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Universal Dynamic Decoupling and Quantum Walks in Functional Spaces

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Hi good afternoon everyone is so today I'm going to talk about universal coupling and quantum walks in functional space so this work is done collaboration with that and it's in the back off from rice University and the interesting details you can find the DNA of people and now also were recommend in you can also take a look at a call and in the paper which also have related topics and so for so since we're hearing a conference for want information processing 1 of the biggest challenges is actually the coherence and decoherence can be induced by system and the past interactions and for example we consider a system of a single qubit the most a generic system and Bassam Tak Shing can be retained the form which is just the sum of the system generators 10 so consider the ways the corresponding past operate so here it's actually in we tried old so and I may see that the sparse operators that had so they may not commute with each other and in addition the system operators they also did not commit to each other so this actually makes the problem a little more hard to keep track of and so and so the idea is actually the press has come from the presence of this system bus actions there would induce a entanglement heating system and the past and which leads to decoherence of the system so the idea of using the analytic up the coupling is to introduce additional so that the old evolution which is combined with the system bars and the system evolution together it will become just identity of the system 10 so with some unitary operation on the past so if we can achieve that then there will be no more no entanglement between the system and the past and so he in in this part here out particularly focus on this universal dynamic the copy which efficiently to couples the system from past for arbitrary system and about interaction so here universal means these bus operators can be arbitrary but with define a norm of course so the 1 and before we talk about these
university hopping sequence let's 1st take a look at this the haggle sequence which we are all very familiar with so the Haggler sequence consists of 2 policies which is in halfway evolution that it's us being and at the end of the evolution of Philips' being back so if we consider this a and higher cost sequence can efficiently suppressed this the facing noise which K induced by the system operator s the tensor so with the bus operator these and let's look at how the EU this system fast Hamiltonian they intimately advance in the presence of this system evolution and it's convenient to to you introduce this problem for income Ottonian which you can think it as a taxonomic linear associated with the the convolution so in the coming from 20 what happens if the 1st term which acts trivially on system commutes with all the operations on the system so there's no th no chance to distance but the 2nd term of the S V which I can enter commutes with the past so the 1st half of the evolution it there's no sign change of this but in the 2nd half there is additional minus so that we can compute the unitary obviously evolution which is consists of 2 to sequentially evolution and we can keep track of the term and it gets this suppression which of the the system bus infection gets suppressed to the 2nd order so actually this example et al a procedure that we would use to keep track of this universal dynamic that the company which is firstly right on the system bus interaction then we write down this popping from Atle Eun and the last that we compute unitary associated with this problem for so which
is outlined here system but public frame hamiltonian and in the end will compute the and for this and and actually other studies about this universal dynamical the cops and lots of sequences up a host and proved and the letter of sequences of conjectured to be universal that here is just the probably incomplete list that consists of a most features but immediately the overwhelming to see all these on the same figure so I tried to come up with a family tree for these then
it could become the sequences so the grand with the great grandfather for the sequence election and actually there could be elected to different kinds of the of the company sequences approaches 1 is called a concatenated enemy could accompanying which then you also mentioned in his sort tutorial on Monday and the other way that we be the company which is a little more efficient basically of impulses to suppress the this teacup into the high order but these 2 sequence to cite this alluring only suppresses the the defacing hoists commit so we might want to protect the interactive it fumbles division bits of noise for that goes to the 2nd generation of this than a critique of the so for concatenated sequence the catch the eye and there's a way Texas oppressed posted facing and the bits it and for the week sequence if he nested wizard to nesting levels which gives you the quadratic dynamical the company for the week sequence and the you can also combine the concatenated with which sequence come up with some content in the weak-coupling all we concatenate the effects with all the media corresponds to the the what which which comes the high-order which which corresponds the law become and and so we can go even further so the generation corresponds to the company sequence which can protect them cubic quantum systems so here if you may have some and the copy system bus interaction which might give off amount Petcube it's like X 1 X 2 times the bus operator and by having this and this did then and the copy sequence plenty of assistance those terms will be eliminated as well so when it matter generation will be nested concatenative because sequence of nested will wreak the companies but the see that OK on the upper right upper left side of the frequency is they all proved to be universal but on the lower left which is still the question we don't know if they're universe or not but the conjecture to be and it's important to understand the sequences because they have much they're very efficient which scales polynomially with all this success so in the rest of the talk I tried to and can you see that indeed all the sequences they are universal so before going to the proof let's 1st
take a look at the and just a recap what's the great dynamical to copy sequences so with that and could become the sequence is actually and equally spaced the company's sequence so the policy of the space in between pulses are not equal they actually obtained by some equipartition over a half and half last year and a half since and then projected along this horizontal axis and for example here and put it to you so you have to pass which kind of correspond to these 2 points on the circle and the for lovely form to suppress the grating noise using the it's noise would use a your economy could the phone we can apply spin-flip operators at these 2 at these 2 added these attacks so what happens is that the system opinion consists of 2 tons and which 1 is actually on the system the other this the face of and the other 1 actually on the system the coefficient of ancient the 1 that induces the defacing of the system which actually use and which has a bit but which changes sign every time you hit this spin-flip operator because it and their commutes so and the messages OK we know what's this F function gets it just the sum of plus 1 or minus 1 and the additional feature is that you may find that if we multiply these 2 functions with different something that's so here and often takes the value 0 and 1 corresponds to to the operational the operator so we multiply these 2 functions it turns out there's a simple rule that it becomes another function with the index becomes a binary some of these 2 something that's it so this is the important feature and you may find that this had a similar correspondence in the system operator we might apply to system operators to be 0 or 1 it becomes another system operator which the indices corresponds to the banners some of these indices because if you found this is true because the user just as the squares activities but you will find that actually becomes a little non-trivial when we go to the case for the quadratic chemical companies which we have a system and so here I will consider and the system of a single qubit which has the S X S Y N S the coupling to the past so now we have a generic you can write for the terms in the system that action and he hourly wage with the with a 2 batteries the role to 1 1 and and so 1st then we consider and that become sequence for this a quadratic than in the copying it consists of contact the nested that will read the company so in each interval of original we've done and the company sequence he put another worry inside here so therefore and so you have like a 1 week sequence at the high and the highest level and the hero played with the 2 you have story that week sequence at the low level and so we can compute these functions as the following and so for example this associated with at the function your such and decide every time it hits this XX operator and the American gets a function of the a with S S X operator and it's why operator and the feature is the again confirm this feature a property is a product of 2 function is another function but with the indices corresponds to the boundary some of the the 2 other team input and he similarly the s function and the system operator functions also have the same feature it's a product of the 2 is another system operates with indices being the batteries on that up to some plus or minus 38 a matter because actually always show that it is equal to 0 so plus or minus 0 it's Europe but keep these 2 features in mind as will use it later and so just a recap what we
how we prove the universe so dynamic with a coupling the last 3 steps we come right a system bus action would go to topping friend and computer unitary evolution so we have already goes through the 1st is that the next step is to compute the unitary evolution and the goal is to show this unitary is essentially at entity acting on system plus some higher-order terms so the way to do it is by doing passing expansion so this is the unitary then the the standard that formula gives us a bunch of terms and here and corresponds to the people and corresponds to a number of intervals associated with the and then in addition and so can plot these occurring in each consists of a bunch of terms and which which he can some all these over but the important thing is clearly you have intervals that will give you the power to to the end because that's the only way you can get property which is a clear allusion and the rest of the term consists of 3 3 components 1st is a coefficient which is associate with the time-dependent function induced from this the company sequence and the 2nd component is the system operator which are the product of these as operators but it's important to keep track of the ordering of the system operator and the last term is a bath operator which is the product of all those bus operator and so here is just the record definition right explicitly so just recorded only always say the system operator the product of the 2 is in that in system operator but was index is a band some so we can generalize this to end product of system operators which is a binary some of the indices it's again system operates and the system operator becomes identity only if some of these indices is 0 so done so if you like if you look at this path extension you will find that OK the goal is to hope and to show that this 1 is equal to some high interject system plus some higher-order terms which means all the lower the terms that means that this function should be the role for all of those and all other terms with any less and Catherine but except for some women but this is about to restrict because actually in this for these expansion if the system operator here is that the entity it actually OK because it does not and poet of our system so we can actually exclude those terms by putting a constrained that this system operator is not equal to identities in those extensions will try to complete the set the which corresponds to the Spanish some of those of us in this function is not equal to 0 so and we can show that that that is FT is coming by polynomial number policies and as we said earlier and but surprisingly you may find that here if you count the number of equations attaches skills exponentially with the number of and was all that you want to suppress because I can take 2 or 4 values and here you have actually impossible choice of office it will go to the fork to the end of 2 to the N so it's kind of a surprise like this will ever work because we have a polynomial number of and variables but explanation constraints but this expression a constant that you have some properties in there you may find that the number of intervals is always less and have to add to this my tells us something that OK maybe we can use this property is suppressed at to show this is all the 6 Commission on constraints can be satisfied so hard
to show it and our mechanistic aspects to show how this will vanish so 1st we use a property just recall early on that the product of 2 functions is neither a function with in this is the the sum of the previous 2 and with this you can we can rewrite this F function in terms of a bunch of intervals but yeah just like we this and squeeze these and the canon into this interval by the making this definition and once we did it essentially woman left behind with this a little and number of intervals and in addition there is no 1 left over a function which corresponds to an the product of all these functions at end so if you look at this and definition of a new definition for this function you can't think it that you can take a perspective this is a sum evolution of a function so we start from identity which is trivial function they multiply it with the F function becomes a function of the One the next step the do it maps from T 1 a function of the One to function of T 2 the next time we do interval from to teach history and so on in the end the maps to a function of T N minus 1 before it hits the lasting 2 that was because the number and with so it just keep this in mind that we essentially have n steps of evolutions of the functions and we would like to find some generic feature notion of functions that are independent of the I want to add to that had and so and you wanted to see it to facilitate the computation we actually it's Qdoba communion to switch to functional basis and the TWAS of functional bases in is the following 1st time-dependent like the coefficients in the public from Hamiltonian it's a this continuous function so that's why we actually I would like to have a functional bases also contain these discontinuities and 1 way to do it is actually to have a step function Khot in between 2 passes and it seems to have altogether Q passes still can stretch their function and in addition there will be some intervals and each interval over polynomial to increase a polynomial power by 1 so that's why I actually also in functional bases was the include a polynomial so this gives us functional basis and then it just like a translator language of this function operator int of function asking the functional bases which would be represented by matrices or vectors the pH of the table a list of his publications in terms of matrices and the effect FIL but maybe I just skip to these details in it so the next step is actually will find that the the the previous matrices like is 1 they actually block diagonal have some block features so we can get rid of this blog features undergo to reduced matrix which will be the size of the matrix will be proportional to the number of passes you apply to the system so in the end and we got it we got to the form that the this functions of the sum of all of these terms with some college and these terms have a feature that the total power of the the matrices if we summed together it's less than N so essentially we're not playing and amount number of the matrices but suppose an arm of the matrices is less in so so
then here is the last day and let us know that
this is 0 and hence there will also be 0 and this can be shown by 1st try some new maps and the late I Faculty of Medicine administrators to find a nice basis for the functions so that we start from some function which is a correspond to this party that every time we multiply the metrics it's increase its change the index back most the 1 then so once we modify the matrix which tend indexed by 1 so it's go from it explore the the original original points and also its neighbor the next time we multiply the metrics it explore the 2nd its neighbors so we multiply the the matrix multiple times and to it heat this respond the boundary corresponds to the vector associated with the other side of the operator and will find that actually for this Warwick case it's a 1 dimensional quantum walk and though the functional basis but convenient to use is associated with the free and similarly in go to 2 dimensions which you find that and which we can get the same thing but the full TIMIT to dimension just recall that will read and quadratic than any critique of the sequence is a concatenative sequence so we need the 2 indices to associate with the 1st level and the 2nd level so this will correspond this will give us a two-dimensional quantum walk and that we can find these functional a comedian basis by some like a free transform with a shift and that's infected because of this shift which makes it a little hard to prove of this quadratic dynamical company but the message is clear is that when we need to take any steps of the measures is before we hit the boundary which gives a non-vanishing value of this interval so and and the will 1 last comment is actually this proof can be generalized to this nested ory dynamic with the coupling and and all the other UTD you CDD sequences and there are also at this proof can also be generalized and to review the cases where the original may have some time dependence but this time dependences analytic functions of time so finally just the
inverse of banditry we can show that all these guys they are universal indeed and just
outlook and so far we have only considered the MGB system which generically care right correspond to SU 2 to the N group but more generally take your right S U N where N is the number of levels and if it's actually not clear if the if the case for few we can achieve a universal dynamic with a teacup sequence so they're happy some ideas of leaving factory would economy the company which I think it's related to just this talk and was the thing about something similar in direction all maybe it'll be nice to have some to show OK maybe there are such a universe of dynamical copy sequence this with level systems newfoundland at the and any questions yes no need of to find and with yes that it will then be added on that discussing the doing the conference so it seems possible and the trick is probably of we need to generalize a formula for and and so here the s is just the number which is probably not but we can we can generalize but still in progress so that was so because it is sitting quantum walks is it possible to consider areas inside of the quantum walk I mean as areas inside the Greek Our scheme and yes it depend on what you want to do but if you want to show its universal then probably you want to make sure all those errors than a show up in 1 work and the name each and once probably important properties this errors they look at all school to nearest neighbor is set out a 2nd year a and that's part it yeah that's something because it if you remember at presentation both my peer-to-peer showed that Udegei suffer severely if you take finite time resolution into account I assume you would have the same problem here and yes and and the legacy of the time resolution probably in principle can be improved a lot because you have with atomic clocks and and of the outset like could probably is in in this sense the may not be the optimum against and such like errors just the 1 there are 2 easy questions know what is the what is the what you reuse of competence of Europe special because if it does not affect the series of combatants there's no point in perturbation and yes so that it can work on this at an at the at in and by by is that if the Hamiltonian system bathing Qatar Hamiltonian found then that will corresponds to the covariance that with the condition the time of the sequence also testimony yes so saturated so why can consider look at from this and there is a bound which may not be optimum which is correspond to him but of the system that's patent agree that corresponds to the time what you the thank you again in
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Titel Universal Dynamic Decoupling and Quantum Walks in Functional Spaces
Serientitel Second International Conference on Quantum Error Correction (QEC11)
Autor Jiang, Liang
Lizenz CC-Namensnennung - keine kommerzielle Nutzung - keine Bearbeitung 3.0 Deutschland:
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DOI 10.5446/35332
Herausgeber University of Southern California (USC)
Erscheinungsjahr 2011
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Informatik, Mathematik, Physik
Abstract We study the universal dynamic decoupling (DD) schemes, which can restore the coherence of quantum system independent of the details of system-environment interaction. We introduce a general mapping between DD sequences and quantum walks in functional spaces, and use it to prove the universality of various DD schemes such as quadratic DD, nested Uhrig DD, and Uhrig concatenated DD, as well as previously known universal schemes of concatenated DD, Uhrig DD and concatenated Uhrig DD.

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