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# Local stabilizer codes in three dimensions without string logical operators

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#### Automatisierte Medienanalyse

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defines as on code of of it is is term is a generator for that 1 can define a three-dimensional torus with 2 qubits per site and you can easily be verified that it provides a that of every ever terms community so spectrum wise everything is no adjust equally spaced and never the debt doesn't cook does not collapse another without limits and it intestinal frustration and we can prove that it has no string like operator follow the meaning of which I will describe in detail later so the key concept of to understand this model is to understand the string segments not not the strings well I will I prefer to define a string in terms of string segments as you'll see later and it has a very it's sort of a generous when the system size is has a lot of powers of 2 the teacher it grows exponentially if you that synthesize has in particular shape then it drops to constants OK thank let us begin
from the theory of very primitive setting this this is the way I found that model from the yesterday's tossed tutorial we know that studying stabilizer codes is equivalent to study the classical close over the operator algebra to the specific of the poly operator and we can parameterise any call the operator of I strings in you can interpret the multiplication of operators in terms of that Modula-2 edition of such a peace tree further you can't even encode the collocation relation into some syntactic structure so every language will each man in establised code can be described in terms of syntactic binary vector space more generally but if you consider our kids with the prime mentioned then it it the discretion became so the linear algebra over that final killed so what what's code in this language the committee said the poly offered at 1st comes to the the null space by no means the where the space on which the this is in fact a form vanishes and we call that the cell as a group and the crest on the symmetric group when you think of the codes as a sum of some representation of a Hamiltonian that the symmetric group of that of that Hamiltonian press to so called the logical operators OK I wish they do that i is there may be some several qubits precise that's very generous having so I don't know some anything about the number of sites number of qubits per and there are some finite number of generators for each side since we I don't want to describe model with in the agenda so I assumed to the translation variants and I want to have no single psychological operator Ridgecrest to the local indistinguishability of ground states and since we we want some of our Hamiltonian where the the best to not move where that string offers an action and so we we want some us some model with no obvious string operators but this is too general so I have to simply by simply pot by herself to the case where T equals to just 1 gender for X. type and 1 Jennifer Z no assumption on there are a number of disperse per site and I wish to have at the General ground space so the product of all terms in a continuous should be identity this this was the you can interpret that the generous editor cutting this way too and the no no single psychological operator condition ture translated to this loss copper on a single site should be identity well this is too restrictive restricted because all we need be use if a single psychological operator is found in and be required that the element should be in this set stabilizes group of setting to be identities more simple and I assume further that the allows proper honest straight line meaning that the 3 three-quarter axis were not faced by animals were body by animals are all identity if they are we PT a single site operators the we can apply this formalism of stabilizer code to this conditions and you it can translate every condition in terms of a linear algebra equations in the last 2 conditions the lost locker on a single site be identity and a loss to operate on the straight line be identity is Trans later to us and rank conditions on the commutation matrix so I started calculation but I might ask the computer busses I assume the 2 types of generators while for his tied when 1 for France type there are 16 quanta operators here and it turns out that the not exact representation of the operators are important the only important and the uniquely Our current unique parameter is the convocation lesion or among those this is there are A-Z is an 8 axis there a 64 so by interested is everything and we need some condition so that our terms defined as that code and that translates to about 27 the new patients which can be easily solved by hand the end we sold that the creation by force so 64 variables and 27 equations and you're left with 37 free variables which is like a hundred billion a not very big number for modern computers so check the all the right conditions that demands the single-site no operator word no obvious string operator when you get not that if it will not be large number of solutions not an those solutions but there are a handful emotions that satisfy all the conditions described before so here comes the first one which has the largest symmetry group of all from and I see that I designed it from this 1st place that they are commuting so you don't have to check it manually but if you reach in there said some proper property that I didn't demand there some duality between Z offered in an ax operator and this duality reduces a lot of calculations practice for example if you flip their body by animals like is easy and I I place the 1st QB and the 2nd qubit and new label Xeon then you get Z from accent ecstasy they are exactly the OK
I they designed it so that it can it it it it's tough foster water it and he has no string but I didn't prove anything it was just a of convening the requirements necessary conditions if it is not sufficient so we have to check but there it really works and there's a of heights and knowing question what we mean by extreme I didn't define string so 1st from it will let us answer the 1st question 1st at the 1st question is musical plus and as risky as being that there's no local observable that distinguish ground states so monikers like this and yet if an operator is local then he when restricted to the ground space it has its action is proportional to the identity element and how do prove that and the clean use any call the operators Support Committee recalls that allows generators then the poly operator itself is a satellite yeah and this implies the 1st statement because any operator years has a lean years of expression in terms of poly operators and this equation is linear because of the duality I don't need to check for all of possibilities I just focus on ACS title of for convenience for is the topic we vote OK all you need is this tool they raise if the question mark is 1 of the this war operators will II means I tensor I actually in an exercise it acts as a I for by convention and I and those that it has to commit with this is easy and was possibilities and you you you should be able to figure out the answer is ideally the the identity that's what if I can post to conditions he has to commit with IC and has the commitment the the I it's nothing it's because it's always be identity in a similar manner if I post lowered to conditions and restaurant should be I get into OK what would what what does that what is good what is a good for so I promised to show that any x type operator that commutes with z tied stabilizer generators are stabilizer themselves so suppose a were really be parts all the big bad finite in chapter commutativity this is the office here since this corner has to commute with this quarter which means this quantity is either the identity or acts on no other option better His XI is this app so you can multiply this generator to here this site is turns out to be excited that you can make this side to be identity by acting on the inside of this box so I could I could keep doing this so whenever I see a protects from this aside then I can always make them as identity by multiplying this so I would get call the operator of support like the that like that but look at this site this site has to commit with ICIAM z i by previous exercise you know that this is so you keep doing that so and then you will you raise the that's inside of vertical square and look at the site and you apply the same logic here you using this edge and you know that this is identity and Everything disappear so I started from the very general of X title figure and impose that it has to commute with this implied that it is a product of the right side so it is it is a satellite right now for those no spring roll is a challenge this squares represents the elementary excitations when there is a single bit late on the 1st kid it were 2nd cubits can have like represents the later to stabilize generators can you make a pair just to defects out of this well the answer is no and I will show in detail how to prove proved that and so here's a all question what to say what what what it would mean by string we know there's history operator in the code because we like operator we named it some offered a string we have stringier for example this looks like a street combination of 2 strings that it's like string 3 OK is a surface restraint is not obvious so we need some did different this definition of for string so here I I I propose most natural definition I think you the problem was that it this discrete lattice 1 game 1 dimensional object is not well defined but you can tell a one-dimensional object you when you see it but it's not a definition and we need to of in general we need to deal with that that toniest and I have no idea so we need to know to some fire-setting if we can deal with an infinite so what the 3 segments it is a finite supported poly operator accreditation at most 2 locations here too is important because any of conceivable string has 2 ends any of 3 more terminology is the anchors mean that the boxes that below stations and with is defined to be the size of the purse a length is the distance between the anchors the point is there's no geometry constraint the strain saying they can be like a sausage or of some weird Nossa or whatever you want the only condition is that it creates only excitation 2 locations but that is not the end of the
story because if you consider I identical it greater than it is trivially segment because it hasn't it creates a new excitation therefore it creates less than 2 the location patients so we need to get rid of such as trivialities in the trivial conditions we rephrase that this whenever you have a string segments it has an equivalent poly operator there are who support part separated by multiplying the stabilizer generators and up if you want to be more sophist catered then you can say that our tree string seconds only creates such trivial charges possible to country globally they will let us see some examples in a trip to the Torah code when your zzz like in this manner than he creates a bit that's at the end 2 locations so it comports with a definition string segments and its enemies yeah in right and you can show that it is non-trivial you there's no of its string segments that you can cut out this region if you have to connect somehow between 2 red I 2 yellow squares he has with 1 because with is defined to be the size of that over a length is popped potentially infinite it isn't puppet not bounded from above because you can put many as many sees as you wish OK but what of
birds then to the eyes and all that we understand really really well it is a it is a it is an example of stabilizer occurred because any classical code this the so I suppose that black region has of late spins than you will create the main wall on this boundary and in order for such a spring segment well assist resentment because it creates a excitation 2 regions where the yellow square common in you can show that 2 anchors must be adjacent otherwise the each state excitation in each square can be created locally so it has some non-trivial the with the size of anchor but length and the distance between 2 actors should be always era so
here's the formal No string rule for my model Our statement is that a stray segment of with W He's always reveal provided that that this length between the test between emperors are sufficiently large and the bound is not too severe it's just a polynomial of well it's just lenient and it and and this will be important when you want to show some of the dynamic properties of this model so how do you prove that we
can just follow the definition because we have the definition who starred with arbitrary spring segment and show that there is a equivalent this connected string segment by multiplying suitable stabilizer generators and the trick is that any string segment will be reduced to the union the union of free flat was and how we show that each flat was tree
so we use the eraser again I'll we just attested from the community with as the operators and since this edge has to independent 1 and that implies you can eat at the lower front part of the crops so you can transform this box to professor here and that trait is used for the reduction for a given general string segment you can reduce our flat ones Latin is parallel to the court in axis then given us out of Latin ones you can argue further using the same argument that using this of the independent edge you reduce to the 2nd figure and yes and some no not well slightly complicated calculations if the 2 adjacent operators are committing this I z and z i and ii at I C then there are only 4 possibilities and since the model is translation invariant you has to make some kind of consistent Our combination of the operators and you realize that after at most after 2 sites it has to be identity eventually a new kid inductively argue of line by line by eating up more and more sites in you end up with a small triangle on the left and then of of thinking of things on the right so we have disconnected from the reader there what the family I know string means you can drag the defect actually if you can drag the better from here to there then the combination of those are represented by strings segment and this is now out so that dragging the appendices it's impossible it's it is protected by the
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