Universal Dynamic Decoupling and Quantum Walks in Functional Spaces
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Jiang, Liang
We study the universal dynamic decoupling (DD) schemes, which can restore the coherence of quantum system independent of the details of system-environment interaction. We introduce a general mapping between DD sequences and quantum walks in functional spaces, and use it to prove the universality of various DD schemes such as quadratic DD, nested Uhrig DD, and Uhrig concatenated DD, as well as previously known universal schemes of concatenated DD, Uhrig DD and concatenated Uhrig DD.
 Hi, good afternoon everyone. So today I'm going to talk about universal dynamical decoupling and quantum works in functional space. So this work is done in collaboration with Adelaide Mbakov from Rice University. And if you're interested in details, you can find it in archive paper. And I also will recommend, and you can also take a look at a quote under Lidaz paper, which also have related topics. So first of all, since we're here in a conference for quantum information processing, one of the biggest challenges is actually decoherence. And decoherence can be induced by system and the bus interaction. And for example, we consider a system of a single qubit. The most generic system and the bus interaction can be written in the form, which is just a sum of the system generators tensored with the corresponding bus operator. So here it's actually, we try to roll, so you may see that these bus operators they have a hat, so they may not commute with each other. And in addition, these system operators, they also do not commute with each other. So this actually makes the problem a little more hard to keep track of. And so the idea is actually to suppress, so in the presence of the system bus interaction, they will induce entanglement between system and the bus, which leads to the decoherence of the system. So the idea of using dynamic decoupling is to introduce additional system evolution so that the overall evolution, which is combined with the system bus and the system evolution together, it will become just the identity of the system tensor with some unitary operation on the bus. So if we can achieve that, then there will be no entanglement between the system and the bus. And so in this talk here, I'll particularly focus on this universal dynamical decoupling, which efficiently decouples the system from bus for arbitrary system and the bus interaction. So here universal means these bus operators can be arbitrary, but with a finite norm, of course. So before we talk about this universal decoupling sequence, let's first take a look at this high echo sequence, which we are all very familiar with. So the high echo sequence consists of two pulses, which is a pie pulse in the halfway of evolution that flips the spin, and at the end of the evolution, it flips the spin back. So if we consider this, and high echo sequence can efficiently suppress this defacing noise, which can be induced by the system operator Sz tensor with the bus operator Bz. And let's look at how the system bus Hamiltonian, an interaction Hamiltonian evolves in the presence of this system evolution. And it's convenient to introduce this toggling frame Hamiltonian, which you can think of as an interaction Hamiltonian associated with the convolution. So in the toggling frame Hamiltonian, what happens is the first term, which acts trivially on the system, commutes with all the operations on the system. So there is no change to this term. But the second term, Sz, which actually anti-commutes with the pulse. So the first half of evolution, there's no sign change of this term, but in the second half there is additional minus sign. So then we can compute the unitary of this evolution, which consists of two sequential evolutions. And we can keep track of the term and get this suppression, which the system bus interaction gets suppressed to the second order. So actually this example actually outlines the procedure that we can use to keep track of this universal dynamic decoupling, which is first we write down the system bus interaction, then we write down this toggling frame Hamiltonian, and the last step we compute the unitary associated with this toggling frame Hamiltonian. So which is our plan here, system bus, toggling frame Hamiltonian, and in the end we'll compute the unitary. And for this, and actually there have been a lot of studies about this universal dynamic decoupling. And lots of sequences are proposed and proved, and lots of sequences are conjectured to be universal. So here's just probably an incomplete list, but it consists of most of the features. But maybe it's a little bit overwhelming to see all these on the same figure. So I try to come up with a family tree for these dynamic decoupling sequences. So the great grandfather for all these sequences is actually Han Eko. And actually there could be like two different kinds of decoupling sequences, approaches. One is called concatenated dynamic decoupling, which Daniel also mentioned in his tutorial on Monday. And the other one is the URIG decoupling, which is a little more efficient, basically use order of N-palses to suppress this decoupling to the higher order. But these two sequences, but this URIG only suppresses the defacing noise qubit. So we might want to protect the entire qubit from both defacing and bit flip noise, so that goes to the second generation of this dynamic decoupling sequence. So for concatenated sequence you can actually, there's a way to actually suppress both defacing and bit flip. And for the URIG sequence, if you nest it with two nesting levels, which gives you the quadratic dynamical decoupling for the URIG sequence. And you can also combine the concatenated with the URIG sequence, come up with some concatenated URIG decoupling or URIG concatenated decoupling. So the ordering here corresponds to the, which comes the higher order, which corresponds to the lower order decoupling. And so we can go even further. The third generation corresponds to the decoupling sequence, which can protect m qubit quantum systems. So here you may have some decoupling, system-based interaction, which might evolve like a multi-cubit, like X1, X2 times the bus operator. And by having this decoupling sequence for m qubit systems, those terms will be eliminated as well. So natural generalization will be nested concatenated decoupling sequence or nested URIG decoupling sequence. But as you see, that's okay. On the upper right, the upper left side of these sequences, they are all proved to be universal. But on the lower left, which is still open questions, we don't know if they are universal or not, but they are conjectured to be. And it's important to understand these sequences because they are very efficient, which scales polynomially with all the ones you want to suppress. In the rest of the talk, I'll try to convince you that indeed, all these sequences, they are universal. So before I go into the proof, let's first take a look at just to recap what the URIG decoupling sequence is. So URIG decoupling sequence is actually an unequally spaced decoupling sequence. So the spacing between the pulses are not equal. They actually are obtained by some equal partition over half circle and then projected along this horizontal axis. And for example, here, n equal to 2, so you have two pulses, which kind of correspond to these two points on the circle. And for example, if I want to suppress defacing noise using URIG dynamical decoupling, we can apply spin flip operators at these two times. So what happens is that the system Hamiltonian consists of two terms, which one is actually on the system, the other induces defacing of the system. And the one actually on the system, the coefficient doesn't change. The one that induces defacing of the system, which actually induces, which changes sign every time it hit this spin flip operator because it's an anti-commute operator. So overall, the message is, OK, we know what this f function is. It's just a sum of plus one or minus one. And the additional feature is that you may find that if we multiply these two f functions with different subindex, so here alpha takes the value zero and one, corresponds to trivial operator or the sd operator. So when you multiply these two f functions, it turns out there is a simple rule that it becomes another f function, where the index becomes a binary sum of these two subindexes. So this is an important feature. And you may find that this has a similar correspondence in the system operator. When you multiply two system operators with alpha to be zero or one, it becomes another system operator, which the indices corresponds to the binary sum of these indices. Because in some sense, it's trivial because sd is just the sd square as identity. But you will find that actually it becomes little non-trivial when we go to the case for the quadratic dynamic decoupling, which we have the system. So here we will consider the system of a single qubit, which has the sx, sy, and sd coupling to the bus. So now we have, generically, we can write four terms in the system bus interaction. And here I'll label it with two binaries, zero, zero, up to one, one. And so first, we consider the decoupling sequence for this quadratic dynamic decoupling. It consists of a con-cad-nested O-rig decoupling. So in each interval of the original O-rig dynamic decoupling sequence, you put another O-rig inside here. So therefore, you have one O-rig sequence at the highest level, and here for n equals two, you have three O-rig sequence at the lower level. So we can compute these f-functions as a following. And so for example, this associated with sd function, which you will change the sign every time it hits this sx operator. And similarly, we can get the f-function associated with sx operator and sy operator. And the feature is, again, you confirm this feature, the product of two f-functions is another f-function, but with the indices corresponds to the binary sum of the two inputs. And here similarly, the s-function and the system operator function also have the same feature. It's the product of the two is another system operator with the indices being the binary sum, but up to some plus or minus sign. But here it doesn't matter because actually later on we'll show that it equal to zero, so plus or minus zero, it's still zero. But keep these two features in mind as we'll use it later. So just to recap how we prove the universal dynamic decoupling. We have three steps. We write the system-based interaction, we go to the top frame, and we compute the unitary evolution. So we have already gone through the first two steps. The next step is to compute the unitary evolution. And the goal is to show this unitary is essentially identity acting on the system plus some higher order terms. So the way to do it is by doing Dyson expansion. So this is the unitary. Then the standard Dyson formula gives us a bunch of terms, and here n corresponds to the little n corresponds to a number of integrals associated with these expansion terms. And in addition, so you can plug in this Hamiltonian, which consists of a bunch of terms, and which you can sum all these over. But the important thing is here you have n integrals. That will give you power t to the n, because that's the only way you can get the total t, which is the total evolution time. And the rest of the term consists of three components. First is a coefficient, which is associated with a time-dependent f function induced from this decoupling sequence. And the second component is the system operator, which is the product of these s operators. But it's important to keep track of the ordering of the system operator. And the last term is the bus operator, which is the product of all those bus operators. And so here is just like the definition right now explicitly. So just to recall that early on we say the system operator, the product of the two, is an again system operator, but with index is a binary sum. So we can generalize this to n product of system operators, which is a binary sum of n indices. It's again a system operator. And this system operator becomes identity only if the binary sum of these indices is zero. And so if we look at this Dyson expansion here, we'll find that the goal is to hope and to show that this one is equal to some identity act on the system plus some higher order terms, which means all the lower order terms show. That means that this f function should be zero for all those lower order terms when n is less than capital N. But except for some, but this is a little too restrict because actually if for this expansion if the system operator here is identity, it's actually okay because it does not perturb our system. So we can actually exclude those terms by putting a constraint that this system operator is not equal to identity in those expansions when we try to compute these f terms, which corresponds to the binary sum of those alphas in this f function is not equal to zero. So as we can show that f is, f t is determined by polynomial number of pulses as we said earlier. And but surprisingly you may find that here if you count the number of equations, it actually scales exponentially with the number of, with order you want to suppress because alpha can take two or four values and here you have actually n possible choice of alpha. So it will go to the four to the n or two to the n. So it's kind of a surprise why this will ever work because we have a polynomial number of variables but the exponential number of constraints. But this exponential number of constraints they actually have some properties in there. You may find that the number of integrals is always less than capital N. So this might tell us something that okay, maybe we can use this property to suppress it, to show this is, all these exponential number of constraints can be satisfied. So how to show it? So I would like to list the five steps to show how this will vanish. So first we use the property, just recall early on that the product of two f functions is another f function with indices being the sum of the previous two. And with this you can, we can rewrite this f function in terms of a bunch of integrals. But here it's just like we try to squeeze these f, independent f into this integral by making this definition. And once we did it, essentially we only left behind with this little n number of integrals. And in addition there is one leftover f function which corresponds to the product of all these f functions at the end. So if you look at this definition, a new definition for this f function, you can think it, you can take a perspective. This is some evolution of a function. So it starts from identity which is a trivial function. Then you multiply it with the f function, it becomes a function of t1. The next step you do an integral. It maps from t1, a function of t1 to a function of t2. Then next time you do an integral, it maps from t2 to t3 and so on. In the end it maps to a function of tm minus 1 before it hits the last integral. That will spit out the number. And we, so it just keeps this in mind that we essentially have n steps of evolutions of the functions. And we would like to find some generic feature of functions that are independent of these beta 1, beta 2 up to beta n. And so, and in order to see it, to facilitate the computation, we actually, it will be convenient to switch to functional basis. And the choice of functional basis is the following. First these, the time dependent like coefficients in the toggling frame Hamiltonian, it's a discontinuous function. So that's why we actually would like to have our functional basis also contain this discontinuity. And one way to do it is actually to have a step function in between two passes. And since we have all together Q passes, there will be Q such step functions. And in addition, there will be some integrals. And each integral over a polynomial, it will increase the polynomial power by one. So that's why actually we also, in our functional basis, we also include the polynomial. So this gives our functional basis. And then it's just like a translated language of this functional operator in the functional basis which will be represented by matrices or vectors. So here is just a table that lists these operations in terms of matrices and the vectors. So, but maybe I'll just skip these details a little bit. So the next step is actually we will find that the previous matrices like these ones, they actually block diagonal, have some block features. So we can get rid of these block features and go to a reduced matrix which will be, the size of the matrix will be proportional to the number of passes you apply to the system. So in the end, we got to the form that this f function is just a sum of all these terms with some coefficients. And these terms have the feature that the total power of these d matrices, if we sum together, it's less than n. So essentially we are multiplying a number of d matrices, but the total number of d matrices is less than n. So then here is the last step. And the last rule that this is zero, and hence the f will also be zero. And this can be shown by, first we try some numerics and the later on is like, okay, a nice way to illustrate is to find a nice basis for the functions so that we start from some function which is a, okay, it's correspond to this dot here. And every time we multiply d matrix, it's increased, it's changed the index by at most one. Then so once we multiply d matrix, it changes the index by one. So it goes from, it, it, it explore the original, origin points and also its neighbor. The next time we multiply d matrix, it explore the second nearest neighbor. So we multiply the d matrix multiple times until it hit this boundary. The boundary corresponds to the vector associated with the other side of the operator. So and, and you will find that actually for this, a, oric case, it's a one-dimensional quantum walk and the, the functional basis convenient to use is associated with the free. And similarly we can go to two-dimension, which we find that, which we can get the same thing. But for two-dimension, just recall the oric quadratic dynamical decoupling sequence, it's a concatenate sequence. So we need the two indices to associate with the first level and the second level. So this will correspond, this will give us a two-dimensional quantum walk and then we can find these functional, convenient basis by some like a Fourier transform with a shift. And it's, it's actually because of this shift, which makes it a little hard to prove of this quadratic dynamical decoupling. But the, the message is clear, it's just, and we need to take n steps of d matrices before we hit the boundary, which gives a non-vanishing value of this integral. So, and, and the, one last comment is actually this proof can be generalized to this nested oric dynamical decoupling and, and all the other CDDC, UDD, UCDD sequences. And also this proof can also be generalized. And to these cases where the original may have some time dependence, but this time dependence is an analytic function of time. So finally just in this family tree, we can show that all these guys, they are universal indeed. So just an outlook and so far we have only considered the M qubit system, which generically you can write correspond to the SU2 to the M group. But more generally you can write SUM, where M is a number of levels. And it's actually not clear if these case, for SUM we can achieve a universal dynamical decoupling sequence. So there have been some ideas of using cyclic, UDD, which I think is related to yesterday's talk and we're also thinking about something similar in this direction, or maybe it will be nice to have some numerics to show, okay, maybe there are such universal dynamical decoupling sequences exist for this M level, M level systems. Yeah, thank you. Thank you again. Any questions? Yes, Leonid. To finance with policies? Yes, actually we are discussing during the conference. So it seems possible. The trick is probably, okay, we need to generalize the formulas, right? And so here the F is just a number, which is probably not enough. But we can generalize it. But okay, still in progress. So yeah, we will see. So because you're considering quantum walks, is it possible to consider errors inside of the quantum walk? I mean, as errors inside the URIG scheme? And yes, it depends on what you want to do, right? If you want to show it to universal, then probably you want to make sure all those errors doesn't show up in the quantum walk, and the one probably important property is these errors, okay, always go to nearest neighbor instead of to second nearest neighbor or so on. Yeah, that's something we can think of. If you remember the presentation of my beer, it showed that UDD suffers severely if you take finite time resolution into account. I assume you would have the same problem here. And yes, but I guess usually the time resolution probably in principle can be improved quite a lot because we have good atomic clocks. But that's not the matter of precision of time, but you have the system which has operates on a clock which has a finite rate. Yes. It's not the matter of precision or stability, it's the matter of resolution. Okay, I would say probably in this sense the UDD might not be the optimum against such like errors. Just one trivial, easy question I assume. What can you say about the radius of convergence of your expansion? Because if it does not, if it has a zero radius of convergence, there's no point in perturbation theory. Yes, so actually there's some work on this, but here one simple answer is that if the Hamiltonian, system-based interaction Hamiltonian is bounded, then that will correspond to the convergence. But you need the condition and the time of the sequence also. Yes, so actually so far I can say okay, there is a bound which might not be the optimum which is corresponding inverse of the system-based interaction taking the inverse that corresponds to the time. Okay. Yeah. Let us thank him again.
 Hi, good afternoon everyone.  So today I'm going to talk about universal dynamical decoupling and quantum works in  functional space.  So this work is done in collaboration with Adelaide Mbakov from Rice University.  And if you're interested in details, you can find it in archive paper.  And I also will recommend, and you can also take a look at a quote under Lidaz paper,  which also have related topics.  So first of all, since we're here in a conference for quantum information processing, one of  the biggest challenges is actually decoherence.  And decoherence can be induced by system and the bus interaction.  And for example, we consider a system of a single qubit.  The most generic system and the bus interaction can be written in the form, which is just  a sum of the system generators tensored with the corresponding bus operator.  So here it's actually, we try to roll, so you may see that these bus operators they  have a hat, so they may not commute with each other.  And in addition, these system operators, they also do not commute with each other.  So this actually makes the problem a little more hard to keep track of.  And so the idea is actually to suppress, so in the presence of the system bus interaction,  they will induce entanglement between system and the bus, which leads to the decoherence  of the system.  So the idea of using dynamic decoupling is to introduce additional system evolution  so that the overall evolution, which is combined with the system bus and the system evolution  together, it will become just the identity of the system tensor with some unitary operation  on the bus.  So if we can achieve that, then there will be no entanglement between the system and  the bus.  And so in this talk here, I'll particularly focus on this universal dynamical decoupling,  which efficiently decouples the system from bus for arbitrary system and the bus interaction.  So here universal means these bus operators can be arbitrary, but with a finite norm,  of course.  So before we talk about this universal decoupling sequence, let's first take a look at this  high echo sequence, which we are all very familiar with.  So the high echo sequence consists of two pulses, which is a pie pulse in the halfway  of evolution that flips the spin, and at the end of the evolution, it flips the spin back.  So if we consider this, and high echo sequence can efficiently suppress this defacing noise,  which can be induced by the system operator Sz tensor with the bus operator Bz.  And let's look at how the system bus Hamiltonian, an interaction Hamiltonian evolves in the  presence of this system evolution.  And it's convenient to introduce this toggling frame Hamiltonian, which you can think of  as an interaction Hamiltonian associated with the convolution.  So in the toggling frame Hamiltonian, what happens is the first term, which acts trivially  on the system, commutes with all the operations on the system.  So there is no change to this term.  But the second term, Sz, which actually anti-commutes with the pulse.  So the first half of evolution, there's no sign change of this term, but in the second  half there is additional minus sign.  So then we can compute the unitary of this evolution, which consists of two sequential  evolutions.  And we can keep track of the term and get this suppression, which the system bus interaction  gets suppressed to the second order.  So actually this example actually outlines the procedure that we can use to keep track  of this universal dynamic decoupling, which is first we write down the system bus interaction,  then we write down this toggling frame Hamiltonian, and the last step we compute the unitary associated  with this toggling frame Hamiltonian.  So which is our plan here, system bus, toggling frame Hamiltonian, and in the end we'll compute  the unitary.  And for this, and actually there have been a lot of studies about this universal dynamic  decoupling.  And lots of sequences are proposed and proved, and lots of sequences are conjectured to be  universal.  So here's just probably an incomplete list, but it consists of most of the features.  But maybe it's a little bit overwhelming to see all these on the same figure.  So I try to come up with a family tree for these dynamic decoupling sequences.  So the great grandfather for all these sequences is actually Han Eko.  And actually there could be like two different kinds of decoupling sequences, approaches.  One is called concatenated dynamic decoupling, which Daniel also mentioned in his tutorial  on Monday.  And the other one is the URIG decoupling, which is a little more efficient, basically  use order of N-palses to suppress this decoupling to the higher order.  But these two sequences, but this URIG only suppresses the defacing noise qubit.  So we might want to protect the entire qubit from both defacing and bit flip noise, so  that goes to the second generation of this dynamic decoupling sequence.  So for concatenated sequence you can actually, there's a way to actually suppress both defacing  and bit flip.  And for the URIG sequence, if you nest it with two nesting levels, which gives you the  quadratic dynamical decoupling for the URIG sequence.  And you can also combine the concatenated with the URIG sequence, come up with some concatenated  URIG decoupling or URIG concatenated decoupling.  So the ordering here corresponds to the, which comes the higher order, which corresponds  to the lower order decoupling.  And so we can go even further.  The third generation corresponds to the decoupling sequence, which can protect m qubit quantum  systems.  So here you may have some decoupling, system-based interaction, which might evolve like a multi-cubit,  like X1, X2 times the bus operator.  And by having this decoupling sequence for m qubit systems, those terms will be eliminated  as well.  So natural generalization will be nested concatenated decoupling sequence or nested URIG decoupling  sequence.  But as you see, that's okay.  On the upper right, the upper left side of these sequences, they are all proved to be  universal.  But on the lower left, which is still open questions, we don't know if they are universal  or not, but they are conjectured to be.  And it's important to understand these sequences because they are very efficient, which scales  polynomially with all the ones you want to suppress.  In the rest of the talk, I'll try to convince you that indeed, all these sequences, they  are universal.  So before I go into the proof, let's first take a look at just to recap what the URIG  decoupling sequence is.  So URIG decoupling sequence is actually an unequally spaced decoupling sequence.  So the spacing between the pulses are not equal.  They actually are obtained by some equal partition over half circle and then projected  along this horizontal axis.  And for example, here, n equal to 2, so you have two pulses, which kind of correspond to  these two points on the circle.  And for example, if I want to suppress defacing noise using URIG dynamical decoupling, we  can apply spin flip operators at these two times.  So what happens is that the system Hamiltonian consists of two terms, which one is actually  on the system, the other induces defacing of the system.  And the one actually on the system, the coefficient doesn't change.  The one that induces defacing of the system, which actually induces, which changes sign  every time it hit this spin flip operator because it's an anti-commute operator.  So overall, the message is, OK, we know what this f function is.  It's just a sum of plus one or minus one.  And the additional feature is that you may find that if we multiply these two f functions  with different subindex, so here alpha takes the value zero and one, corresponds to trivial  operator or the sd operator.  So when you multiply these two f functions, it turns out there is a simple rule that it  becomes another f function, where the index becomes a binary sum of these two subindexes.  So this is an important feature.  And you may find that this has a similar correspondence in the system operator.  When you multiply two system operators with alpha to be zero or one, it becomes another  system operator, which the indices corresponds to the binary sum of these indices.  Because in some sense, it's trivial because sd is just the sd square as identity.  But you will find that actually it becomes little non-trivial when we go to the case  for the quadratic dynamic decoupling, which we have the system.  So here we will consider the system of a single qubit, which has the sx, sy, and sd coupling  to the bus.  So now we have, generically, we can write four terms in the system bus interaction.  And here I'll label it with two binaries, zero, zero, up to one, one.  And so first, we consider the decoupling sequence for this quadratic dynamic decoupling.  It consists of a con-cad-nested O-rig decoupling.  So in each interval of the original O-rig dynamic decoupling sequence, you put another  O-rig inside here.  So therefore, you have one O-rig sequence at the highest level, and here for n equals  two, you have three O-rig sequence at the lower level.  So we can compute these f-functions as a following.  And so for example, this associated with sd function, which you will change the sign  every time it hits this sx operator.  And similarly, we can get the f-function associated with sx operator and sy operator.  And the feature is, again, you confirm this feature, the product of two f-functions is  another f-function, but with the indices corresponds to the binary sum of the two inputs.  And here similarly, the s-function and the system operator function also have the same  feature.  It's the product of the two is another system operator with the indices being the binary  sum, but up to some plus or minus sign.  But here it doesn't matter because actually later on we'll show that it equal to zero,  so plus or minus zero, it's still zero.  But keep these two features in mind as we'll use it later.  So just to recap how we prove the universal dynamic decoupling.  We have three steps.  We write the system-based interaction, we go to the top frame, and we compute the unitary  evolution.  So we have already gone through the first two steps.  The next step is to compute the unitary evolution.  And the goal is to show this unitary is essentially identity acting on the system plus some higher  order terms.  So the way to do it is by doing Dyson expansion.  So this is the unitary.  Then the standard Dyson formula gives us a bunch of terms, and here n corresponds to  the little n corresponds to a number of integrals associated with these expansion terms.  And in addition, so you can plug in this Hamiltonian, which consists of a bunch of terms, and which  you can sum all these over.  But the important thing is here you have n integrals.  That will give you power t to the n, because that's the only way you can get the total  t, which is the total evolution time.  And the rest of the term consists of three components.  First is a coefficient, which is associated with a time-dependent f function induced from  this decoupling sequence.  And the second component is the system operator, which is the product of these s operators.  But it's important to keep track of the ordering of the system operator.  And the last term is the bus operator, which is the product of all those bus operators.  And so here is just like the definition right now explicitly.  So just to recall that early on we say the system operator, the product of the two, is  an again system operator, but with index is a binary sum.  So we can generalize this to n product of system operators, which is a binary sum of  n indices.  It's again a system operator.  And this system operator becomes identity only if the binary sum of these indices is  zero.  And so if we look at this Dyson expansion here, we'll find that the goal is to hope  and to show that this one is equal to some identity act on the system plus some higher  order terms, which means all the lower order terms show.  That means that this f function should be zero for all those lower order terms when  n is less than capital N.  But except for some, but this is a little too restrict because actually if for this  expansion if the system operator here is identity, it's actually okay because it does not perturb  our system.  So we can actually exclude those terms by putting a constraint that this system operator  is not equal to identity in those expansions when we try to compute these f terms, which  corresponds to the binary sum of those alphas in this f function is not equal to zero.  So as we can show that f is, f t is determined by polynomial number of pulses as we said  earlier.  And but surprisingly you may find that here if you count the number of equations, it actually  scales exponentially with the number of, with order you want to suppress because alpha can  take two or four values and here you have actually n possible choice of alpha.  So it will go to the four to the n or two to the n.  So it's kind of a surprise why this will ever work because we have a polynomial number  of variables but the exponential number of constraints.  But this exponential number of constraints they actually have some properties in there.  You may find that the number of integrals is always less than capital N.  So this might tell us something that okay, maybe we can use this property to suppress  it, to show this is, all these exponential number of constraints can be satisfied.  So how to show it?  So I would like to list the five steps to show how this will vanish.  So first we use the property, just recall early on that the product of two f functions  is another f function with indices being the sum of the previous two.  And with this you can, we can rewrite this f function in terms of a bunch of integrals.  But here it's just like we try to squeeze these f, independent f into this integral by  making this definition.  And once we did it, essentially we only left behind with this little n number of integrals.  And in addition there is one leftover f function which corresponds to the product of all these  f functions at the end.  So if you look at this definition, a new definition for this f function, you can think  it, you can take a perspective.  This is some evolution of a function.  So it starts from identity which is a trivial function.  Then you multiply it with the f function, it becomes a function of t1.  The next step you do an integral.  It maps from t1, a function of t1 to a function of t2.  Then next time you do an integral, it maps from t2 to t3 and so on.  In the end it maps to a function of tm minus 1 before it hits the last integral.  That will spit out the number.  And we, so it just keeps this in mind that we essentially have n steps of evolutions  of the functions.  And we would like to find some generic feature of functions that are independent of these  beta 1, beta 2 up to beta n.  And so, and in order to see it, to facilitate the computation, we actually, it will be convenient  to switch to functional basis.  And the choice of functional basis is the following.  First these, the time dependent like coefficients in the toggling frame Hamiltonian, it's a  discontinuous function.  So that's why we actually would like to have our functional basis also contain this discontinuity.  And one way to do it is actually to have a step function in between two passes.  And since we have all together Q passes, there will be Q such step functions.  And in addition, there will be some integrals.  And each integral over a polynomial, it will increase the polynomial power by one.  So that's why actually we also, in our functional basis, we also include the polynomial.  So this gives our functional basis.  And then it's just like a translated language of this functional operator in the functional  basis which will be represented by matrices or vectors.  So here is just a table that lists these operations in terms of matrices and the vectors.  So, but maybe I'll just skip these details a little bit.  So the next step is actually we will find that the previous matrices like these ones,  they actually block diagonal, have some block features.  So we can get rid of these block features and go to a reduced matrix which will be,  the size of the matrix will be proportional to the number of passes you apply to the system.  So in the end, we got to the form that this f function is just a sum of all these terms  with some coefficients.  And these terms have the feature that the total power of these d matrices, if we sum  together, it's less than n.  So essentially we are multiplying a number of d matrices, but the total number of d matrices  is less than n.  So then here is the last step.  And the last rule that this is zero, and hence the f will also be zero.  And this can be shown by, first we try some numerics and the later on is like, okay, a  nice way to illustrate is to find a nice basis for the functions so that we start from some  function which is a, okay, it's correspond to this dot here.  And every time we multiply d matrix, it's increased, it's changed the index by at most  one.  Then so once we multiply d matrix, it changes the index by one.  So it goes from, it, it, it explore the original, origin points and also its neighbor.  The next time we multiply d matrix, it explore the second nearest neighbor.  So we multiply the d matrix multiple times until it hit this boundary.  The boundary corresponds to the vector associated with the other side of the operator.  So and, and you will find that actually for this, a, oric case, it's a one-dimensional  quantum walk and the, the functional basis convenient to use is associated with the free.  And similarly we can go to two-dimension, which we find that, which we can get the same  thing.  But for two-dimension, just recall the oric quadratic dynamical decoupling sequence,  it's a concatenate sequence.  So we need the two indices to associate with the first level and the second level.  So this will correspond, this will give us a two-dimensional quantum walk and then we  can find these functional, convenient basis by some like a Fourier transform with a shift.  And it's, it's actually because of this shift, which makes it a little hard to prove of this  quadratic dynamical decoupling.  But the, the message is clear, it's just, and we need to take n steps of d matrices  before we hit the boundary, which gives a non-vanishing value of this integral.  So, and, and the, one last comment is actually this proof can be generalized to this nested  oric dynamical decoupling and, and all the other CDDC, UDD, UCDD sequences.  And also this proof can also be generalized.  And to these cases where the original may have some time dependence, but this time dependence  is an analytic function of time.  So finally just in this family tree, we can show that all these guys, they are universal  indeed.  So just an outlook and so far we have only considered the M qubit system, which generically  you can write correspond to the SU2 to the M group.  But more generally you can write SUM, where M is a number of levels.  And it's actually not clear if these case, for SUM we can achieve a universal dynamical  decoupling sequence.  So there have been some ideas of using cyclic, UDD, which I think is related to yesterday's  talk and we're also thinking about something similar in this direction, or maybe it will  be nice to have some numerics to show, okay, maybe there are such universal dynamical decoupling  sequences exist for this M level, M level systems.  Yeah, thank you.  Thank you again.  Any questions?  Yes, Leonid.  To finance with policies?  Yes, actually we are discussing during the conference.  So it seems possible.  The trick is probably, okay, we need to generalize the formulas, right?  And so here the F is just a number, which is probably not enough.  But we can generalize it.  But okay, still in progress.  So yeah, we will see.  So because you're considering quantum walks, is it possible to consider errors inside of  the quantum walk?  I mean, as errors inside the URIG scheme?  And yes, it depends on what you want to do, right?  If you want to show it to universal, then probably you want to make sure all those errors doesn't  show up in the quantum walk, and the one probably important property is these errors, okay,  always go to nearest neighbor instead of to second nearest neighbor or so on.  Yeah, that's something we can think of.  If you remember the presentation of my beer, it showed that UDD suffers severely if you  take finite time resolution into account.  I assume you would have the same problem here.  And yes, but I guess usually the time resolution probably in principle can be improved quite  a lot because we have good atomic clocks.  But that's not the matter of precision of time, but you have the system which has operates  on a clock which has a finite rate.  Yes.  It's not the matter of precision or stability, it's the matter of resolution.  Okay, I would say probably in this sense the UDD might not be the optimum against such  like errors.  Just one trivial, easy question I assume.  What can you say about the radius of convergence of your expansion?  Because if it does not, if it has a zero radius of convergence, there's no point in perturbation  theory.  Yes, so actually there's some work on this, but here one simple answer is that if the  Hamiltonian, system-based interaction Hamiltonian is bounded, then that will correspond to the  convergence.  But you need the condition and the time of the sequence also.  Yes, so actually so far I can say okay, there is a bound which might not be the optimum  which is corresponding inverse of the system-based interaction taking the inverse that corresponds  to the time.  Okay.  Yeah.  Let us thank him again.
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