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Fault-tolerant quantum computation with high threshold in two dimensions

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so we're going to continue the discussion of fault-tolerant quantum computation for erosible and Mark thank you very much for the introduction yet welcome to all thought tutorial lecture on quantum error correction we're getting a bit more specialized now namely we will be talking about 2 dimensional architectures and fault-tolerant quantum computers with the geometric constraints namely when only nearest neighbors can interact OK so what is the it was 2 and a mycro the clicking utilize the OK but you recognize that this is this is a common not posing as a donor you know what that is you think well that must be topological quantum computation but it is not we will be talking here about fault-tolerant quantum computation conventional setting so we are not naturally protected against errors we have to do something to protect against errors and in an active way so our systems will be 1st saying not topological they will just be that is the use of conventional cubits but the methods that we choose to correct errors in those systems will be topological and what is certainly true is that of the experimental setting criteria the type of Cu carries a physical realization that we use to realize quantum computation will influence the architecture that we are going to end up with and will influence the methods that we will use to implement fault-tolerant quantum computation quantum error correction so let me discuss this in and slightly more detail so for
example we could think all of all OK being carried by photons well in that case you're not likely to have a geometric constraints well the most problematic things about photons is to have them interact so you need to think about that but you cubits they could live anywhere and then you could use optical fibers so whatever that to shuffle between places so if you're discussing photons as your carrier of cubits you can afford a global architecture everybody can interact with everybody else now if you move into all the systems like cold atoms in optical lattices or I traps now suddenly shorter distance gates become preferrable over longer distance gait in the well something that would be very nice would have would be to have 2 arrays of superconducting qubits well the natural arena for those guys is to live on a sidewalk on some other waiver so you're two-dimensional and now local interaction of short range interaction is very preferrable so you would be thinking about a local architecture and I will make precise the setting the local setting that I wanna discuss here in this lecture but before I get to that let me just brief briefly remind you of what we need to
know about fault-tolerant quantum computation for the purpose of this talk and we've helped the front about already in and was talk that we just summarize very quickly so fault-tolerant quantum computation is the part of keeping up a quantum speed-up in the presence of so the coherence it is the debilitating quantum computation street so you need to do something to prevent it and we are very fortunate situation here namely we can do something against decoherence in the circuit model the sort of system models and that the very important or most important result in this area is the so-called threshold here and it tells us that they have we can manage to reduce the Arab parliamentary operation below a certain threshold then we can compute arbitrarily long and arbitrarily accurate OK so that leaves us with 2 questions namely so what is this value what is this threshold that we have to beat and also what's the operational cost of doing on this and well these are the 2 questions that I would like to address today for this particular local setting so let's go back to the to the setting and that let me specify what I will be talking about so again we're looking at a two-dimensional setting say just a grade of can which only nearest neighboring cubits follow to interact and that is as I said relevant for superconducting qubits quantum dots optical lattices and to a lesser is that extent also for segmented on trips off so you have more option so let me get back I skipped the outline slide so
that the let me go that now so I hope I just gave you as the motivation for what I'm hopeful while going to talk about those 2 dimensional architecture still the road relevant in many physical systems and what I going to do with my talk today is I just wonder presented you 2 constructions to topological constructions for handling goods geometric constraint of nearest neighbor action interaction in 2 dimensions so the 1st constructed and going through presented you it is based on the so-called surface codes which have been rapid of years now and the other construction is considerably new works only about 1 year old has been suggested by hacked being is the so-called topological cholera subsystem coats and this 2nd construction is considered considerably more intricate and therefore I will just between those 2 constructions I just will pull out several elements of the 2nd construction namely the twists color and subsystem codes subsystem cos you've already heard about and talks lecture this morning and will discuss those aspects separately and simpler setting before we then get back and discuss the go the full example and I will conclude with some marks so that's the outline I for my tutorial today so yeah so what I wanna do with it is I yeah I wanna show you structure and then hopefully you can find some duty in them and maybe that inspires you to come up with your own construction and maybe contribute to this field so let me begin with the 1st of those
constructions with which is based on on the surface of so I'm so this surface codes have been invented by out of sync Italian around 1997 and they a stabilizer codes of more precisely the see assessed codes which makes them already nice but then in addition of the stabilizer generators are close to local which makes them very suitable Our fall architecture no will it expand on that later so this is the 1st idea that goes in here though serves codes are perfectly suited for two-dimensional nearest neighbor settings but yeah just error correction is just half of the story we also need to think how to do quantum gates in such a setting and it turns out that there's a topological our way of doing quantum gates in this setting namely what we will be doing is we will be punching holes in the cold surface and this way creating binary and this will provide us with encoded cubism that surface and then we take those holes and move them runner-up the fairly similar to what is done in topological quantum computing and vibrating those holes in the castle this we will be able to realize topological quantum gates so those are the 2 aspects of the construction that I would like to illustrate here and some greater detail OK so let me begin of defining the surface coats so what we need for them is two-dimensional lattice with the cubits living on the edges of the so the cure it's live here what we next need to define for stabilizer cold out the stabilizer generators and the encoder poly operators so force units see ISESCO to stabilize generator come in 2 kinds is that type and X times and this set type stabilizers are associated with the plaquettes namely we pick up a cat and then the corresponding stabilizer generator is a fourfold tens of product of sigma z is on the edges around the plaquette with this and likewise the how outside balises or associated with the sites of the lattice and well in fact the consist of the tensor product of 4 poly operators for for spin-flip operator sigma Sigmex Our next to a site like this and you can easily check that all those opportunity commute as they are supposed to holding the micro right can you hear me from the back yeah OK good thank you OK so so much about the stabilizer generators now what are the encoded poly operate like in this surface here where we have specific boundary conditions I haven't talked about the boundary at that would do that later the encoded polity operators are have a geometric shapes namely that of a strain so in case of the encoder poly operators in it it runs from left to right across the entire lattice and the for the encoded poliovirus operators Sigmex that runs from top to bottom across the entire lattice so now what is this topological about these well so the 1st topological last that you see we could consider different types of surfaces say without binary but homological logically nontrivial once with the genus with a non-vanishing genus but is with handles in them and then it turns out that the number of cubits and encoder Cuba's so they can put into such a surface depends only on the genus that is it depends only on the topology of the surface but not on the sites another topological aspect that I find here with those codes is when I said this strain of sees corresponds encoded that operate A-Z sees half using me so but it's understandable anyway so it's not just that string it could be any stream that runs from left to right and then in a more mathematical fashion icon of the forum such as training but by in a homological fashion that is by adding the boundary of sin the resulting string would still represent the same encoded operator sigma z so this encoded operator is represented by a whole homology class of strings OK so the the quantum
error-correction no-fault quantum error correction with those colds is well studied and you could ask well it's nice that they already stabilizer codes we know that as soon as we're dealing with a stabilizer codes code the whole machinery of discussing quantum error correction becomes available to us but now we have this topological picture this picture of homology in addition so weak we may ask ourselves is anything in addition that we get all of this topological picture and the answer is yes and Andrew alluded to this before we can map of the successful new may discussed the success of quantum error correction in this setting in terms of a statistical classical statistics mechanics model namely a so-called random the CAD said to gage model so we can study quantum error correction by drawing phase diagrams of those that make models which of course doesn't make it any easier but it's a nice connection anyway and also as Andrew has pointed out for this but particular topological quantum called the surface code quantum error correction of the 2 media precede a classical procedure needed to be run for quantum error correction is actually efficient and we have known this for a long time this minimum perfect chain matching that Andrew alluded to is also known as admins algorithm and dates in sixties so of that has been even before quantum computation the necessary mathematical tools have been invented to deal with this setting but the other the error correction other topological quantum codes is more difficult than this and this is a very recent advances in quantum error correction theory of that has led to efficient the quantum error-correction protocols for those of the topological quantum coats and you hear about this this conference into talks by Pettibone being and do you do trust and she's from Sherbrooke OK yeah that was a little bit what I want to say is that the error correction for those codes as well was studied and we have sufficient algorithms for quantum Error Correction with them OK so let me now
leave error correction for a moment and think about and called cubits in those surfaces so as I already said we can encode cubits by going to home a lot of by going through a homological
non-trivial surfaces but maybe that's too complicated for experiments so that's not what 1 is doing here we wanna stick with the play but the plane itself does not offer the that the support for any cubits so what we need to do is we need to introduce some boundary so we can use the segment of a plane the special binary to encode 1 cubit and you've seen this before but 1 qubit isn't enough so what we're gonna do is we take we are punching holes into the cold surface and I will I I will explain on the next slide what that is and then the way we handle the situation a pair of holes will constitute a cube this here already in this drawing and you see more of that is the encoded polity operators so 1 encoded poly operated the encoded see here will encircle 1 of the holes and the other the polity operator was stretch between so let me define now that in more detail what those schools are so those holes are just missing stabilizes so so stabilizes as I said earlier come in 2 kinds namely we have side stabilizing plaquette stabilizes so that's what we call a primal hold this just they cite stabilizer that's not enforced so we removing a generator from the stabilizer the stabilizer shrank number of Cuba stays the same so this gives room to have well encoded cubits likewise for the cat operators what we call a dual hall is just a missing plaquette stabilizer so we remove this plaquette operator from the stable and then pairs of plaquette missing plaquette stabilizes passive 2 will holes was supported dual Cubitt pairs of missing sites stabilizes that is pairs of primal was support primal cubic so let us now look at those qubits
those encoded cubits so what you see here is to primal holes with their binary and now encoded is a strain that runs from 1 hole to the other whereas the encoded x is a string that loops around 1 of the holes that was for the primal Q now let's look at the dual cube it followed by Cocke to do will holes the X. type string is running between the holes and does that type string the code is that the snooping around 1 of the OK so when we look at those those crashes we can infer the following rules namely an X. type chain cannot and in the primal whole but it can and in a dual is that type chain can and and primal holes but it cannot end in the dual that has to loop around around it so these are the the topological rules we have to respect here in we wanna use them
namely a 1 explain to you how seemed works for this scope OK so what you see here in this drawing is just the the code surface with the holes punished into it so here we have 2 primal holes that has the support for Q would be we have to do a port for the dual cube at and this axis just time so here goes I also just moving parallel so nothing happens is just storage
what can only break memory we also OK anything interesting happening here and indeed this configuration is suitable to realize the Controlled NOT gate and actually this is something that I can show to you with what I have introduced OK so that was as follows so let's consider this string here this X. type string running between 2 dual holes OK so let's go to all table that we love that we discussed earlier so we find yeah and X string running between 2 doll holes is actually possible and it represents in Codex operator so what we know the I do with this thing is we slide it forward in time and see what happens in respect those rules namely the endpoints of that string must always remain attached to those do holes and they can never end in the primal OK so let's get going you move this thing for nothing much happens that at the beginning so the ends remain attached but no this primal whole starts coming in the way and the string has to avoid it so the string must recede and Bollore's out so now the by now the the primal whole has really come very much in the way which means that that this x types training now has to loop around it before it can and in the other do a whole so finally this loop here cuts off and at the end of the day we we are left with with an X. type string around a primal in a closed loop and a string thing between the the to do a whole so now we go back to old table up here and try to figure out what that is so that as it was before is an X. and collects operator on the dual cube it and this loop here is an ex operator on the primal cubit so we see that under forward propagation this operator has changed from a local operator to the 2 local operator nonlocal operator and this is the signature on entangling gates and in completely the same fashion we can examine the all the free poly operators I mean the the all the accident to reserves and refine those propagation relations in exactly the same way and this tells us that we're dealing with the C. Northcote here and so we can ask what can we do here with this cold and topological matter and how actually that much it's certainly not universal but what we can do is the scene out in a show new simplified version of it in a a minute ago but this 1 is a bit more complicated because you wanna do is see on also between to Cuba is of the same kind so that the the previous not involved the dual in the primal puberty 1 show that you can do is you know also that freedom to Cuba is of the same kind well anyway we can do this not we can prepare and that we can prepare a measure and that eigen bases and we can prepare and measuring the excited places yeah but that's evidence that it's not universal put its at least to start so how do you make this universal so there is in fact only 1 additional operation that you need for universality here in the sky and well it is the preparation of this state that that Andrew mentioned it's the eigenstate of golly operator Sigmex plus Sigmar why and if he can prepare the state with high-quality can provide the stable high-quality then we can run universal quantum computation in fact there exists a procedure not doing precisely that this is magic state distillation is a protocol that breaks but a plot of topological construction but we need to include its to achieve universal that is the 1 non topological element in the construction OK so this is pretty much what I want to say about the gates so let's get back to the trash roles no that's now discuss the characteristics of the sky so let's begin with the threshold and so and Andrew has already told you that we can talk about hour all of various levels of detail so that surface support of here and in fact so the the the code the the the the of the trash shows from memory they carry over as thresholds fall fault-tolerant computation the code so there's no deduction for the fact that you actually computing which is something that has to be shown of course but this is how it turns out so the the most elementary level of a at which you can provide a trash as if it if you assume that you error corrections absolute is actually perfect well then you could if you assume that phase flips and spins of arrows actually dependent independent then you arrive at the tragedy of about 11 per cent so next thing that you can do is take into account the errors of crime prevention and in a phenomenon phenomenological fashion Banyan treasurer goes goes down to about 3 per cent but then if you really all spilled up into gaze and allow a gay thick each individual gated treasurer goes down even more but not that far so what we end up with if we assume an error of each individual greed is a threshold of about a per cent now with this reason approvement of the threshold value and this is not so bad so let us compare this with with other things so but 1st of all I wanna say and actually you think this is now the region where you can actually begin to meet with experiment so you think that in our chart of 1 per cent 1 in hundred gates failing this is something that's probably occur within experimental reach so you think problem how scaling up systems is much more difficult such securities should be in range let me let me also compare this number with with a number that I just learned this morning there from Andrew namely the threshold for classical fall told computations 9 % actually I expected it to be a bit higher but anyway so they're not so far away from that number and finally let me compare this number of 1 per cent with the highest known for fault-tolerant quantum computation is is a rough result but many no and without a geometric constraint and that number is about 3 per cent so so that that number tell tell you to compare those 2 numbers is that the geometric constraints which at at the beginning seems fairly severe actually doesn't constraint too much it's a factor of 3 that you're losing OK so I think this is good news so let's celebrate them before we go to the next slide which will be about overhead all lectures and this is
a slide about the error model well I don't wanna go into it couture a detailed we can talk about it if you have questions about it is basically what I said that we have a an hour for each of the involved states and outcomes of this slide about the overhead so this is perhaps a bit more mates so far the computer scientists overheads of 40 logarithmic so I made that doesn't sound too bad for the experimentalists especially if you said in the back this is 10 to the 7 this there's 10 to the 9 and so forth so these are all very very big numbers so let's see what's plotted here so what we are plotting is the cost of an individual fault-tolerant quantum gate as a function of the total size of the circuit so say if I wanna do 10 to the 8 logical gates which I might look here for the cost of an individual fault-tolerant quantum gates so these numbers are indeed very very big but I should say 1 thing those curves are at the point where our Bayes error rate is 1 3rd of the error trash so if error rate is lower than those curves will look better so the scales will change so the set of all that is put a logarithmic and what this means in practice because of the scene not dinner computation with 100 gates isn't isn't very much cheaper than the cost of a fault tolerance you know if you have 10 to the 10 gates in the computation so the is mild but you starting at a high level that's a problem of the red curve you was for the scene so the the other curves the green and the blue they are for AIDS the 1 qubit gates of requirement of state distillation in this protocol so we see that they are higher so the messages that well again the 1st for the computer scientists that the scaling is unaffected Scalia means that remains 40 logarithmic the exponent remains the same but in absolute terms those procedures all very very costly so if we can get by without them that would be a very good thing so the drive off the overhead in absolute terms and this is just a pile here so I leave it up to you to contemplate a so that they could type of construction on top of that all right so so so I have 1 more slide and this so I wanna get back to the point of that those curves might look a lot better if you go to lower error probabilities but and the full this topological scheme I didn't have a plot that would actually show it maybe somebody else at this conference will show such a plot later let me just discuss what a phone in the literature so this is for O'Neal the scheme of the highest threshold and this is not topological at all what he plotted in his paper The at the same figure namely the cost of the Office E not per that because of an individual tolerance known as a function this time of the Bayes error probability enough of various computational sizes again 10 to the 8 sound something we might wanna should foreign OK so you see 3 per cent is the trash also add 1 one-third of the threshold you're getting something like 10 to the 11 but if you are at 1 hundreds of a threshold you already down to 10 to the 3 10 to the 3 is still a very big number but it is a lot better than 10 to the 11th so I mean I cannot guarantee that we will see the same kind of traumatic crops and all topological scheme but it is something to look for so we need to to do more than analysis OK so with this comment I wanna leave at the surface codes and wanna move on to discuss the another type of code
namely at the Hatteras topological color subsystem coast and the other side initially in the the construction is rather spicy and has many interesting elements in it and so before I wanna put them all together I wanna discuss those elements in simpler settings OK so the 1st element of Bobby's construction that I consider all twists and we can discuss twists in the more familiar surface code what we need to always of advantage to to look at the surface code in a slightly different way so what we will be doing for the court surfaces to manipulations 1st on all cube roots the on horizontal edges we will we will apply holder more transformation and then rotate the cold surface of by 45 degrees so we we we then end up with the letters well I come to that so this is the 2 due to changes that I'm making our and here you see the resulting the operators said were the full plaquette operators inside operators and the now look more most similar to each other in fact they look exactly the same except for their location OK so that is now a new Latin step we will be considering OK now we
tolerate it just black-and-white for the moment so what used to be the sites in the old lattice now becomes white plaquettes and what used to be but the old plaquettes will become door plaquettes in this lattice so this is now this coloring is what we now wanna discuss because
likewise I mean what we inherit of course from from the previous letters are all those closed string operators and then that give us an error syndrome so all that that we can so the with the associated string operators we can measure the errors and so here you see a dark strain so this goes around what used to be a site so box strings measure the SI type stabilizer lights strings measure the plaquettes evil OK so now let's take some sort of all of this picture again and now let's take the color back out just remember that 1 of
the faces was called dark and now we're gonna do the modification drone and this well actually lead us to twists so what we do so the twists are analogous to what were the holes in the previous construction the difference is that now and non-local money from manipulation of the lattices needed to introduce the 2 A so we are cutting the lattice open and modified the edges along the cut now and the end point of that cut is what twist is going to be so now after we have introduced the twist into the lattice let's back in
out doesn't work anymore so we we fail to end up with a consistent coloring of that lattice so we can still and talk about a coloring locally but there is a global obstruction the to the to the coloring now namely along this cut here we have a neighboring faces with the same color which is forbidden by our coloring I just I just wanna say 1 thing so found but there's nothing physical about the location of the cut I mean locally part of the lattice looks perfectly fine the obstruction to the coloring is a global property so we can move this card anywhere on the is the only property that must that we must require of the Khot is that it ends in a twist OK now let's discuss those stringlike operators again that go around the in loops so what we find now with the cut present that those string operators don't close anymore in another OS so then we also have well what those if they don't close anymore what invariant meaning can be attached to their measurements so that's the that's the weird situation another property that we observe is that the string operators when going around a twist change color but this training does not correspond anything that we would wanna measure so here's an operator that we can measure it has to go around the twice in case and so I leave it I leave the twists our here for a while but yes so the point is the twists in this more elaborate construction will now I take the rule will take the role that the holes had earlier in gay when we will get back to them so so much about the twists and I now continue with subsystem of tonight I think I can be brief because I already mentioned them this morning but I would nonetheless refresh your memory about so the so you can be introduced the my exam so maybe that's the that's a before we can extract let's start with an example so those stabilizer codes so we need to define the stabilizer and encode upon the operator so this is what they look like so we have again we have a lattice here the cubist living on the sites and the stabilize operators that type of just 2 columns of poly operator sigma z and they could be anywhere likewise the X. stabilizes all are all their generators 2 rows of poly operator Sigmar X and India the encoded poly operator so just 1 column of individual poly operators sigma z and 1 row encode polity operators sigma X and then of course it does matter where you all yeah and now if this was an ordinary stabilizer cold you would imagine that those several hours operators is what's being measured to extract the Sundram but that's not the case namely what you measure the of operators with very small support and that's in fact the point of the whole construction you get by with measuring to local operators so at 1st you think OK I'm getting the idea was to be we measuring those operators and their translates and then by classical the post-processing the measurement outcomes in me by multiplying them together well we can infer the outcome that we had obtained had we measure those stabilizer generates so far so good but then you notice out those operators stone commute so what does that mean so that's that looks that doesn't look right so if I measure an operator 80 and then operator B that doesn't commute I'm not losing the not the the measurement outcome of the 1st measurement does this out this measurement the 1st measurement still makes sense the now added seems a little by the peculiar situation and actually but what really matters is whether those operators commute with the encoded poly operators and with this in fact that they do so yes those operators are not necessarily commuting but they will always commute with what we are interested in so while we are measuring then we are not affecting those operate so that's good about but it is still worried that explain a little bit more but that was just an example and that has now become a little bit more abstract about those codes so here in in the case of these codes then there are a couple of types of operators that you would find a group theoretical framework for so well there's the stabilizer Group as usual you know there you go the operators that we measure in this scheme the out they generate what the gage group and that contains the stabilizer group as a subset and then there is the centralizer of the gage group that is all those poly operators that commute with everything in the gage group and they were contained in addition to the stabilizer all the encoded colony operators so it is still uncomfortable with the construction and that we often that a bit
of additional information here in in part already mentioned this so yes of those operators are not commuting in the the mess up certain things in our code so in in addition to this system Cuba that we are interested in protecting we have additional cubits which we call gage cubits and when be measure those operators will be here we are affecting the gage Q. but not the humans we are interested in so when of this whole construction well there is a gain in there is a loss the gain is that we end up measuring operators and very local nonlocal operators are hard to measure so that's good about the construction but there is a price to pay 2 2 ways of explaining it so for example I mean we are this calling the gage cubits over here I mean those are Q was that could in principle be protected but we choose not to protect them so all rate goes down on this matter very much but there's another way of expressing this we could also say well tentatively those gage cubits Our could be prepared and fixed states sigma's the eigenstates and then those there and couldn't poly operators would turn into additional stabilized so by choosing to measure those very small operators what we do is in fact the reader's card stabilized this we choose to learn less information about our arrows then we could have potentially have learned in this setting and so we could potentially affect the ultrashort for such coat yeah this is what I want to say what subsystem and now let me
come to the final ingredient of color so the standard to this color encodes is color codes which also and an invention by a bomb being and marking the God of so what we're looking at here is a hexagonal lattice and the 2 cubits live on the vertices of that lattice and they underlying lattice graph has to be trivalent for the construction to work the stabilizers or associated with the faces of the lattice and for every phase we have to 2 types of operator type and exercise OK and again we are dealing with a topological quantum code namely the number of coda Q with that we could can put in the court surface without boundary is just 4 times the number of handles on that surface and is independent of the size of the surface OK but there and so this red and black is not where those codes or called color codes so
now color comes into play so what we can do here with this with this with those codes is to call the faces such that no neighboring phases have the same color and what color it does for us it is a convenient bookkeeping device so as always in those topological quantum costs the operators that you're interested in namely the stabilizer generators and the encoded poly operators correspond to strings and no having colored the faces we can call of those string operators as well string operator that goes across the green faces and therefore we call it a green string likewise this is of string operator goes across the blue faces and so we call the blue string and so on and so forth so and and as before those string operators can be deformed without affecting the operator that 3 read that I represent that's this rule over here but now there's something in addition and now you see why color becomes useful for those codes those strings can also supplies up and then merge again and continuous 1 strength so what we have here in addition this is something that we didn't have in the surface code is Lotus sees work where multiple strings can meet with 3 strings can meet but there is a constraint not every triple strings can meet in the vertex they must have different color the muscle if different color and took care of that you see that color becomes a useful bookkeeping device here so that is now put all these ingredients together and and let me review those topological color subsystem codes you can also be begin with a hexagonal lattice pretty much the same as we had FIL for the color code but now differences beginning to appear so we have we have to place or cubits 1st and this is done as follows so for every site in the original lattice we put 3 cubits here back in this fashion this is all indication of Cubans and now this being subsystem codes there are a number of questions to address we need to what is the gage group what is to stabilize the what is the centralizer scene called upon OK so let's begin with the gage group so that as in the case of the Bacon Shor code the the elements of the generators of the gage group will be too low and they will be on those short links here they will be operator Sigmar Zezima z and on those long links here they will be Sigmar why Sigmar X but there only 1 such generator for each long link so there's an orientation question involved here you they can't figure it out from this little diagram here so you have the Y here and an X over here likewise you have for this said she have the Y sitting here and the exit OK so that was the gage group the next we
need to talk about the centralizer which would give us the stabilizer is but take a set but a subgroup but also all the encoded operators so at this point as a graphical the device of bookkeeping otherwise we introduce the notion of a consistent shading that which is the following thank you what is something like this so we we can and introduce shadings for all of the holiday operator separately so the polity operators sigma z is only shading the edges in our decorated lattice the identity shades nothing and the polity operators sigma acts signal more more widely shade edges and also chorus of those faces and our consistent shading is the shading where every triangle of phase here is consistently shaded but there's other completely shaded or not treated at all and every such urgent and of consistent shadings well I can using this table translate back into poly operator and and every such consistent shading will correspond to an element in the centralizer will give us an encoded poly operate lower code it could be the identity yeah so this is the way of representing this centralized and in that light it's now there is a level of abstraction or a clarification we can translate every shading into a string that so is a one-to-one correspondence and what we basically do is we we relocate every cube adhere to the the to the next site which is in this example is just here and then so that means all those edges are relocated and that gives us those strings and of uses a string and now the color of those strings is inherited from the color of those plaquettes over here so let me just the yes so let me see can yeah so for example this training becomes this training it inherits its color from from this could green plaquette over here so if you're looking let me see that if you're looking at those 2 line elements they become the screen string and why is it René Green because this line element and this line element they've become mapped to the same string segment and red plus blue discrete OK still as a little bit of explanation What's color comes in next thing we have to talk about it as a subgroup of the
centralizer the stabilizer OK so like in the conventional the surface code we have to stabilize areas for each plaquette there's a set type stabilizer which might happen and stabilizer generator involving X and Y poly operators which looks a little bit more complicated so that has that describes so far too subsystem aspects of this coats now well all interests and what is at risk after all a twist is a face with an odd number of edges around so what you can see here whenever we have a twist in the lattice this messes up or coloring scheme so that corresponds with the twist that we saw earlier in this simple example of the surface so as soon as we have twists there's no longer a consistent coloring and like before we have cuts going out of the 2 It's too was infinity and along the card we have faces of same color meeting of the thing is that for the twists there only remains 1 stabilizer generate and the the other 1 is inconsistent so the number of cubist hasn't change number of stabilizer generators goes down so we make room for 1 coder cubits that is so far my discussion was based on the shadings and that's no obstructed a bit more and go from the shadings to the strain so here's what happens if you move a strain around a twist so let's do this in the example with the twist is of red color so arrest ring doesn't change at all agree in string is converted into a blue strain the blue strain is converted into a green stress so when we have the string operated going around with in general the colors of the strings permute the and now let me finally come to explain where the important cubits and how we do gates here in this fashion in this world now with a lot more preparation necessary look very similar to what we saw earlier in the case of the surface so 1 example of getting encoder cubits here is using for twists of for 1 encoded cubit and what you see here is the encoded poly operator sigma acts and this string operator corresponds to the encoder poly operators is what I was not explaining earlier so and when when strings cross into its changed the order of the operators you get a plus if those strings of the same color and minus if the strings of different color so that's why those string of reproduce the commutation relation that you expect for may see know if you take 2 important cubits each consisting of Fort Worth of different color you can perform an entangling gates in this following fashion so fairly similar to what we saw in the simple example before but this time it is the twists that are being moved novel so we take twist number 3 and 4 of the 1st of the 1st Human and taken on a journey and possibly through here between the phone a set and the 2nd effort was of the 2nd human and put them back into their original positions so this is this turns out will perform entangling gates and we can analyze it
in a very similar fashion to what we saw earlier so on the best information the encoded that operator becomes this thing so that then how can we manipulate the string over here so on this green strain doesn't actually see those defects what is the factors is of importance only 2 blue and red strips so we can actually retracted and you end up with a situation like this and I'll furthermore we can introduce the stabilizer generator over here whether Suffolk is so this is perfectly fine and convert this situation to this situation so now we can go back to our table up here and see that there's that I've operator has been converted to what it was initially times an X. type string operator are on the 2nd cubic again the important point will be here is that a local operator has been changed to to local operator referring to different cubits and this is the signature of entangling gates and if you I mean you can deal with the other power the operators the same fashion and you figure out that this is and X controlled spin-flip gate so comparable to the scene of but how heavy improved over the previous construction of the surface well it turns out that with this code we can now implemented the end higher Clifford groups I in a topological fashion the does make us universal but we are better than what we want that and what we were before and we can therefore expect that this cold would give us significantly lower overheads I don't know I mean I haven't done any simulation in this regard but maybe we actually hear something about that can here this conference so this is the material that I want to review today and I hope I made you curious those construction so let me summarize so I mean
it it gets even how many minutes of a left 1 minute OK let me just close up it gets 1 of than do so as and we already mentioned the other 2 Eurocodes area in which you can do the inside that universal by themselves to do you don't need any not topological helper constructions but they're not stimulus hose probably more difficult to realize OK let me just
summarize if you have a 2 local architecture into dimensions topological quantum coast of probably the best way to go about this scenario we we're seeing very very high threshold numbers overheads at this point 1 of the reduced and therefore so we live in quantum colors but in principle the method that a show on you is of particular relevance I would say to carry arrays of Johnson Junction humans but also called of atoms and optical lattices and to a lesser extent so to segment on track thank you very much for your attention we and cost you might so the question was how do I actually envisioned those breeding operations what would physically need to be done and this is not topological quantum computation so in our case moving those holes or twists is actually a set our operation so what you just need to do is a bunch of measurements so we would be say I have no I have no board here and so what you would do at 1 side of the whole say let's talk about holes you would do measurements that destroyed the stabilizes and on the other side of the whole you would to stabilize the measurements story pair the cold surface so it's a little bit like I think snakes walk like that so you're extending your whole on 1 side repairing the surface of the offer immovable this fashion so this is standard operations is all done in either case it's both poles and twists this is all done by 1 and 2 cubic case you need Nothing in addition yes also I think that it's the OK can just a thing it is just 1 repeat the comment the comment was that in the surface coal states you can still do the entire Clifford group without magic state installation that was the comment OK thank you very much for the yes the cite you history of the life we use the as the of among that you don't have a you know this was the example that you have registered there'll the response yes the harm there has allowed the use of the self at the end of the year was that I don't know I mean I'm not sure I think it's probably a better question for actor I mean I was not claiming optimality here in any way I mean I'm not sure I'm not even sure that this fall twists per cubic is optimal so maybe you can do it with less the main point I wanted to make about that those codes is that you can do the Entiat Clifford grew the a topological fashion so how much that buys in the and in terms of overhead I think has to be analyzed in more detail yeah factor abstract I see that that is that he was discussing high enough to generate those more like despair editions of the missing link their combinations of this 1 elementary trees totally Union to a minute there really is that yes you have cited OK thank you again the end of now continue your tutorials with Dillard will talk about dynamical 2 cups the you didn't have at that point
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Titel Fault-tolerant quantum computation with high threshold in two dimensions
Serientitel Second International Conference on Quantum Error Correction (QEC11)
Autor Raussendorf, Robert
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 10.5446/35304
Herausgeber University of Southern California (USC)
Erscheinungsjahr 2011
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Informatik, Mathematik, Physik
Abstract Fault-tolerant quantum computation with high threshold in two dimensions Quantum computation is fragile. Exotic quantum states are created in the process, exhibiting entanglement among large number of particles across macroscopic distances. In realistic physical systems, decoherence acts to transform these states into more classical ones, compromising their computational power. However, decoherence can be counteracted by quantum error-correction. In my talk I first give an introduction to fault-tolerant quantum computation in the setting of two-dimensional lattices of qubits in which only nearest neighbors may interact. Such a geometric constraint is, in many physical systems considered for building a large-scale quantum computer, imposed by experimental reality. It is relevant for arrays of superconducting qubits, optical lattices and also for segmented ion traps. Efficient solutions for achieving fault-tolerance in such a scenario are topological. I will review some of the known constructions based on surface codes and color codes.

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