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A Class of Quantum Double Subsystem Codes

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at system called thanks for that but at the person but at that time so this is joint work with Thom Endre Dougherty at the University of sitting and some so these con double subsystem codes and can be introducing today are essentially a generalization of the bacon shawl collars on using the mathematical formalism all of the quantum double models introduced by a the kid tires said to the Sandel goes a very briefly the bacon chilled quote and and look at the condo model signed with the simplest con- double there's a z tutorial that and then we'll move on to the non-abelian quantum so they can show
code is of interest because it
has very it's been shown to you have a higher and and the reason yeah and so they control quoted is is shown here we we place our cubits on the vertices of this directive to lattice and the EC stabilizing here is a tensor product it will x operators on these an a horizontal people in line like this and it it stabilizes the tensor product then operators vertically like this and we call these bacon strips for obvious reasons so we the reason this code has a high error thresholds because we can infer error syndromes indirectly using these 2 body
measurements of a gage operate I'm all of which commute with the stabilize operators found on the page shown before by taking part said these gage
operators we can obtain the syndrome associated with a stabilizer without measuring the stabilize this is good because the stabilizers are 640 operators would be an prohibitively difficult measure so these to operators required some circuits major and we can also paralyzed mentioned so you can measure all of these at the same time there also exists a simple fault-tolerant gates at the major these body operators which is nice and as a result of these gage
operators error correction and making sure takes a particularly simple form if we have a z on the central PV here the lantern commute with this I strip here but if we apply 2nd Z on the keyboard shown here what we have is a product of Annex stabilizer and engage operators now the skate operating commit
commutes with the stabilizer here and so we've effectively corrected the error by turning it into a transformation on this gate as subsets so what this means is that we have divided the body SUBST system the logical subspace I'm into a gage subsystem which we don't care about into which we encode no use going to make information analogical subsystem yet so we only need correct errors up to rotations on the gage space so
now can we generalize this bacon Chilcoat can we have come up with the code which keeps this property of its gage subsystem and world so and with
it the 2 yeah sorry to body gage operators and this error correction procedures so we wanna consider generalizations of this code using the quantum double models and the reason for this as a show
later is that is a nice connection between the stairs some stabilizes in the stabilizer group of the toric code and the baking chocolate so the toric code is defined on molar lattice embedded in the torus all we do is we associate EQ but with each of these edges z stabilizes we cope with cats and tensor product is that operators around faces on the lattice here and AIX stabilizes cartons approximately X operators around vertex and of course that that the kids and vertices commute since products stabilizes also stabilizes we have a link between toric code code in the baking chocolate as it turns
out we can multiply adjacent wickets and vertices to create I z bacon strip acting only on horizontal edges and if we do the same for these vertices like this we get an expected strip acting on the vertical edges the other that overlapping edges cancer and this produces bacon strips of reading the same
thing what we get is 2 independent codes acting on either the horizontal or vertical edges only we can then go on to decompose these bacon strips and to gage operators are in the usual way these codes a completely independent and because they act on disjoint edges so a natural question to ask at this point is half out from we bushes can we generalize to come up with tons of bacon your quote for the non-abelian models as well so I did this and let's look at the non-abelian
quantum double this is more complex and non stabilizer generalization the Torah could here that we place huge debts that is the level systems as opposed you so on edges but the states of these cheap debts are labeled by elements of a non-abelian finite group so we can think of these codes using a kind of electromagnetic al analogy in which our birth rate has become a kind of electric operator as shown here and magnetic up were output can operates become a kind of magnetic operator these are labeled by elements of this non-abelian group the operators are not completely analogous to the plaquette and vertex operators I showed for the Torah code in that they don't directly detect errors this is because errors in this
code actually thought of it excitations known any of which are arson particles found in two-dimensional condensed matter systems so the air syndrome in these codes corresponds to the charge carried by that the annealed in these models is possible it had air anyone's with electric and magnetic charge at which are not reducible to electric only all magnetic only the learners style so in order to fully describe the model we need operators on adjacent vertices and Paquette's we go was the site these operators generate what's as the quantum double algebra which this is just some linear combination of these Hg operators them son
these the charge associated with these any eons actually corresponds to the irreducible representations of this quantum double algebra and these irreducible representations of quantum algebra will by a representation of a particular normal subgroup of this finite that cycle group and US conjugacy classes of the same and these correspond to I think that at
1 an electric that that the representations of correspond to the electric charge and the conjugacy class mean a charge so it's possible right down operators which project onto the charges respectively and we can also write down operators which projector onto the dynamic that is electric and might make a challenge these at most similar to stabilize in that they allow us to detect errors in charge so we have a
problem It's not possible to simply multiplied together and suspend vertex operators as good as more structure and in fact there is no longer label by group owns the label by an a of corn level so we need to do something else from bacon strips such as this year is a loop around the torus among contractible loop around for us we need some method of creating operators such as this what we need to the ribbon operator which are
responsible for the transport of any on the round the lattice We've come in 2 flavors jewel ribbons and directory Jewell remains on the right here and are formed from this operating here which is a sort of left group multiplication operator and as director events in red are a sort of proof projection operator since the dual ribbons as for Konate sites with the same vertex different but cats they're responsible for transporting magnetic charge around the lattice and so but opposite direct ribbons transport electric charge and they come connects sites consisting of the same put kept but different vertices so using these operators it's possible to construct a
kind of analog and kind of non-Abelian now all of these bacon strips using a particular operation in the algebra associated with Rubin operas a kind of a concatenation operation these operators
not simply products of the vertices and with that found there there are independent of them and note the similarity get to
be the operators that we got before by multiplying the gets vertices of the rhetorical so we wanna construct
operators to detect charge out within this code so what we're really interested then is in projectors onto dialogs but the protect eProject onto electric amid net charge that we need operators which consists of both the red and blue the electric amino take charge them so we can do is we can
combine these these kinds of operations these combined direct you rivers there is a sort of link between sites in their regular quantum double and these operate and insofar as we can you generate their own quantum double algebra and this operator does not affect charge along the strip because it meets with the wickets and vertices know that this is a combination of the bacon strips from the horizontal and vertical quota showed earlier had that we could get by Moss Point the cat vertices this means it we've now got 1 to she'd code them so now we want to
consider projective versions of these so we get something that looks like this something ugly and unlike that that as was the case we constantly put this up and the gage terms as we did for the toric case it just doesn't work with but to many things in there and indeed we would want to because this it doesn't make sense the structure of the code is different and these
ribbon operate is actually the new quantum double found found that thanks for what that yeah and so what what they what this code does is it defined in Eq citation space which is a subset of that of the original on double model and the they the difference between the 2 constitutes the gage subspace of our model the and so we can define operators on this case subspace as those which commute with the air for generally these effectively roman operators which create charged with in the area of the projectors whose total tribes sums 0 and we can write down to body operators which act on this stage subspace so on the Net and yes
so it's possible to come up with a a three-dimensional version of this goes all by writing down that three-dimensional inventory all right so the questions every worked out any like exposed examples in a non-abelian group like S 3 to see where you actually get if you were to use your construction Qandil construction on a non-abelian group using its irreducible representations upright lots of a 3 1 option allele is it is it interesting as there is something you can set like you know there are various the quantum technologies that can realize Q traits repudiates generally and they might be able to realize this more readily than a qubit type implantation perhaps I don't know so I was just wondering if there's any Nycz geometric structures these so I'm not abelian groups eventually get certain including gates you know you have a larger repertoire of Dayton Euzenat abelian group for example just by doing any operating so this is 1 if you had any so you work through it all sorry not on and but the other questions all right let's think the speaker and all the previous speakers the
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Metadaten

Formale Metadaten

Titel A Class of Quantum Double Subsystem Codes
Serientitel Second International Conference on Quantum Error Correction (QEC11)
Autor Kumar, Prashant
Lizenz CC-Namensnennung - keine kommerzielle Nutzung - keine Bearbeitung 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt in unveränderter Form zu jedem legalen und nicht-kommerziellen Zweck nutzen, vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/35287
Herausgeber University of Southern California (USC)
Erscheinungsjahr 2011
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Informatik, Mathematik, Physik
Abstract We introduce a family of codes which are a generalisation of the Bacon-Shor code using the quantum double models introduced by Kitaev. In particular, we show that such codes are possible for non-Abelian quantum double models, as well as three-dimensional quantum double models. Our codes possess a structure which generalises the gauge subsystem of the Bacon-Shor code to non-Abelian models. This means that we retain the ability to infer error syndromes using two body measurements, as well as a generalised subsystem error correction procedure.

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