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Fault-tolerant quantum computation with asymmetric Bacon-Shor codes


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Title Fault-tolerant quantum computation with asymmetric Bacon-Shor codes
Title of Series Second International Conference on Quantum Error Correction (QEC11)
Author Brooks, Peter
License CC Attribution - NonCommercial - NoDerivatives 3.0 Germany:
You are free to use, copy, distribute and transmit the work or content in unchanged form for any legal and non-commercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor.
DOI 10.5446/35290
Publisher University of Southern California (USC)
Release Date 2011
Language English

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Subject Area Computer Science, Mathematics, Physics
Abstract Bacon-Shor codes are quantum subsystem codes which are constructed by combining together two quantum repetition codes, one protecting against Z (phase) errors and the other protecting against X (bit flip) errors. In many situations, for example flux qubits, the noise is biased such that Z errors are much more common than X errors; in these cases it is natural to consider an asymmetric Bacon-Shor code where the code protecting against Z errors is longer than the code protecting against X errors. This work provides fault-tolerant gadget constructions to achieve universal fault-tolerant quantum computation using asymmetric Bacon-Shor codes and controlled-Z gates as the only two-qubit gates. The qubits can be arranged in a constant-width ribbon and all gates performed on single qubits or neighboring pairs of qubits. In the presence of biased noise, these constructions allow for powerful reductions in the error rate with modest resource overhead.
dominates over other sizes noise like bit flips or whatever else and we expect this to be true for many useful systems for example superconducting flux qubits and so we wanna be able designed error-correcting codes and then fall tiling gadgets that can take advantage of this diffusing bias in the noise so we use making sure codes which we've heard a lot about this conference or just briefly review their properties so they're a family of Quantum error-correcting subsystem codes they include a single qubit into an M by M block of physical qubits and that asymmetry in the title refers to the fact that will consider codes which are it's a wider than they are tall and without corresponds to is that we have independently tunable different levels of Xeon X error correction of for example this 3-by-5 walk which are depicted concoct up to to Z errors and then a single Exora in addition so we had as is of this form where the phis stabilizes our product of X is on 2 adjacent columns or the long to different rows and logical operators in our pack of X is on a single rollers he's on a single column we also have this this gives structure of gage choose which we don't use to encode information and we can think of that the gage cubits as being generated by translation of of these 2 patterns there's these in a in a vertical alignment Rex's in a horizontal alignment and with this essentially corresponds to is the fact that the only that that the parity of that busy information in a single column matters because any other information we can remove by application of the gage qubits analysts property which we've heard about is that using these vacant shot using the gage qubits we can actually build up a measure of a say a stabilizer and so we can do you just use to body interactions to to build these up so the error-correction measurements some pattern of Z areas that has occurred then we can measure using the XX gage operators we can measure the at the parity information in each adjacent to that's and we can build this up into a stable either and then we can continue doing this approach Hairer Collins the Iverson information and then using this enumeration meaning we can apply a correction so in this case we apply based on this to the Polizzi in the 1st 2 columns it doesn't matter where and then after we apply this fraction in may look like we still have a lot of errors but actually all these are remaining operations are just gage of freedom so there's no information encoded and is but according to Miller in
general the error correction will fail of more than half of the columns of the code have on number of serious errors or more than half the rows having an item a of X errors and because we changed we have different length and width of the the block we could be a different production so it's so
we've seen how these codes different power to treat serious versus X errors and 9 usually how can also design faltering gadgets that provide protection again treating of zeros in giving more protection to years the next areas and some the key ideas in this construction on that we use a fundamental gates at which is compatible with this idea of a bias noise we use the teleport seen not gate as encoded on and we applied magic state distillation to Ritchie don't full universal set it so what I mean by a bias compatible hits at is that we want our our serious to be more common X 0 so if we have a gate like that Margate which transforms Bezier into an accident will automatically lose that bias even if this year started out as more common every time we had had a mind there will start to get more more x areas and we also 1 try to avoid cascading errors and gates as in this in the out where a single 0 can propagated to zeros on the upward horse ammonia single ex-air compatriot to 2 axes so far from that will choose just 3 operations Our preparation of qubits in logical plus state the controls consoled phase gate and measurement and the X basis and the control the it has some nice properties so this yeah would just odd we just commute through it and X here will produce on the output X air and also 0 and the other out so errors can only spread as far as but you but they come from or Cuba that's directly connected to that came through seize control ticket and also depict these picture early as follows so the apostle indicate this plus preparation this indicator controlled the gate and so the measurement
so we'll start with this fundamental gates at which I described and will use our they can short codes to implement this this other gets and at that point will have some weaker level of noise and love lost the bias in the noise and so to reach arbitrarily low noise we can top of this an additional code and will use of magic state oscillation I seen injection distillation to provide a universal set of its we might also be interest in the case where after they can sure code you know the errors an arbitrarily well if it's low enough for our purposes and in that case we can just inject and distill directly into the Universal so to do that
teleported control not cable used the circuit here and to see why this produces a a can control not just consider the case for example where there is the 1st input is the 1 and the 2nd 0 switch expect the 2nd but to be flipped by the control not and in the circuit the the control qubit comes in on this block and comes out here and the target qubit will come in on this block and exert on the 4th block so these are all of logical operate operations so if we measure the 1st zz measurements then battle projected onto that portion of this code with this state with even parity on the 1st 2 cubits and then when we measure the 2nd 1 we product and even parity of these last 3 so we end up with just the state and if we measure the intermediate to or just left with that's exactly what 1 with we've put the 2nd the circuit will also have to perform the error correction so we basically Teleport the information on the fresh until every time we go through controlled market so we have 3 bonds we need to do we need to
do plus preparation and the way we can do that 1st by preparing each key in the each individual to bit of the code block in a plus state this'll commute with all the X-type stable operate what stabilizes and the crack thing but it doesn't commute with that the z type operators so to fix that will measure those and we can do that by introducing some ancillary qubits preparing them also in the plus state and then coupling them with these controlled phase gates and after we measure the controlled phase gates if all these results were 0 then we've prepared exactly the state plus but if these results are different than we've actually prepared some other state but it only differs by local poly operations on individual the qubits and we can just keep track of that and we should
repeat this measurement multiple times for fault tolerance so to do it and that is actually very
simple adjust we can just measure each of
these that's in the x basis we form groups according to the the column and compute the parity of the result of each column and we simply take a majority vote of the outcomes the parity outcomes of each column so this is a very simple acts measurement and we wanted to use the measurements as
well and wanted use in a nondestructive way so we wanted just take out that the parity information and not disturb the the rest of the state we could imagine doing this with a single ancillary qubit for each role which we prepare in the plus state I interact with controlled the gets and measure I and then we take the majority vote of the the real outcomes this I have a problem because I is not fault-tolerant science is a single layer on Celexa on 1 of these and sold qubits to propagate errors on all of the documents so a possible solution is to use a structure like this where these where we prison with this essentially as it prepares a cat state along this block this talk and we we use a catheter we but the cats in the parity information and the cats in measure it but then there's a trade because now we have a very large cat state and the larger is that there's more chances for it to have errors so
we can imagine some intermediate trade off where we have some smaller cat state and now at there are 1 to many areas but but there's a 3rd they can't probably as far and so perhaps there's some tall off especially since the PX errors are assumed to be less common in any case
to show how this on measurement works we prepare all the in solos impulse states and measure these intermediate ones I was prepared cat state of the on these key that's 1 for each
row and then we couple the they're qubits the cat state and complementary to the cubism I cat say and I will tell us that the parity information again we have the problem with these us these measurements are arms are not all 0 then we haven't prepared exactly the constant we've wanted but we know how it differs from from the cats that work that we do what and to do these longer it
z z and I the party measurements of 2 blocks 3 blocks we can as imagine extending this cat state to the adjacent blocks their range kind of adjacent to each other in a in a ribbon so analyzed the there we will
study this under at a local stochastic of bias noise model so that the 2 separate it's epsilon forward the phasing errors and secondary epsilon prime for arbitrary errors descending weaker I will define the bias as the ratio of these 2 their strength I as I mentioned a key Foucault is ensuring that the cats are prepared correctly because if there's some errors during those measurements and and we the wrong the reason that the cats is something that is actually not that can actually lead to an error almost immediately and we need to be very careful analysis for not and once we taking care of all these things we can arrive at an analytic upper bound on the effect of noise strength is just a polynomial in x 1 and epsilon prime and degree is given by the code parameter of the block size and and the number of times we repeated to these different kinds of measurements and so we can search for a given strengthened bias we can search for them the best parameters that minimize the effect of strength and that's done in this part
so here we have 5 different values of the bias when from 10 1 10 100 thousand and 10 thousand in Seaford for advice bicycles 1 these codes actually don't in this range they don't do any better but then not including at all but once you go to higher and higher bias they do better and better so I biases 10 to the 4 we have a threshold of pseudo threshold around 2 times 10 to the minus 3 and on that in error rate of 10 minus for weak and ineffective their strength which is below 10 to the minus 13 and so again each of these points it represents a different the belly of all these code parameters the site the length and width and repetition rates and so it the optimizer at each point I can also look at
the the the resource requirements for this code so in this part and again we have the standards for different values of the virus and we can see that they as you go to higher and higher bias you can you can do better and better and this black curve show some data from a survey of codes at for depolarizing noise and I and numerical study of how how well these codes perform in terms of this X axis which is the number of control not for the black curve or controlled the gates for the other curves the number of those gates in a given rectangle verses the of the logical theory Oceania physical error rate of 10 to the minus 4 and that's if you're a fan had biased sit on the floor but you Naudet you'd probably get very close this black curve users and changing different 2 different kinds of codes if he's there if you take advantage of this bias you can actually the 2 codes which for for the same overhead in terms of number of gates give you increased amount of protection so summarize we design fault on
gadgets for these are it's in the chip-making Chaco's we have approval upper bound on the error rate which achieves a significant reduction in their strength a modest number of gates and because of the structure these they control cause we can actually possibly also lay out these qubits and it's an and geometrically local fashions long thank you you and question repeated if my brain working I can probably work that out from your resource slide but for for us e to the minus 4 where you had really low error rates with an eye Africa with the bias was how many levels of concatenation where is it just the 2 the 2 layers do you need more so it's a for these for this part it's just the list is the bake ensure code and there's no 2nd level concatenation but presumably conditioning similar with with a planar code and if you just I want to encode a single Cuban by the code you could make it by mentions asymmetric red how does this scheme compare with that have you thought about that so little yes I haven't done any analysis on that but it's definitely true that you think of others schemes where you where you have 2 more independent control of these by different size and and length yes although the gates to give an operations OK so then so you mean this picture so the eye but essentially what you're doing is you're preparing of you you doing transversals are controlled phase gate with it with a twist between 50 blue good thing which is the the dative it and then the other qubits are a 2nd code which is kind of the opposite so if it's the 3-by-5 which about 5 in the other is 5 I 2 is prepared in the plus state but the desire these intermediate qubits are just used to prepare the cat state and the reason for preparing the cat states are too to prevent this problem of from this previous library or a single act ceremonies and solar qubits completed who is the most players on the so this thing here and the rest of the speakers today we are


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