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Dynamical Decoupling a tutorial
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00:00
and you try to avoid the topological twists that we saw Robert do was the mike and the pointer so I've been a switch gears a quite bits and we heard a lot of abstract approaches to doing quantum error correction and fault tolerance this tutorial will be about a much simpler approach to reducing the coherence that this is the the technique of of dynamically coupling it's not by itself a stand alone complete solution but it died takes as a waste words are higher level error correction and fault tolerance so our 1st always say that there was a really nice tutorial it was given balance of the all of 4 years ago and you can then go to the website and and check it out slides and movie as so what am I doing here well that I would like to go a little beyond the widow of Lawrence a covered back up the back in the day in fact there has been progress in this field quite a bit and so I will cover some essential introductory material and then I will talk about the things that were not available for years ago but at least not at the level that they are today that's a high order the coupling dynamically coupling to arbitrary levels of precision under under certain some certain assumptions then also talk about the coupling along with quantum computations
01:35
OK so the origins of of dynamically coupling of with the Hon spinecho that cell not a new field it started 50 years ago on actually 150 years ago 1958 and so the axon a when he discovered the occult out in his young days this is a more recent picture of his and and here's the echo its base of resurgence of of quantum coherence after decoherence is taking place excuse me only try to fix this noise like you know this what happens when you run
02:17
Windows on Mac I should have known better I want to live
02:43
with it OK so the the explanation of of the Hanukkah which has to do with getting us spends to to Rico here has
02:54
that this a simple explanation no 1 can understand in terms of letters of running on a racetrack which each running year she represents a spin on the loss here if you liked all the spins Estonia together all the runners Estonia together and then they run at different speeds and each spin sees a different local environments and so of course after a while they have completely gone out of phase but they would you can do is a simple trick you can flip them all around 180 repulsed so here I am am leaving this guy's elitelevel doing but of course he runs a lot faster so a monolog and he catches up with me and here we are about to the beginning we've already go here there
03:39
is a more precise way of looking at it which I stole straight from Wikipedia you can go to the website and if you like but this book blue it's the it's the same
03:49
idea and there is a a collection of spins all putting along the the Roth of Poulton can you perform a 90 repulsed look them into the x plane they go out of phase you apply 180 repulse they all flip around and then the slow ones the fast ones catch up with slow ones and they all rephrase at a certain point in time at which i which time you have the echo so all of this is to say the same thing we can know reface lost quantum coherence to some extent here's all modern version of the of the experiment I do to idea suitor whose who's so with us today so here is the the survival probability or the magnetization if you like of a collection of of a of a carbon13 Cubitt couple to all the proton spin bath and if you don't do the whole article you observe this free induction decay so you lose coherence rapidly but if you apply the non spinecho you can restore column extent the point is that this is only to some extent and of course if the evolution time is short then we get a lot of the coherence of the back but i if we wait too long then the the coherence the case is the strong decoherence and therefore were in trouble before trying to clone compute and so much of what I wanna talk about today is how we can extend this coherence time way beyond the that the rapid decay that you're seeing here OK so study
05:29
get serious so I all on introduce the formal setting we have the hamiltonian model so this is the the nonMarkovian noise model that was described an earlier talk and we have a system about that evolve jointly under this noise Hamiltonian age and the noise Hamiltonian consists of the pure about terms a a free no noise system term so this is the the undesired terms in the in the on in and then there is of course the the system bath interaction and together these 2 other things that bother us assume the call that the error Hamiltonian and that the complete Hamiltonian is the sum of the basalt on a posterior Hamiltonian and it generates the unitary evolution operator and so this evolution operator generated by the the air about complete Hamiltonian which includes in the battle is what when the call for evalution so far we haven't applied any pulses we haven't done any dynamical coupling or article or anything of that nature right so now in in this talk I will assume that all Hamiltonians are bounded in the operator norm as so what the up enormous is just the largest singular value at all the largest eigen value for emission operators and so the the norm of the error Hamiltonian all sometimes called JD than all the possible to use data and the sum of the 2 all I'll call epsilon but I have to say that this assumption of bounded Hamiltonians is is not necessary for the analysis of dynamical the coupling and in fact there are examples of where of course this assumption is violated in particular if you have an oscillator bath the norm of the the bath Hamiltonian as well as system athermal tuning can can be formally infinite I so if that's the case it doesn't mean that all we can do the locally a coupling just means that this type of analysis that I'm I'm showing you here and needs to be modified and instead will introduce correlation functions in about spectral densities and fill the functions that is indeed possible it will be covered I believe in my critics and good solvers toxic kind of approach to to doing dynamical the coupling but I will assume for simplicity that the the the Hamiltonians all have bounded norm and that actually does describe well situations involving standouts as post oscillatory that's OK so what is the medical the coupling well basically coupling is is just over of rude way to treat the the system best you just interrupt it all the time so which you doing is you apply a set of instantaneous unitary operators piece of J only to the system you apply pulses unitary operations at times t sub j in between free evolutions so here is the the 1st pulse p 0 then free evolution of 4 times tau 0 and then again another Paulson another free evolution and all the possible foolishness such etc. and at the end you have a unitary evolution operator for the system and about but remember the pulses are applied only to the system so pictorially years the 1st pulse a free evalution subject to the the Hamiltonian which includes about them their Hamiltonian another policy evolution at such at such and in fact pay this is Dougal coupling at the simplest level all dynamical coupling sequences that can be described in this socalled bang bang manner will we bang on the system with with these pulses IDE the only complication in more realistic treatments of them ample happening is by when we have to include the the fact of life that pulses have fun at which in which case of course I you cannot treaties the the the pulses as delta functions as as in doing here but I'm not I'm not going to worry about that in this talk and there will be talks for example Lawrenceville as an and suitors aware this assumption will be relaxed of course it's it's a very important thing to do to include font pulse width affects everything I'm at in this talk is under this scenario to see how simple this is compared to and the the more elaborate constructions we we heard about before and there's no topology a no braiding and at this point in fact is not even going to be any encoding although later on I to invoke some encoding I pay so all it is really is just a this idea of applying pulses of certain types at certain intervals of separate by certain intervals to the system and and that is supposed to like in the on spinecho experiment supposed to restore coherence to the system in fact do more than restore coherence we we should be able to protect an arbitrary qubit Artur Hubert State so indeed the the way that different an ugly coupling pulse sequence is different in different differ is by the the choice of Paul type of types and intervals that so for IQ that's typically we can take the pulses from the Poly Group the identities thing Maxim was was the but we don't have to we could we could take all the elements of S U 2 as the pulses and the the other important choice is the question of of the pulse intervals so those of degrees of freedom we have this problem it's desirable that to ask to optimize the choice of pulse types and the choice of pulse intervals and this optimization is something that's been done and I will talk about 2 to some extent so that many examples and the well all these acronyms here PDE the already the CDD etc. and we'll we'll talk about those in a more detail are they all these different acronyms refer to different choices of of pulse types and some in some cases also calls intervals so you you buy yourself a lot by choosing different pulse types were different pulse intervals all right so how good is the medical dynamically coupling get what here again as the the general scheme but this formula basically summarizes of which you can expect in general from dynamical couple so it at the end of this evolution we have this unitary operator here at the final time T you can always write it in this form it's a unitary so I can always write it as the expense so I might I turned the Hermitian operator that there is a sum of of 2 terms of 1st H knot here age not is the component of the original Hamiltonian that commutes with all the pulses both if you have a piece in the original to commuter all pulses at on modified so therefore it just carries the original timescale t but then you have contributions from my new terms which did not commute with all the pulses therefore they they got modified but in these new Terms of age it's about by effective at each have a new type skills as with them so I'm just going to write that generically as T the at Alpha plus 1 and alpha is what we call the a coupling water of different types of errors in these in these terms are undesired sold the classified as theirs and we have different error types so the problem we we want a solvent then coupling is sort of the minmax problem we would like to maximize the smallest of the the coupling orders which I'm just gonna call and but while minimizing the amount of work we have to do which is the the number of pulses that were applied answer for the least amount of work that the fewest pulses I'd like to carry these errors to the highest possible water in time that's what we'd like to do with the name the company if if we could make this infinite than that and we be on a very happy because then all would be left with is this this term here which we can design in such a way that it contains only harmless only harmless components or even useful components such as components allows to to perform computation so that this is the basic problem in an uncle coupling how do you maximize the order of of each of these areas while minimizing the effort in terms of the number of pulses I wanna make a accord
13:55
aggression because the these errors can be computed using the menace of the dice a series of this being a tutorial thought it would be nice to like to say a word or 2 about the Medicine licenses and I'm sure everybody's familiar with the Dyson's sciences magnets might be less familiar I you run into it when you start to study them ugly coupling all or on average Hamiltonian the remark so pay 1 of the technical tools that we use in this business is called a Magnus series and as I said it is the the following it is an approach to solving the firstorder linear differential equation which is the Schrödinger equation for for most of us now in the in the jth in the system that setting it so ages the the Hamilton Forces about and it's timedependent because we've transformed into the socalled totalling framework rotating with the pulses of everything has become we were went astray were telling the pulses the Hamiltonian become talent that but so while the the Dyson serious familiar to to a body sure is is an expansion as as an infinite sum of the unitary operator of the Magnus series is an expansion inside the exponential and this has nice features that all comment on in in a moment but you can well you can say that write down what the terms on the license to this this should be familiar but in the Magnus series the the terms of the of this expansion on rapidly grow more more complicated the firstorder term is easy the 2nd order is involves 1 commutator that the 3rd order involves a double commutators etc. etc. and in fact is that there there's an explicit recursive recipe for writing down general terms in the magnet series and so it said technically it's possible to science right down the hall expansion and certainly to bounded and and do useful things with and yet the 2 expansions are related you can see that the the 1st order term is the same and 2nd order magnets term can be related to the 1st 2 terms in the that's serious etc. that can be done for all for all for all terms so the reason we care about the magnets series in ugly coupling theory is because it preserves unitarity to all order orders and maybe that's because the expansion is not inside the exponential so that's that's a very useful feature so it allows us to to bound things in a nicely unpack it does have a limitation which is that they have the convergence radius or like the the case of the licenses are the a sufficient condition is this the the integral of the operator norm of the Hamiltonian on should be less than pi that's that's a sufficient condition soul are technically what this means for dynamically coupling calculations that is that we should we cannot take the time here to be too large that translates into the into the that out of the coupling of a sequence of not being too long OK enough with this digression on Mendez and Dyson Belote talk about the the different pulse sequences that I mentioned earlier this is just a preview of of what they do all for what what they buy us so on the 1 hand we have the price we pay which is just the number of of pulses and just talk about the single qubit case that's a already rich enough or try to do here is protect a single qubit the price we pay is just the number of pulses k is the number of pulses and the payoff and is the the coupling water the the minimum of of all the coupling orders and the expression that I I wrote on before so paint the simplest of all sequences called PD your periodic an overly coupling and all of talk about all these in detail but the the simplest sequence uses up to 4 pulses and would advise you in a the coupling order of 1 meaning that you you remove the 1st order term in the Magnus years and that's what the Hon spinecho dots and that's why the decay so rapid you can do a little better by the symmetrizing of and basically this means that time symmetric sequence were or a palindromic sequence where you you run this sequence and its and a copy of of it in in reverse so you know if you make a copy you will when you double the price and the payoff is that you go up the 2nd order so that you can with a symmetrized version is palindromic version in in remove the 1st 2 orders in the 9 suspensions then there are a pulse sequences of which I can get you any order so that you can go up to order of arbitrary order n the the price you pay in the case of concatenated a coupling is exponential this is a recursive construction it's has some similarities with concatenated error correction I just in the sense that it's it's concatenated it's recursive and then there is a a sequence called overly coupling which was introduced of several years ago and it it works just for a single error types so for single qubit that means you can only removal of the pure dephasing war the whole world of pure amplitude damping of but not general errors and so there with just pulses you can be coupled order and and then there is an improvement of of UTD which allows you to remove arbitrary errors on a single human and with the order n squared pulses you can get toward order so that's the kind of exponential improvement in in the ability to to do that coupling of course there are can in nothing is nothing is works under under all conditions here and the most important caveat is that again as I said I had to assume that the operator norms abounded Hamiltonian operator of bounded which translates technically of you from in the frequency domain a translates into a statement that the high frequency that in the high frequency cut off of the bath spectral density at this point OK so let me now talk about these
20:12
various the coupling sequences back and let me start with the simplest one PTT periodic overly coupling which is this sequence that with a constant number of pulses give this 1st already coupling and here this this
20:27
really is an outgrowth of of the horns pinnacle of a nonspherical can be described in this language so so let's say we have a free evolution operator f from the college we all the system about together for some time now you know the construction is to big pulses of ball the design from a unitary symmetrizing group so it this is a group that has elements jeez G sub j unitary operators those are pulses and the way we're gonna apply these elements is in this manner so you just cycle of the group you apply its elements 1st element and its conjugate to such sector and then you take this sequence you repeat it nets this repetition is 1 called periodic an overly coupling no we identify the pulses as the products of 2 successive group elements that so this would be 1 and J in general the the inverse of pulsed minus 1 of elementary response times 10 selects element and we also identified a G. K with with G 0 OK so now if you that I use the fact that you can move the UNITA's up into the exponential of essentially a form of the Mac's expansion or in this case is actually just the Baker camp allows the pipe the calculation you see that is to 1st order in Tao died 2 divided by K is this Tao and you have this sum here and then there are some some 2nd order corrections and so clearly that what happened is that we've taken the original Hamiltonian and we've transformed it into what I so called H knot before which commutes indeed this term commutes with all the pulses it's been some symmetrized this is a projection technically a projection into the commute and of of the group G so this this kind of expression indeed commutes with all the pulses so so it's it deserves this this name age not not as as before but they this is that the term that 1st order in T and then we have these higherorder terms which is still active there's but the point is that we've gotten rid of things that don't commute with the the pulses to 1st order all that's left is just this 1 term here that already commutes with all the pulses of so to be the chief 1st order the couple there is no water teacher left that doesn't commute with us with all the pulses however the highorder terms again you can compute them using nice expansion of Dyson extension but basically they involve commutators of double and triple in higher order commutators of of this form
23:25
OK so pay in this language what is the harmonic well let's say that the noise looks like this so typically the Unicos is described as for a pure dephasing so when it was just as he tensor easy term but let's let's consider a general air Hamiltonian a single qubit for the Hunchback whether a coupling group is just the 2 group II and and sing max Watson and now you go through this construction where you identify with the pulses so we multiply the group elements you find that there are actually only 1 pulse of P 1 is acts and P 2 is also X. so the pulse sequence according to the previous construction were really
24:06
writings things in this way the pulse sequence is just that staff that that is free village that so were doing free evolution where Lumira from right to left were fireballs free evolution pulse fruition that's that's basically the of the harmonic of sequence or urine as picture apply the calls for evalution postrevolution it takes total time to tout to free evolution of segments and when you write out the of the unitary operator for the system you see that you have a 1st order terms the firstorder term is the thing that commutes with the group so obviously x commutes with with that so that this 1 is not be coupled and then you have a higher order terms and the important thing is that Y and Z got eliminated from the 1st order so they only appeared as to 2nd order and I put primes here would you barely see but these are not the original Knuth operators of course they are modified due to these commentators in higher order so the point is that with the Honecker we got rid of of y and z of signal signals we got rid of those errors to 1st order and this is a an interesting connection an important connection in fact with errordetecting codes you are you see that in a way the idea is that that got removed to 1st order of the errors it can be detected at all by the stabilizer group and that actually is a way to to generalize dynamical coupling with all I'll say a little bit about later on alright it's now next example the Sokal universally coupling group II that's when we are dealing with when we want to be protected Cubitt against an arbitrary system that's Hamiltonian on a single qubit so we expand the group no we we have a polar group rather than just an identity X we have added I X Y and Z and we we do this multiplication and we find that there are to false operators of exons OK so now the pulse sequence is this z of the effects that's the pulse sequence and this pulse sequence has the interesting and important property that it completely the couples a a a cube it to 1st order from an arbitrary versus the math interaction right so here here again as the pulse sequence now it takes time for Tao and if you so right the unitary operator you see that the the communing term now is just identity so there's nothing left but the only thing that that the commutes with with all forces his identity and a so no errors left to the there's nothing useful left there's nothing we can do with identity but but then where's left a 1st order all there's a pure the 2nd so we've we've completely the couple is Cuba to to firstorder every error has been detected by about this but as stable as a group a liaison have dealt with the PD in using 4 pulses of 2 in the case of the harmonic we the couple the firstorder now let's move on to the symmetrized ET or palindromic I DD and this there's a result that's been known for a very long time in in community which is that any palindromic timereversal symmetric pulse sequence is automatically 2nd order with respect to to the base sequence that that you're on a tight of in fact it's better than 2nd or all even terms in the magnets series manage if you make your control Hamiltonian times that this this each year should be the control Hamiltonian the tone in generating the other pulse sequence so with by symmetrizing we can remove all even terms and minus series and therefore if if we pick a sequence eliminates 1st order it automatically in this construction also lamented 2nd order and therefore we get and and 2nd order the coupling so
28:06
here is what it looks like again same same noise model the this is the the general somebody with hamiltonian we take the same the coupling group but now we take the pulse sequence and we also right in reverse and pack that if you multiply that the 2 the operators here they cancel so actually you you're left with 2 free evolution operators here so this is in the middle you have a a a segment that takes time to tout that's that's what shown in this in this picture here as you know the total sequence takes time a town and the unitary operator it now and I have no longer include 2nd order to so the 1st order term is just identity and everything else shows up to 2 3rd or right so we have 2nd order the coupling of for the price of making this sequence twice as long OK so pay that's not good enough because we need to be able to get to arbitrary order so how do we get to arbitrary order there basically to general techniques that I'm aware of for for doing that was concatenation which gives rise to the concatenated coupling receding and the other 1 is a pulse interval optimization which gives rise to a whole new family of of pulse sequences so concatenation go like that you wanna think about the of the air hamiltonian the noise as some level 0 type of Hamiltonian OK so that's why it has the the superscript 0 here where we're going to iterated sums again let's pick the the general single qubit Group as the the coupling rope they let's call the pulse sequence that we use before the the effects of the of excess let's call that pulse sequence p 1 it's going to be the love 1 in the inner recursion in the in the concatenation so it so here it is again same as before on using seen this already but now let me think of of this Hamiltonian that appears here but to 2nd order 2nd technically it's little point is those have the right units and the squared but the thing of this operator nevertheless think of Trout think of it as a Hamiltonian which has the same structure as the the noise that we started out is that we want to decouple right so that suggests that if if we somehow iterate this pulse sequence we should be able to remove this just like we remove the original noise and so the way to do that is to insert the pollen to itself inside every free evolution period we're going to insert the same pulse sequence facts so written out of the Impulse equals language rather than having a free evolution period we take the original pulse sequence p 1 and stick it in between the original posts so this clearly is is firstlevel recursion and then so it's while it's hopefully it's already selfevident but it can be shown rigorously that this indeed removes and the 2nd order term completely ill users just with a 3rd order term Everything is changes the the the pure the the the bath operator of has also changed the air Hamiltonian here is nowhere Hamiltonians superscript to everything changes but importante point is that the order in time has grown from 2 to 3 by by this recursion why because it's the same error model back to what worked at the lowest level should work again if we can catch so you can keep going and pay in this this is the general form of the recursion and then if you do this level k you you will have to couple to order so that the 1st nontrivial turned it shows up as water K plus 1 so this month it can indeed be made rigorous this kind of construction and and of you can prove bounds and that's a problem but but this this really captures what's going on with concatenation you you actually are able to to go to an arbitrary high order in time provided here then keeping that the total time fixed as as this picture suggests what I've done is I've fixed the total time for which the pulse sequence evolves in I perform a concatenation by shrinking the intervals tap so at every level concatenation making the interval is shortened 5 factor for each time 1 the object's of correctly that that this is in some sense it's unphysical I cannot keep making these these interval shorter and shorter because that in the short interval the more energy any tool to put in there so another way to look at it is to say but let's Let's keep the interval fixed let's grow the total time and to another total time rose like a grows exponentially but in this case you find that there is an optimal concatenation level at some point but depending on the parameters of the problem at some point it it doesn't help to to concatenate any further because essentially the bath is it has too much time to act and to to inflict damage and the the optimal concatenation level so it can be shown to be the inverse log of the norm of the error Hamiltonian plus enrolled about some of times the the fixed pulse interval time right so that that does make sense of the larger that the norm of the error Hamiltonian the smaller is the optimal concatenation level and likewise for the of the best Hamiltonian OK so this is concatenation
33:58
and what it does is for the price of using an exponentially growing number of pulses you can with you want in other words if you want that the coupling order and you need to use an exponential number and N pulses to get the OK so I this this is nice but but an exponential 100 and so there is an alternative way of looking at things so far would've done is if kept the pulse intervals const but what if we tried to somehow optimize the pause intervals that is the that by some and the answer is yes and this was observed by but Oregon vapor in 2007 but as I said before force for the case of all Arab so what security facing and when he showed was that path using only the order impulses we can really get at the coupling water and there will
34:54
be there will be several talks about this undergo into too much detail but let me let me give some background about and no OK so again CDD concatenation requires an exponential number of pulses can we do better and and the optimization problem was solved is to maximize the smallest the coupling water while minimizing the number of pulses OK so for purity facing word showed that we can do this with an pulses and for general single acute as as as all argue briefly they can be done with with n square pulses and this is the the construction of the word constructions so this is not a proof that it works it's just a way for you to visualize a how will we choose how we have to choose these pulse intervals and and proving that this 2 intervals is optimal is is nontrivial so there will be a couple talks and posters on the topic so the a construction which buys this order in the coupling with only impulses that refers to to this Hamiltonian where we have purity phasing pay in the way that you choose the pulse intervals is has a very nice geometric of interpretation so that the pulse times the times at which you apply the pulses a given by this formula it's a Javier runs from 1 to n so that's the total number of pulses these are the times at which we are supposed to apply pulses and with this formula means is nothing but the following you you draw a semicircle and have you divided into N plus one equal angles each the 1st analyst twice that a 1 the angles are given by the by twice this this so I j pi over impulse 1 and then you dropped out of recall and the point at which pay these vertical that intersect the the time axis that's where you apply the poles so the only pulses were we're really for that so the 6th order at work sequence that has pulses exactly at these these time intervals at the lowest order this actually reduces to what is known as a C. P. and you see in the in our community so OK so this is this is the way that the word sequence works technically you are you apply the pulse of that at the time points and what that buys you is what it says here is the the unitary evolution operator now has the the component that commutes like before to 1st order but the components that we were supposed to decouple the purity facing term now appears to order N plus what provided you choose your you but if you apply your pulses at these at this point in time That's remarkable
37:49
no pay on a single axis errors or on a promotional thing how about general security coherence right so all the general model in this case or what we can do is we can basically use the concatenation idea again we just have to concatenate or nest ones this is called quadratic were qt and we have 2 nests at 2 different types of of word sequences so geometrically can think of it as follows again take the semicircle just like before divided into N plus one angles and in this case and call and 2 plus 1 n equal angles because gonna be twopulse sequence and so this is the outer pulse sequence there's still apply ex pulses at this moment in time the X pulses effectively will decouple signal Winer sigma z of contributions to the Hamiltonian and then you take each of these intervals and drawing of a semicircle and inside each 1 of these you do the same thing and this is where you apply those epochs so the the points in time at which you apply the pulses are precisely where you see the the red and blue line but if you do this you're guaranteed to essentially they get a d coupling to order then 1 plus 1 or as I with using and 1 plus 1 times in on pulses are you going to get the coupling to water main of N 1 and 2 so that this is a lot better than the using concatenation of because it's basically quadratic in the in the number of pulses whereas concatenation required an exponential and so this will be given in a talk at least by an open problem but that was the end of the talk about but a proof of this as well as a generalization of will be given in this talk and the fact that we can go further and this is described in in a poster by 1 June quote and we can be much more specific than just to say what the that the the coupling orders the men of N 1 and 2 we can actually analyze in detail a text what the coupling order is for each error so if you wanna know what if you if you apply exons the way as described here in AZ pulses in our X pulses you wanna know to what order the air X is the couple or Y or Z so you can look it up in this table which which will be described in a in one's poster and there's some interesting effects here due to the parity of the relative parity of of N 1 and 2 OK
40:41
now we've covered the a small piece of the of the pulse sequence to as of talked about UTD in and unity which are the unity is provably optimal our for single air tightly coupling GATT is almost optimal as far as you know from a numerical analysis of symbolic algebra analysis I should say we know that we the base you set of of constraint equations that can be checked and we can show that it's not possible to achieve and already coupling with fewer than than than squared pulses up to up to a constant of the proof of whether that is truly optimal is that's still an open question so that's that's a really interesting question to us to try to attack that is it possible to do any better than of an order and squared for general and ordered coupling of of a simple cubic body optimizing that's way I missed
41:41
the mark OK so before I start to talk about computation me just show you the the battle of the of the pulse sequences and so it's it's as you'd expect here what I'm showing is numerical simulations now and I should I should remark that these results here these are
42:03
analytical upper bounds that so the real life the situation may be better than the analytical upper bounds and that's the purpose of these
42:10
numerical simulations so at what's being simulated here is a single cubic coupled to a bath and the parameters beta and J. the barrier is the strength of the the pure bath J is the strength of the system of coupling so a single qubit coupled to about the bath operators chosen as 1 or 2 Jupiter operators so now we have a twobody interactions and and threebody interactions here these parameters are are chosen at random uniformly at random between 1 1 1 so that's the the system bath model and now we we start applying pulse sequences and all the different types and so we're doing PDD and and CDD and there's another type Ia that and talk about sorting or that and quadratic duty and this this is just a reminder of the cost that's involved in in each 1 of these pulse sequences what we're doing is we're actually applying a fixed number of pulses and for each pulse number let's say a 100 we compute the log of the trace norm distance between the the initial state that the system was in and the final state that the system wasn't that so we have a single cure and we wanna check to what extent it was preserved this is the hard drive problem that Endre was talking right so we decide to store Cuba so what you're seeing here is how well we were able to store these qubits as well as measured by the science between the US initial and final state of the same and I should say that this is averaged over not only over the the bath parameters year but also over the initial and initial system bastards but what we're doing is we're picking a random a pure initial that's the initial system that's it at random and we trace out the bath get the system state and then we average over many such choices of of the initial system bastard so that's where you see the error bars here others averaging over all these random realizations OK so pay the smaller the better we would like the trace norm this to be small and so with with PDD so you see that the the actually starts to to increase as you increase the number of pulses and so that's not good starts to build up the more pulses use of the larger the air but with the techniques and CDD which is the purple line here the year goes down nicely with concatenated a review gets a little better this is basically a lot lamina talk about this 1 and and qt the green line you see that the error goes on very fast so for using 50 pulses in this model by the way this this is a small simulation it it's a single cubic couple to 4 other Cubans of about represented by above by 4 other cubits but were actually solving the Schrödinger equation entire our hilbert space of of of 1 system cubic for Basque qubits and so to live there for example with 50 pulses so these are the kind of trees some distance that that you get that it's very very small with Qt it's it's also small with the other ones but the improvement is of course the remarkable whiskey so the battle is won but you that's the half that's the message here
45:44
known in the remaining time I would like to talk briefly about how we combine all this was computation after all as Andrew said we wanna do more than the hard drugs so the basic problem we're facing with using dynamical the coupling is that the ugly coupling pulses seem to interfere with the computation and because they they cancel everything of or at least they cancel everything but to the extent that they will leave the for the component that commutes with all pulses so how can we reconcile doing ample coupling Western computation I am aware of at least 3 approaches what is called a couple while computer the is called a couple then compute and the last 1 is that corrected gates and Lawrenceville will give a talk about that on at that 3 to that so let me start very briefly unfortunately for lack of time say a few words about the couple all compute but it's it's an approach that really works well with a with error correction in in in some sense of and so what we want when we do the couple all compute what is a couple compute its as the name suggests we're doing the coupling and were performing computation all at the same time so in order for that a coupling not to ruin the computation we we need the pulses the computation to commute and they are immediately solutions suggest themselves from from quantum errorcorrection from those subsystems for example we could use the code and the stable of was a structure or we could use the double commuting structure of of was a subset so specifically what I mean with use stable role structure is is this let's say that we take our identical the coupling pulses as the stabilizer generators of the stabilizer codes so the the stabilizers of the code are not things were going to measure now in order to detect errors instead we're going to apply these operators as pulses if we apply this pulses they commute with the logical operators of the code and therefore the thing that remember the age age not was the thing that commutes with all pulses what will the left is precisely the logical operators of of the code so if if we do this well we can achieve universal computation because these logical operators now are terms the Hamiltonian than that in the case of the universe all the constraints with the Clifford Group and all that but I need to rethink that because now these these are terms in the Hamiltonian so it's enough and 2 halves of arbitrary single qubit and and to and it's enough to have single and 2 qubit operators in the Hamiltonian in order to achieve universality in that we can so we can in fact construct for interests so so this is 1 idea that again we're using dynamical the coupling pulses which are the stabilizer generators was table of the elements of the code and then the Hamiltonian that's left will consist of will consist of the logical operators of the code the pair alternately in this refers to this and was subsystem was this we can't use a ugly coupling pulses as as collective rotations so think of all of the DFS they're going the of the collective rotations what commutes with collective rotations well Heisenberg exchange interactions commute with collective rotations so we can use this to a to show that we can for example perform highfidelity dates encoded gates for for conducts or Heisenberg interactions or are relevant so that's always say of very briefly about decouple while commuting uses this commutation between the pulses ation in order to achieve their goal and then pay let me talk a little bit
49:43
more detail about the ID coupled then commute approach compute of cell here we're doing the following here the circuit
49:55
and end of let's say the where she trying to perform the following simulation OK so no that coupling yet but we have we have gates in a in a faulttolerant simulation and let's divide the time steps into durations tau 0 and the gates themselves and take some some time they have with delta and we're gonna put them at the end of of each time interval so that the full Hamiltonian is the the the Hamiltonian plus the bath Hamiltonian plastic the control in this case is what implements the gates and I'm thinking about this the purely Hamiltonian and it's too bad person's cover now because this is really important links so in faulttolerant in in fault Ollanta called computation and in particular in the context of the accuracy threshold theorem we will we are interested in the noise strength so this is a is the noise strength which is the normal here of a the error of the of the error Hamiltonian multiplied by tau 0 OK so the norm of the error Hamiltonian times tau 0 that should be less then the the threshold which puts a stop to the minus 4 so if this quantity which we call the noise strength the norm of the error Hamiltonian times this time interval is less than the threshold then fall talent simulation is possible OK now how can we improve this using them will become that so far there is no doubt is the way to make things better using that the the coupling and the answer is yes sometimes and let's it modify the circuit so that every gate is proceeded by a sequence of dynamically coupling pulses so we're gonna make the circuit longer so instead of having a gate separated by intervals tau 0 we make our gate separated by intervals capital T. which is n times 10 0 and sorry that should have been paid actually to be consistent with my presentation so T is K times tau 0 where k is the number of pulses in the delicately coupling sequence so it's a very simple idea of conceptually we simply replace each gate by a pulse sequence then the gate so this is what we call dynamical the coupling protected gets alright so now we know from the previous results that life we choose a that for any pulse sequence of features it well so that we can get a high the coupling order right so at at the end of the pulse sequence and then we will have removed much of the noise and at the the gate sees an effective system of interaction that is whose strength is much reduced that's the basic idea so the new noise strength for me read this off this is aided DDE right so are likely to the previous in see this was the the pure which was the norm of the avatar in times time now it's the norm of this new Hamiltonian here times the time it takes to implement the pulse sequence was the gate and that quality has to be smaller than the than the threshold and so under the right conditions this is effective Hamiltonian will have a very small more and even though you have to multiply by a longer time this number 8 DDE here top then if you didn't do that article a couple so the the idea is again to reduce the noise by applying the the sequence and they if that's better than are not doing the sequence of the new games so let me skip
53:52
this although I wanted to tell jurors talk
53:56
was very relaxin important assumption that was made in in this analysis color local bath assumption and pay so just to show you
54:04
how how much improvement we get and here is a graph that shows the relative nor its strength so the noise strength in the presence of an ample the coupling versus the or strength of other people the coupling as a function of epsilon the epsilon was long of the Appleton post or the battle in so in these units that if ADD is less than data the flow this line then dynamically coupling helps before about this line that the coupling hurts so you see that I the tuple sequences here the x the sequence and they palindromic sequence you see that we can easily get below the level where that ugly coupling words into the regime where where it helps we have a lower effective noise strength of for the circuit and this this means that faulttolerant a simulation might become possible where it was previously not possible on the or in other words if they did indeed dips below the threshold level and then followed simulation because possible with where it was not possible without doing any of the coupling but it is also improves the the overhead to quite a bit I don't I don't have a graph to show you but I can give you reference if you're interested in the overhead is is reduced very significantly and provided when in this region and finally what if we use it better so what if we use concatenated the coupling now all of the when we use the optimal level of concatenated coupling because we're we have to find out the pulse interval so we're in this regime where we have to worry about using the optimal of a concatenation of the pulse sequence you see that the reduction in the noise strength is quite dramatic while for the right parameters and relative to that not doing a 10th the coupling protected gets again this is for the X the X exceeds sequence is the blue line and the red line is the palindromic version of of the same sequence so so these results show is that using a locally coupling in conjunction or doing quantum error correction fault tolerant quote computation in conjunction with the Member the coupling can actually I help quite a bit provided when the right parameter regime but it can in particular it can reduce the noise strength in some cases to below the the structural other cases where we were ready below the federal level it can improve the amount of overhead that that we need to perform the simulation so let me this is my
56:44
last slide let me have some up in some sense back doing dumpily coupling is trying to fight decoherence with with your hands tied because I this is a method of we're all you do is you apply these control part pulses and it's it's openloop it doesn't use feedback and the news measurements so while have there there's an extra resource out there which is doing feedback doing measurements of in order to to improve performance but here we're saying let's not do that so why and that's why why do that well in fact you might argue that it's not a good idea because then locally coupling is is not a standalone solution and it cannot by itself the made faulttolerant lever covered up I'll talk about that on on Thursday so why do we do it what we tie our hands behind their back now and our backs and the answer is 1st of all this openloop approach is technically easier but I think that should be clear by applying pulse sequence is easier than the breeding the or are you and it just doing the performing measurements and and feeding back on information acquired through measurements so it technically it easier we've also seen that we can use that handily coupling at the lowest level to improve performance and maybe even get along the threshold and reduce overhead and finally I what I really like about this approach of the locally coupling is that it it is ready call for experiments and in fact have been plenty of experiments and I testing them for the coupling and the results are are nice and encouraging and I think we're going to hear quite a bit about that later so without all thank you and thanks to you questions of the Daniel and in between of feedback and measurement there's a possibility I would think of having some more knowledge about the Hamiltonian and therefore making use of the extra space in the free evolution for robust control color so open so you have any ideas as to what that extra knowledge might look like fake any if you're referring to realtime acquiring of this extra knowledge of Union offline on prior yeah I agree with you 100 per cent and the fact that there will be you talk about that very subject howdy use dynamical the coupling to the bath specs so old yes that information can be acquired even using them bully coupling and if you if you know something about the bell for example if you know the spectral density the shape of the hospital density I you can do extra optimization and the only assumption is that I needed for for this presentation were these assumptions that the more the Hamiltonian was norm bound by adding use a particular details other than that it's about the best spectra if you know it you can optimize for the history I have a question about year to couple then computer or dynamic DD protected gates and then you showed that you can actually improve by making it longer find some dates but I assume you're assumed that the gates are prefer another is they have no errors that's right now know of that what no I assumption is wrong word by the gates have there's the gates are allowed to have this this is part of a faulttolerant simulations so the get out of errors with the little cheat in this graph is that actually there it doesn't go down forever and in if you look at the which being implied here are these formulas and you see that there is a lower bound in the lower bound involves delta which was the time it takes to implement the gate OK so I actually there's a flaw here which is is given by by the left hand side that so that the the air that you're asking about is basically the associated with the gates of that's measured crudely by the parameter delta the width of the time it takes to implement the gate so I will not call that the mayor of blood we yes still be perfect finite duration right right so the area as is measured in terms of that's times I let's say the the norm of the error of the air Hamiltonian on the gate OK so but gate stands and the gates the pipe pulses they themselves are considered to implement exactly a pipe holds for instance right right you know so it's it's it's the same thing it's so this this parameter delta here delta 10 that is the strength than the noise strength associated with the gate or with a pulse and N is the number of pulses so it gives you kind of a cumulative error with applying and faulty gates pulses a finite width he said that you have a question regarding the you it posted wins so as as you many talks excellently under the assumption that the norm is bounded in our language that they're in PostScript offered a bus has a cut off and of course all experimentalists including of Cerf and many people here but ideal exist among natural noise sources and it is men only the political point will do so proficient noises and put a cut off and showed you a quote books that all the other examples it doesn't work that so so it seems that this is a Gaussian assumption for the you in my question is in what sense to such materials for the other person quotient for the other POS sequence to mention founded yeah so that the 1st of all regarding the word sequence yes and no it this this will work for the right experimental system as as you said in the language of spectral densities there needs to be cut off and 1 can think of interesting solidstate examples where the so I I think it is a useful sequence in that context as for whether the other sequences depend on there being a short cut off the the concatenated sequences that nearly a sensitive this this assumption here was in this presentation was made just for convenience but the same analysis can be done but in the language of spectral densities and you see that the circle that's the that's the cut of does not enter in quite a dramatic way as as in the workers and it all boils down to as we'll see migratory stock of applying a filter function which suppresses lowfrequency noise and if their filter the functions very flat which is what the word sequences basically multiplying a alone lowfrequency part of the spectrum all by very flat function which is everything the high frequencies then you get the strong dependence on the on the cut off but you can come up with alternative actions and concatenated coupling is is an example of that it's not so the best example for about swiftly Softail which doesn't suffer from the from the core of problems the question about this slide compared to the various scheme for single QB and the complexities of the convertor complexity were the the coupling order I wonder how the randomized schemes the fit into the picture this distance you had a decent to be all deterministic I wonder what you could pilot with a randomized sequence which to coupling orders a month chief right and so I did not talk about the randomized their sequences and for lack of time and they are the important the relevance and the there important relevant especially for about that have fast turn dependence if the bath is has a strong time dependence which basically means that the this language it means that the the norm of the bath Hamiltonian the pure baffles on as large in an interaction picture reconfigure the base's rotating faster but many of these are optimized tailored pulse sequences don't work because basically the whatever sequence you trying to apply at the next step of the bathos changed so much that it the water got scrambled in that case you might as well just do randomly coupling so that's the context for ending the Morandi coupling now as far as the order I we believe that the the answer is that it's it's a fixed order that you can achieve I'm not aware that a randomized scheme that achieves arbitrary order but perhaps Lions would like to to comment on that and so well it's a bit tricky to define what the author of suppression useful randomize the capping schemes because they don't naturally lend themselves to modern stack analysis back to from my own experience the other feature that they have is that they become a monster a power forward in many cubic settings for which the the capping group as in a pressure below and the largest size it does the better it because what and the musician buys essentially is a square root improvement in that time for traversing that group so there have been a lot of the numerical evidence that indicates that there can be significant other advantages in using randomization for long times and large scale systems and a quantitative easing that myself at some point on I I don't know if we're going to be stuck to low order there I think it is possible to achieve I ordered that I just because you know what we know that the good at that that mean stick to ski 6 these so randomization in some sense call the all the possible path on any must go there high you there as well but after that they don't have a good answer as of now It's an open problem so should we can think our morning speakers again now thank you thank you and lunches upstairs but have but thank you and I don't like the you have and get and the thank you
00:00
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Titel  Dynamical Decoupling a tutorial 
Serientitel  Second International Conference on Quantum Error Correction (QEC11) 
Autor 
Lidar, Daniel

Lizenz 
CCNamensnennung  keine kommerzielle Nutzung  keine Bearbeitung 3.0 Deutschland: Sie dürfen das Werk bzw. den Inhalt in unveränderter Form zu jedem legalen und nichtkommerziellen Zweck nutzen, vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. 
DOI  10.5446/35300 
Herausgeber  University of Southern California (USC) 
Erscheinungsjahr  2011 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Informatik, Mathematik, Physik 
Abstract  Dynamical decoupling (DD) is an openloop method for decoherence reduction based on the frequent application of strong pulses, designed to cancel terms in the systembath interaction Hamiltonian. DD has been tested and successfully implemented in numerous experimental systems. After introducing the basics of DD, this tutorial will review recent developments concerning optimized pulse sequences, capable of canceling the systembath interaction to arbitrary given order. The integration of DD with computation will also be addressed, from the complementary perspectives of "decouple while compute" and "decouple then compute". The former requires a commuting set of pulses and gates, which can be designed using tools borrowed from quantum error correction. The latter allows DD to be integrated with fault tolerant quantum computation. 