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Smallest Errors achievable by Dynamical Decoupling (and How to Maintain Them)
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00:00
but we coupling and the the organized in the opportunity to and you see it's about monitoring the lines of the company and I think that many people my it did not do this is this better the but it was about this about OK so load OK so here's I
01:05
mostly focus on the dephasing qubits which is the familiar system is like it's been boson defacing characterized by the coupling of the qubit tool without it modes or you could think of a classical fluctuating fuel gaussian random variable along this thing and here the formula that it's clear that that describe the coherence there that means so do call the metrical years I have is the after the decay of the of the elements there's K and the K. value or the lover among the ratio of these 2 is given by this formula on land of omega is related to the spectral density in case of designing fast or the power spectrum for the gaussian random right so I define this object here which is sort of the noise spectrum divide by omega square the of major and to this logarithm of the ratio of the coherent I call coherence error on this error and this is the starting point for me so there are many things that we can learn from the spectrum in what a free free decaying particle free Cuban but I know that the coupling differently would correspond to the sense of decoherence us the cut off frequency is going to respond to the coherence time scale the softness of the cut up is going to respond to approximately defined timescales because going to change the effective the change the cut off frequency an interesting thing that happened the power line of the noise near 0 frequency is going to affect the longtime so here's the model that using sometimes in this talk 40 spectral this it's a power last year and indicate where there's legacy case the softness of the this that the cut off now this is still an explanation became so it's not the 1 of of a square decay that you sign for the talks and a famous cases omega corresponds to a sequel to 1 this is sort of a pinkish or monograph is as equal to minus 1 and a flat result correspondence about you of course this is flat will recover so I don't this is not exactly whiteness is just flat of so here is an example of 2 different frequencies and see what it does to coherence a difference of an order of magnitude in this state the changes the behavior of here this is this should be very familiar to you from all of the things that and that's the spectrum as you promised this was done OK so let me give you a couple more slides on the freewill mission so here is different with applied this is the list but it's clear the functional time and here I have the logarithm of clear slots so it starts from something small and then grows and the different cases this to Prall make all make flat and something which is almost like 1 over what exactly and there's different behaviors in the the 3 top cases where the noise is all make war so there is an unbounded unbounded including 1 in this case there is clear saturation or residual coherence so this is like only and in this case we get a clear saturation this guide never uses its coherence there we should users and there was something the post on so that the message is that different environments different partners will correspond rent coherence behaviors especially in a long time and for sure short time as well so the more you know about
05:06
the expected and the ordered environment in general the more you can do to suppressed so I just give you to sort of general approaches to this optimization for in the optimization for suppressing Here's the 1st example is about continuous control and you constraint totalenergy acting on the system and you try to compensate the noise for a 1st order and knowing the the phasing spectrum this is a work by Gordon Kristen New York of a new 1 by the christian group or you could just go back to the coupling model and try to find the best timing for pulses to optimizing the sequence for a given spectral as so the water might there's a did then the at NIST was rated that they didn't optimization even empirical based on this but they are and also the scene and we did something like that based on the power loads of respect honesty and this work is also based on the part of the spectrum so there's all these optimization methods and improve have you can come up with to improve the the coupling with pulses or just continues control or is is suppressed years but is a fundamental limits and the best error that you would achieve it the control so I'll call this user contrib because what he chain and system is the Hamiltonian just supplying at Repulse university do measurements or recently in propene contain so here's the premise of
06:34
that the Governor's which to we my focus that you know very little about the environment except that you probably know something what kind of frequency you know the algebraic structure of the noise which might going to be phasing is proportional to the you might know the noise I have some limitations in control which is very important for example you you have limited pulsar timing you cannot apply your files your pauses infinitely fast you normally don't get dust upon that the ship policies also so you have limits on the speed of your past and you also wasn't there can then you apply opposing you get something it's I will focus only on the 1st of which is the bare bones constraint this thing there's only make things worse there's tons of efficient unequivocably no I'm going emission somewhat you to go that relate to this premise OK so the pulse sequence has a starting point than in point starts at time 0 and at time t the impulses in between and the true ways of characterizing the control question is to think about the bound for rate which is that the pulses can be applied too close to each other so there's a male separation between the times which is sort of matter how infinite traits that is going to break your system so there's a technological bound and the minimum interval the sort of obvious constraint is that in the decoder postings the tides are going to be marked with a resolution this part of his beyond what we would normally consider but when you're playing many policy then you try to make a point is very close to each of these both of these countries on board and they're very similar act so again the air that I'm going to look at is the of the the decay of the offdiagonal terms and here is how the control for of system looks like it is basically something that fluctuates it didn't get updated so sorry this should be a signal x to power y t plus 1 just something that fluctuates between signal x and identity and by the streetsweeping plus minus plus minus 1 and y of t but is appropriate so there's very minus 1 the poster being gives you to offer OK so this is a letter like this setting on my problem and it's very simple and before again like that the approach that we give you 1 more introductory science so they're all would like to think that the 3 objectives and of the coupling the first one is probably the most heavily research money is to just that you digestibility for a given time and when you tell yourself that I cannot as many parts of 1 and there's a perturbative solution to this problem that we all these named sequences the other the problem is that you would try to do the best video in the shortest time and you have 2 different strains because you cannot apply things too fast in fact you have you started building blocks that policies or like intervals and want to combine these building blocks to get the best get this sort of like a dating problem where you want to do some operation quickly and I see the other problem and those have all these 2 problems the last 2 works is that you want to do something nicely and what do forever would take as you have a few times already you you can maintain coherence for a long time so what is the best you can do and if you can do it for a very long time to see all these would the color of acronyms and that's just the signature of the the NMR influence so it would not so we see that for the function and people the that might might said that he prefers functions to the k graph you should be careful for over what you wish for and socialism so here is the overlapping today so the air captured by an overlap between this spectra measure and if it a function and find of often is really the Fourier transform of the sequence provided ordered phase of the actual provided and there's a very nice compact formula for the for the function for a given sequence is just the sum of the Spanish and the different asleep term tj if you remember the where the timings of a sequence of write the argument of the function so you can actually come up with something very similar to this for the for a continuous control case in that case the year you get an integral what is that all still be related to the free transform the face of the continuous appropriate that is for defacing it's easy to drive the approximation which acquired the born approximations and it's used for continuous control optimization theory so let me just say a few general things about this that a function so you could say there are 2 ways of thinking about the subject once the usual idea that an overlap between the for the function this spectrum measure that you want to suppress some part of the spectrum thereby by making the function small in those areas the like that or you could think of it as the weighted and 2 more of the Center and waited until the weight here is the spectrum edge OK or that out of the mathematical term for it these the thing would overlap is that if 2 frequencies a small 50 cut off frequency this small you don't need to worry about the other parts of the spectral density beyond the cut off just need to make a for the fucking small tried this and and this is the essence of perturbative become that that if all of us see this small just need to have a look at the expansion of the universe unions 0 and this is the essence of all these namely covering the so for example the and you get all the suppression which I think mean find in terms of the magnets patch here can define from the the center so for a free evolution the order of the suppression is 0 because the flea market proportion or echo this sequence of equally well equally since bosses want recipient geeks tool for UTD NCE could get any desired and at times here that the sequence usually manifest themselves as pseudo peaks India if you plot for the poor and Fig. 1 of our lives in many of these I mean we can just plug them play with it and you see all of these things there's is nothing simple intuition doesn't tell you what OK so then bring in the year that control constraints here which for me the main what's important is the middle interval between the that a fundamental time limit in the control that you get and you it would be a good idea to use it as a unit so let's just do it is that a unit of time define these the murkiness parameters delta and the rewrite the for the function I get a polynomial in him but this vital which is somewhat say their function in terms of the negotiating or men went the spelling and if you have the the constraint in the year up pause time you get a normal on here otherwise you get something which is called condiments polynomial which is a polynomial with noninteger export I know sustained existed before I what we do that so it turns out that the fourdigit constant you can expand the million and factory to look like so starting from from 0 and going up to type and plus 1 times the resolution time the true 2D and possibilities of have different sequences the 2 to the n different sequence of that you put up like in this along this line to put the and slots for act like will not applying tuples and if you write the polynomial And you if you try to and if you factor out you see that each sequence corresponds to a choice of a little would on which is just this simple pronominal work efficient could be 1 or minus and according to police and there of interest to mathematicians
15:02
so in band a coupling interested in the for the function which is just a play on it is a very simple object in fact the only interested in how it behaves near 0 where which response to it is equal to 1 context that suddenly is
15:20
open only for the function and we are interested in the sport and 3rd order that increase the this a circle here to show and there are many routes here but you don't see them because there they are of density 0 so there's a very large concentration of roots right on the equal to 1 and the rest of it just obviously you can see of many Terence that can are trying to get out of this picture and while we went out would this was
15:50
so as we said the error of is an L 2 norm and the to you can show that it's larger than and in 1 over squared but some anomalies H so the error of at the coupling sequence is always larger than some of the 1 norm of some figure so that's nice way of thinking about this because I had a polynomial it's and complex functions unwilling to look at how it behaves in the unit circle and I'm going to and the interval of from 0 to become of frequency is going to correspond to an arc and the unit circle so just find your balance this way but it so people have repetition have thought about similar problems and that's a good thing so there is this little conjecture that the L 1 norm of and the people number than the unit circle grows at least not this was proved in 81 why might be opinions independently by conducting then if it was extended to the case where you don't have that the regular Polonia you have the most polynomials but you put the condition for amino separation of the exponent is with proving by and then more recently a similar result was obtained 1 at a lower bound with obtained for the suboptimal use circle by the were which after a suggestion Thom uttered the prove that you could say something about the most luminous of part of the inner circle is he said that's why I have a method but at the interest in just so this is the more familiar control which I know this is this conclusion which is the undesirable control that is when the controller timescale is too slow for the coupling in that case the L 1 norm or 2 nor corresponds to a full integral over the whole unit circle Digital pulsing when they're so by at least what which made it implies that if such is the case then the coupling is not going to be able to the air for decoupling will behave at best as long as they were in the number of what so if the middle switching time towns the spectral density cut off his eyes into by the carbon of this is from miracle I'm not saying that this is Hansen but it's nice to prove it and Tao here again is not the total time is only the interval ineligible and get so the control times still his talent and commands are the only factor that be so that's sort of nice but you need the year the Artur regime which is the regime where the coupling works so what do we get there this is what you get there is a 1 word and here and here is the the inverse exponential function which is and central similarity that is when the inca when you decrease how it goes widely to 0 but now it's widely so this is this is not part through the roof of innovation and vitamin it gives you a lower bound on the error for the coupling which is a nonperturbative result again I repeated but there's no that there is you can expand this functioning but emission here this is an essential things that and hinted at indirectly proves that there is no attraction for the that means that when a celebrity so the tension here has to do with it what to you the best theory can get with the given resources and it says that if a given resources are better than something then you can get errors you in finite was not a fucking approaches to this is telling you that if you're resource which is the minimum switching the interval is finite you can never get 0 and I would it can be very small but not here so there is no here so there is some thinking that maybe you can break this is all for the for a finitedimensional environment but you're not sure there's a similar result that you can get for dinner for the cases where you have a continuously their function for continues control and can get the function we saw approximation and that a similar result acceptance that the minimum interval you get the maximum industry so you get a similar look result for continuous control so that was sort of anticlimactic so let me just there to explain to them that show you how you actually see that there are many different sequences or sequence families like CDD UDB Europe began order alpha used to the alpha posters take to the upper house times would you did the you too good to order out for you need all of alpha policies and it takes all of the Square Times Tao time now telling you these the duration of the 1st interval so if you think differs in the Royal plug in the formula and look at the total time it was carried off a score they can Country but it but the UTU this far fewer pulses and shorter time density of but after a few about better was sometimes you need to put a longtime then you are willing to apply many balls like me we are hearing area thousands of policies hundreds of all of you who are willing to do it as 1 of their coherence times extend so maybe they will actually like units necessary but the reality is that I mean you know you want to apply to many parts they made because they're pot and the other thing that your system will heat up things like that but anyways so few of them not necessarily that that could be better the depend on your control so here's a plot of UTD errors when you constrained amino switching and the the different UTD sequences and this is this axis is the minimum interval in dementia say that this is the error and this is and a lower bound that we found you see this interesting variety not here again which corresponds to the back to this factor know what happened is that there is no treasurer you have to push it back to the so there is no single we see how value that expect everything to go down there is no that's common inflection and it turns out that the optimal path the optimal duration time for a sequence for you the sequence there is proportional to excludes would want to support so if you want to do longer than that or then there you will not work in this case when it is a nice my son now in this election very similar to the rest of the lower bound be proved and the fact that I showed you the sequence to mind to at a petition colleague he was excited because he thought that it related immediately to some conjectures in number theory but I don't want but it's so what let me just show you this 1 so if
23:29
you fix seek out here and you start applying higher and higher order duties sequences isn't always always an optimal value for the order of a sequence that works which is this picture if
23:42
it's a certain time and then you start writing all possible UTD here that you get a certain order 1 2 3 4 with slightly longer longer durations and there's always an optimum but the sequence so the different colors correspond to different on softness in the specular there's the cut off and in all cases of using an only part so there's always the optimal value it's not that is not gonna work if it might be that the optimal UTD sequence might be like this to be its included among these always better use sequence depending on the sequence on the control capabilities time the Court of Requests so the correct frequency determines the best sequence you can so that was searching for Insiders specific family of sequences which was the you sequence but you don't need to be constrained to be family you can visit while the family and then you get what we call the bandit set up that that good or bad and this is found through numerical search it is always adapted to respect to measure and the bandwidths of the back and you just can't figure positive to be larger impact and that's their main concern and then you keep increasing policies to get better and better video and longer time the search space is the search space of the number of intervals that you have just have more intervals you get more vigorously space and it grows expression number of that's not good but I was it's doable you could so here in this graph is a little complicated but the nice thing about it is that corresponds to this lexicon qubit with the some experimental experimentally verified mountains there exists a problem in perspective and the power of 3 and the court after that tree radiance of over picoseconds apparently the experimental every will post with sort of from 2 seconds and the intervals our OK so you could do like point 1 because in the the interval of point 3 so these this is the curve that actually before except that the we already passed the the error increases as you add more positive you these this the INS factors they correspond to the optimization solution fonts will be a and you fix the switching time and then you start looking for sequential decision tree intervals for intervals 5 intervals so on and just explore longer and longer time no change the constraint 2 comma decimal 1 picoseconds again the start increasing after the numerical search actually improves the core here EU using longer more pulses dream is improving the Koreans I'm mentioning at 0 dB here only the the sequence so that was discovered by 1 so the the generated by my peers that they discovered the idea and what they did was they fix the total time and they just added more and more pulses so if you add more parts your inevitably going to shrink the intervals so you can always match subsequences is going to Connes each constraints and develop the these are the the view that we almost at the constraint that the so there are also similar to the media in the audience so what open question is that if you have say a flatworm norm equally hard cut off some given the spectrum the measure this possible to find the best sequences that satisfy the constrained models which account constantly and I don't know if it's possible it will be graded so it can do it because then you could permits them into other spectrum measures that so I want to I just quickly go back to a different measure of complexity 1 of the the control of the complexity that mentioned here was the time between the pulses or the pulse rate that can apply and assist the outer major is not he would have died you have response and under certain of light if it's just to kinda sequence or so simple equity when this easy just apply sequences it's very easy you can do with simple circuit but the more sophisticated this sequence becomes harder to do it on board Latin African into computer to program the controller so you'd like we're you don't want to like just for keeping the coherence high you don't want to apply like bigger electric circuit on 5th but in which the so you have to improve the video sequence and in electronics people have been doing this since the seventies using watch which different shorter than just their family a piecewise constant binary value functions and they provide us is on semiautonomous Autopilot enormous false bunch well you can just define a sequences it indefinately dear recursively you start a sequence and double up the running time of it and then we decide if you want to sync prepare for the 2nd part we want to invert the propagator so from the W. D and you can make wh when and twin plus 1 the difference is that this in w twin plus 1 the 2nd propagator which is your memory for the year 1 or minus 1 is inverted and to achieve the inversion in actual 1 this is a provision to actually achieve the burden you might need to add a pause or not articles depending on the parity of the sequence there for example you are a way to define them was in terms of the government for functions which are which is defined like this there are oscillatory constant of square shapes with the frequency that correspond part of 2 so you take the number n and you look at the way presentation and then you were the voice function for the interview and like this it looks complicated delivered of this course are easy to make it in this way or that they have here's the the the polynomial for the watched a couple so then had the idea of what the company I mean the way reach me was true a paper by they face at the this Maryland and in Kras Monroe's were did this for down the 1st moment date or don't lineage mitigate and they want to suppressed so frequency remember I'm not sure what that's what they use is a life and I look at what I said all this is like a dynamic you created gates of that indicate and suppressing years the live close yes I no this is concatenated though the cup FIL that that could have made that's equivalent to the was so here is the 1 providers for what 0 231 0 propagator 1 is like a single article in the middle tool is more QoS tree is like this you can t and so on and it 7 it's really easy to make this on a circuit this is from an old paper what function the Fourier series the year numbering that there he's using here is different from the numbering of using here the different orderings for what I don't know maybe this is easy I don't understand anything of this they look like for this 1 so it's easy to make them according to the electronics people well
31:32
there is why it's a easy to make them is that they correspond to binary finer frequency that is frequency 1 frequency to 4 against war and there is a tumor the European electronics and that's quite a nice and just have to combine these 1 if produce any of the what function that you want so there are the attractiveness money every DDE functions which is that as I said if you wanna go to say from an interval 0 from time 0 2 it's time to to the end time stop you would require you would end up with 2 to the end totally be totally and possible sequence but if you only look at the Y sequences that apply in this running time there's only 2 to the end of and they shrink the search space exploration the search space is what we're looking for we're finding the BAT the sequences which nice if not that so let's say use the what functions to have a look at the search space so these plots correspond to the error values for applying DWash function from orders want toward a 5 12 Gy headed for 2048 but I must of my presentation side to renew this this search theory about reactive the green points correspond to the local minima which in this is look and minimize the minimum there in that time range so between 8 and 16 of 16 3 to end these are the winding representations of these the indices of these what functions to cite this so they so 1 is a kernel this 1 is simply gee this wall is CD tool and this is the street this 1 is to treat repeated suturing fewer twice reviewed more and more and more so they all request 1 the 3 so you want me to go beyond the treaty disheartened trees and use a soft part of so just how many sequences we have a nominee was acquitted vehement those achieved a account of the heart this chasing together about for each point pattern and a set of optimal value but again the search space is very quick we can look at a very large span sequences in this way and here is some of the larger than the watch faces you don't get to see the you get it off to fall within there's a CBD which is repeated and then concatenated again so that all of these companies of concatenations and issues here and then here you are just trying to find the best yes Eicon capital repeat or content made again that's the idea so this idea of
34:20
reputation is very nice because repeating sequence effectively changes the spectral density of the environment so you could start with an Ohmic environment and you apply sequence and a repeated what happened there when you're looking at the long time behavior you're looking at 0 frequency a different approach and sequence and appealing it you can adjust the power of the expected so Here's the formula for repeating some sequence that is if the corresponds to the function of something they repeat it and and if this often behave nicely and if we look when and costing this is the reminder that then you get this adjusted for the function for when you repeat the sequence heading It's similar to the old it's working you get this extra Sanskrit Indian I think they would this and substitute the order of the suppression and some other of the power of and get this condition for clear saturation so let us quickly look at this if it's free evolution of ways Europe then the condition for coherence so saturation is this great event so you need to be released a problem to being in the Baker the debate which if you use like echo sequence then you add 1 here so you can be that you were the power and so did agree as such so I really in the clear centuries when saturation means a guarantee of coherence and that's what's really interesting because you can not just reduce the errors but can maintain that for a long time and we've seen this in almost all of the copying parts heard in this way of life that people can actually get T 2 kinds of key growing until you hit people and that's the idea you just keep increasing so if we go back to my
36:23
wife Châu functions and look at the value of the error I did forcing the sequence which is the right 1 and if I repeated infinitely many on the number of times which is the green stuff the blue region is the region where it the years to large at a what the green was and the different grasp was what a different values of the so obviously repeating a sequence increases the but not necessarily by a lot large FAQ and sometimes you can this some sort of thing going on here I can only as for illustration
37:04
so the reason why we think that the it's important to think about longterm coherence and think about bandit that up to longtime back the cooking is that they would come in at just a part of the noise and the limits that we have found for the coupling do not depend on it with the only depend on the control capabilities so if that is really true then you could be a you should be able to get a nice quantum memory that operates at the best available Fidelity's for a very long time and if you take if you if you ended up repeating a sequence to achieve this long time limit then you have something like a nice because if you're repeating is because human family that anytime you don't need to wait for a sequence to finish to some of the have low latency quantum It's so obviously the foreign because you can add sample like you long and elaborate sequence in the middle of review always in a bit larger in the middle of the repeating a sequence would always be a sample at the very end of the sequence security at any point working but to which you sequence shall we repeat act so here's a tool as sequences that they're being repeated 1 of them is UTD 6 some of them if you had nearly 10 is the best sequence for this switching time which is play were 64 units the scene 1 2 6 is not so as you see good but single repeats the 10 does much better and these but if you keep repeating the error for the test being repeated keeps increasing it by that saturate but I can't money 6 was 6 which starts with a bigger saturates at a much smaller so the choice of the sequence to repeat it is not for and you can actually get for the mind to the different in the while doing the lectures and the and here is just a comparison of the BAT the works as PA l or bold for a long time so being these that you take the for the sequence and the European many times using a formula it's the error and for that this graph corresponds to this previous picture so the green curve here has a minimum value actually the 6 while the red ones cost was to a single cycle has a military 10 what issue it took 10 undefeated you then about 10 months FIL which for so done so years that that you know that the company can effectively just the spectrum and the only 1 and the main the main main finding is that the limit is always bounded by a nontrivial function of the noise band on our control capabilities and we have found ways of maintaining that small areas for long times and I have to mention the wife of the company's sequence which are based on the simplicity of generation in terms of electronic could erase it to generate using minded circuits so here is our acknowledgment and our collaborators thank you very much for your attention thank you so unexpected consequence of DG theory is the ability to prove theorems in number theory the top 10 years closely any questions thanks and like really how you sector calls for your sequence and I think that's very helpful for us we he found the same correctly you always assume perfect pulses for a yeah this is very limited to be the simplest what military data here do you know what will be the Walsh function all his it or the war of verify our because itself IT strategy and look very like Uber fractal but in the the other thing is that they are a and over a complete basis for interval them to the null that letting the because the like based the year that it doesn't have any particulars of there there's no combatants OK thank you you more questions so that the for this lower bound on dynamical a coupling is it somehow related to the fact that we're not extracting entropy and trying to pinpoint the difference this and fault tolerance but I believe that that's the difference here because a lot of people never entropy there's military but the me I really cannot prove as the big thing OK so that's what I was getting at is is that you see a path to a connecting this to no entropy extraction that this your approach once you know what the reason the idea that that in this was adjusted to me by daily plan that there are cases where the norm of column putting codes do not work which is if you just have a cup into an single qubit Hamiltonian and you know there's such a nonMarkovian system that was going to not work for that but the coupling works perfectly for and yeah he where just have a Markov environment and the coupling doesn't work at all and since that in 1 case have case you might also good threshold which I'm not sure but there might be a tradeoff or some sort of entropy the argument from here from the Michael Dennis of the Amir the 1 just a comment on that I mean if your environment is sufficiently simple you might be able to to get 0 error literally just by reversing the head the mission of the environment exactly alright so thanks again the and as if the surface code to people could really take another DD talk very go blasted he thought for today
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05:04
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National Institute of Standards and Technology
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Titel  Smallest Errors achievable by Dynamical Decoupling (and How to Maintain Them) 
Serientitel  Second International Conference on Quantum Error Correction (QEC11) 
Autor 
Khodjasteh, Kaveh

Lizenz 
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DOI  10.5446/35326 
Herausgeber  University of Southern California (USC) 
Erscheinungsjahr  2011 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Informatik, Mathematik, Physik 
Abstract  We explore the fundamental limits on coherence preservation by dynamical decoupling methods in terms of control time scales and the spectrum/bandwidth of the environment. We focus on a decohering qubit controlled by arbitrary sequences of pi pulses. Using results from mathematical analysis, we establish a lower bound for coherence loss in terms of the minimum time between the pulses and the spectral cutoff frequency of the environment. We argue that similar bounds are applicable to a variety of openloop unitary control methods while we find no explicit dependence of such lower bounds on the total control time. We use these findings to automatically generate "bandwidth adapted dynamical decoupling" sequences that can be used for preserving a qubit up to arbitrary times with the best fidelities theoretically possible given the available control capabilities. We also introduce "Walsh dynamical decoupling" schemes that are optimized for digital sequence generation. Our results imply that fact that, unlike in quantum faulttolerant architecture, errors cannot be reduced indefinitely using reversible control methods yet a small error can be maintained for a long time. 