges that are not the image of anybody here. And they are dual respectively to the charges with the same number. Through the morphism, if you want this map to the trivial charge in the stabilizer. So there are many annual models that we could have, but all of them can be obtained by combining four basic ones. So the first of them correspond to what we find in the TORI codes with these two bosons with unique mutual statistics. Then the second one is the same but with two fermions. So that, well, the three non-trivial charges are fermionic. And this can be found in topolecule subsystem color code that we discussed by Robert yesterday. We could also have a single boson. This can be recovered from what I call the subsystem TORI code, which would be a TORI code where to the stabilizer, to get the gauge group, we add all single qubit X or C operators, whatever you want. And we can also have a single fermion. And this is what we get if we look at the Kittives HONICOM model as a subsystem code. So in these two cases, actually, we have no encoded qubits. But this fits into the classification. And actually, it's important to keep it, even though it's not useful from the encoding point of view. So once again, we can construct a framework of single operators. And we would like to recover the stabilizer and gauge generators as plaquettes of these largeses. And we can do so. So in the case of the gauge group, it's possible to find a set of generators for the gauge that include these plaquettes. So this would be coming from the largeses that where strings have this chart. And this is the dual chart. Well, I should better say it. Can't you get this case? So the thing is that this is the chart of this plaquette in this basis and this set of generators, similarly for this one here. And then we also have generators that have these charges which are not unique. And they come from plaquettes made up with strings that have a chart that corresponds to these guys that were not dual to this other gauge. And similarly for the stabilizer. So the idea is that the rest of generators of either the gauge or the stabilizer will have a trivial chart. OK, so the idea is that with this, we put the codes in a standard form. And therefore, we can see that all codes can be understood in terms of these anions. In addition, we could put these largeses back into a finance system like a torus. And then once is, well, what do you expect that you have again, that the homology gives you the logical operators and so on. So each of these pairs will contribute two qubits in the code in the torus. And these guys contribute nothing. And in addition, they have a problem, is that when put in a finance largeses, we need to add either a gauge or a stabilizer generator that are global. And you need to add one of these generators and per generator of the first homology group. Yeah, so the lesson is basically this that we can understand two-dimensional topological subsystem codes in terms of anions. And yeah, so with that, let me just leave some questions. So first of all, we have been considering just the bulk. But we could also look at the defects in the codes. So we could have a code with several different bugs put together. Either of these could be just a vacuum at the end of the code. So we have boundaries and we have point defects. But we can also classify this potentially using this tool. So we could go to a setting where we have two semi-infinite planes each with a different representing different bulk. So this is translationally invariant. Then we have a boundary that is translationally invariant in this direction. And then we have some point defect. And this way, we could classify all such possible things up to local transformations and so on. Of course, we could consider more general two-dimensional codes. And finally, maybe the most exciting is going to 3D, because as we know, as we learned this morning, I think, there are much more exotic things waiting for us there. So the classification is more than just homology. OK, so that's it. Thank you. So questions? I have a question. So what's the defining property of the 2D topological stabilizing code? So do you assume the locality of the operator and find a distance, because you didn't specify it at the beginning? So all you impose is the locality of the stabilizer generators. And then that, so you will have a family. And then you impose that any operator. So you could do it in different ways. But the idea is that there will be always a large code in your family, so when you go to large enough number, where any operator will be local in the sense that it doesn't affect that. And so if your coarse-grained operators are limited in width by their distance? So the coarse-graining we only do in the infinite lattice and so on? Yes, but they are limited in size by the locality of the stabilizer generators. Do you mean whether? What's the limit on the required coarse-graining? Oh, OK. Let's see. So there is a limit, but yeah, I cannot tell you. There are many exceptions to proof that are just like, OK, there is some coarse-graining that will give you this. But it's a bit involved. So there are always, I mean, you can put a bound on what coarse-graining you are going to need depending on the size of the generators, but I cannot tell you like, expression. It is possible to give it. For the standard stabilizer case, was the condition that the centralizer of the stabilizer is modulo phase is the stabilizer? In the infinite lattice. Yeah, so yeah, that's exactly what I'm asking about, sort of the encoded operators. I mean, what about this? I mean, do you put a sort of a size? I mean, is it on the entire group or elements of bounded size? So this appear, encoded operators will appear when you go back to some finite code. Yes. But not, of course, in the infinite lattice. So the infinite lattice is just a picture. The idea of going through the infinite lattice is that everything becomes local. So everything is automatically local, so you don't have to care about. That's why I wasn't careful in defining locality or anything. So the thing is that whatever definition you take, it will basically work. And then once you go to the infinite lattice, everything is local. And then it's easier. OK, thank you. Other questions? I think the speaker. Thank you.
stabilizer codes. So topological codes are given topological stabilizer codes, including subsystem operators. This is the main message. Okay. So now details, it's possible to show that either p locally equivalent to a finite number of copies wouldn't be true. This property, it can be not the third code. Okay so we have several copies of to depend between all these string segments. We obvious meaning in terms of states or anything have two sets of two groups of charges and we