Zeno effect for quantum control and computation
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Dominy, Jason
Paz-Silva, Gerardo
Rezakhani, Ali
Lidar, Daniel
It is well known that the quantum Zeno effect can protect specific quantum states from decoherence by using projective measurements. Here we combine the theory of weak measurements with stabilizer quantum error correction and detection codes. We derive rigorous performance bounds which demonstrate that the Zeno effect can be used to protect appropriately encoded arbitrary states to arbitrary accuracy, while at the same time allowing for universal quantum computation or quantum control.
 So, yeah, I'm going to talk about the Xeno effect for quantum computation and control. This is joint work with Rarotopas Silva, Ali Rizikani, and Daniel Lidar. So we've got sort of the typical situation that we've seen all week here. We've got N qubits and some uncontrollable bath, unknown bath that's going to act on these qubits. As far as notation, we're going to split the Hamiltonian up into a system bath and interaction terms as usual and have a unitary propagation on the composite system. All right. So the issue is we've got decoherence, of course, because of that bath, and how can we deal with it? And we've seen a number of methods this week, passive methods like decoherence free subspaces, active methods, closed loop methods like quantum error correction, and open loop methods like dynamical coupling. And we're going to add one more open loop method, which is the quantum Xeno effect. So this goes back to the 1970s. It's a well-known work protection of given states with projective measurements, although other work since the 70s has extended that to more general types of measurements as well as protecting subspaces instead of just given particular states. So what we want to do is protect an arbitrary state or subspace using the quantum Xeno effect. And we want to try and understand, try and quantify how well we can do this with finite resources. So the Xeno effect typically is looking at infinitely many measurements, but what happens if we do finite number of non-projective measurements? So just to review the Xeno effect is this idea where you can do many measurements, and by doing that, you basically slow down, freeze the dynamics of the system so that the watch pod never boils. So we're going to do these M measurements in some finite time interval, capital T, and I want both M and T to be finite. So I mentioned we're going to do with non-projective measurements. We want weak measurements. So what that means is a weak interaction between the system and the measurement apparatus or the probe. And the way that we describe this is with sort of an interpolation parameter, this parameter epsilon. Oh, I had it for a second. There it is, which parameterizes the strength of the measurement from epsilon equals zero, which is basically no measurement at all, to epsilon equal infinity, which is a projective measurement. So how do we do this? How do we set up a Xeno effect and do it with the weak measurements to protect arbitrary states from this environment? So the basic idea is we're going to incorporate some ideas from quantum error correction. We're going to make a stabilizer quantum error correction code, encode the initial state. We're going to then weakly measure the stabilizer elements from that group. And then simultaneously to doing that, we can think about doing some degree of computation and control by using the logical operators. Since they will commute with our measurement operators, we can kind of slide those in there. And I want to emphasize the point that we're not doing error correction. We're not closing the loop and correcting the errors. We're just doing frequent detection of those errors. So okay. Again, we're taking an unknown state. And some known code for some finite time. And again, this system Hamiltonian that we talked about. These two parameters I want to emphasize, J0 and J1, are going to be parameters that we're going to use in this formulation. So we're going to grab the norm of the system and bath Hamiltonians together and the norm of the interaction term. And then we've got this super operator from the Hamiltonian. And we make this assumption that the bath interaction is at most D minus 1 local. So that ensures that our system bath interaction term is a linear combination of errors that will be detected by this code. So we encode into the stabilizer group G. And then the protocol is just this very simple thing of free evolution punctuated by measurements. And we do that m times over the course of the time interval T. And the projection, well, P here is the measurement, sort of a weak projection, if you like. It's the measurement of either all of the elements of the stabilizer group or just the generators of that group. So those are two different protocols that we can look at. So in order to sort of evaluate how well we're doing, we describe this distance metric between the final state, rather, having applied this Zeno protocol compared to what the dynamics would have been if we had no system bath interaction. So we just had the internal Hamiltonians of the system and bath, but no interaction term. So that's row not in this notation. So I mentioned that we want to try and quantify how well we can do with finite resources. So we're going to try and decompose this distance, try to develop a bound for this distance that we can easily compute with the parameters that we've got at hand. So I can't really go into the details. They're kind of painful and shocking and, well, tedious, to say the least. But just to give it a lot of line here, we can decompose the dynamics using the Dyson series. We can use basically the algebraic structure of stabilizer codes, these error subspaces, to decompose the terms of the Dyson series into those various subspaces, and then apply a bunch of triangle inequalities and so forth. And what we get is a triple sum in which the sum looks like a combinatorial type of a quantity. So there are binomial coefficients and exponentials in there. And so that series is what's known as a hypergeometric series. And that brings us to the mathematical none here. So this is sister Mary Selene Faizenmaier, who did some nice work back in the 40s where she came up with some paper and pencil methods that were quite simple but really nice for evaluating these types of hypergeometric series. And what she came up with was a way of basically searching for linear recurrence relations that are satisfied by these series. So she was interested in special functions. These series show up in a lot of places, including special functions. So she was interested in evaluating that. But these methods have been developed and generalized and optimized and turned into packages for Mathematica and Maple and these computer algebra systems. So for at least certain subclasses of these series, you can just basically plug them in and get out an answer. It turns out that our series is a little bit different. It has non-standard boundary conditions essentially and that introduces sort of an inhomogeneity into that thing. But most of this comes out right from this work. And if we use these types of methods, we can get what's going to look like probably an ugly expression for this bound. And it's not going to get any prettier when I show you what the notation actually is. But the nice piece of this is that these are closed expressions and they involve only the half dozen parameters that we talked about, the norms of the Hamiltonian terms, the number of measurements, M, the final time, the size of the stabilizer group, and the strength of the measurement. So we can just plug those things right in. So we'd like to try and analyze this a little bit further and understand, is it a good bound, for example, and what is it really telling us about the behavior of this method? Or are we actually realizing a Zeno effect using these weak measurements? So what we can show is that actually for any measurement strength greater than zero, at least, we can expand around M equal to infinity, M being the number of measurements again. And we can see that we get this 1 over M behavior going on here. So it is converging to zero as M grows. We are realizing a Zeno effect no matter what the strength of the measurement is. And I'll point out that basically the strength of the measurement shows up in this term here. So if the measurements are projective, epsilon is infinity and this term becomes zero. So that vanishes. And so this term is the only one that shows up in the effective limit. In the other limit, when epsilon is going to zero, this thing dominates. This becomes this gross to infinity. So the rate of convergence of this thing, even though it's 1 over M for any arbitrary epsilon, the rate of convergence is decreasing as epsilon decreases, as the strength of the measurement decreases. So I mentioned these two variations on the protocol. One where we are measuring the full stabilizer group and one where we are measuring just the generators of that group. So there are trade-offs between the two. Obviously, if you're just measuring the generators, it's an exponentially fewer measurement that you have to do at each iteration. But with the caveat that the behavior, at least of this bound, shows slower convergence. So this, the Q here, which is related to the size of the stabilizer code, is replaced by a 1 in the case where you go to the generators only. So it's a little bit slower convergence. Come on. So we just plot out what the bound looks like for a certain set of parameters here. So we've chosen J naught, T, to be 1, and then we've generated to be 4, and the size of the interaction to be 1 tenth of the size of the system in bath, the Hamiltonians themselves. And so you can see the bottom surface here is just when we take the limit of epsilon going to zero. So this is the limit of strong measurements, this purple surface here. And then the middle surface is the full stabilizer group. It's converging very quickly, and when you do the generators only, it converges a little bit less quickly. So that's recap here. We've got this method or this protocol for protecting systems from the effect of the environment. We've got this rigorous distance bound that we've been able to come up with that characterizes how well we can do with finite resources in this scheme. I didn't really show it, but what you can show is that it doesn't really matter whether the Hamiltonian is time-dependent or time-independent, the Dyson expansion will work either way. So you've got the potential for application to adiabatic quantum computing. And I mentioned that we could potentially also use the logical operators of the code to do computation while we're protecting these things. And I'll just mention briefly that what I showed you here was a result for non-selective measurements so that that final state that I talked about is basically the average, stable average over all possible measurement outcomes. But we'd like to know what happens if we consider all of the individual possible measurement outcomes, all the possible outcome states. Is it just the average that's converging in this nice way, or is it sort of a concentration of measure that's happening so that with high probability you're getting near the ideal state that we're looking for? So thank you for your attention. If you repeatedly do the same weak measurements, then that converges to a strong measurement. So it's not surprising that if you keep the measurement strength constant and let the number of measurements go to infinity that you get a Xeno effect. But I have a question. If you measure all of the elements of the stabilizer versus measuring only the generators, from measuring the generators, you would, because that's, you know, that gives you the full information about the whole stabilizer, you could deduce from, if it were a strong measurement, you could deduce the outcome of a measurement of any element of the stabilizer just from measuring the generators. So is the effect of it converging more rapidly when you measure the whole stabilizer just that you're doing more measurements? In other words, would just measuring the generator, if the ratio between the generators and the whole thing is n, if you just did n times as many measurements of the generators, would that be just the same as measuring the whole stabilizer? It's a little hard to tell. Just because what we're looking at here is this bound that we've created. Certainly in the bound it doesn't look quite that clear cut. But I think there's some saying that the additional measurements are contributing in that sort of way, that you're not gaining any new information really. But yeah, I think that's pretty close. Do measurements, if you are dealing with the same analysis? Yes we did. For the purpose of the analysis, yes we did. So many body measurements are not always easy to come by. I'm wondering if you thought about physical systems where this could be implemented. For example, when I was watching your talk, Kadeo Mabuchi's proposal from this morning came to mind. But have you thought about other physical systems where this might work? I have not. I've easily adapted. Largely because I'm not a physicist really. We have considered other possibilities of extending this beyond the stabilizer code formalism to potentially the bake and shore code and two body measurements in that case. In complete opposition to the spirit of this conference, do you have an anti-zeno effect here also? Can you get an anti-zeno? Again, it's hard to say because we're only looking at this bound. It's certainly possible that in the introductory part of that protocol that something bad is happening but it's kind of hard to see in this analysis.
 So, yeah, I'm going to talk about the Xeno effect for quantum computation and control.  This is joint work with Rarotopas Silva, Ali Rizikani, and Daniel Lidar.  So we've got sort of the typical situation that we've seen all week here.  We've got N qubits and some uncontrollable bath, unknown bath that's going to act on  these qubits.  As far as notation, we're going to split the Hamiltonian up into a system bath and interaction  terms as usual and have a unitary propagation on the composite system.  All right.  So the issue is we've got decoherence, of course, because of that bath, and how can  we deal with it?  And we've seen a number of methods this week, passive methods like decoherence free subspaces,  active methods, closed loop methods like quantum error correction, and open loop methods like  dynamical coupling.  And we're going to add one more open loop method, which is the quantum Xeno effect.  So this goes back to the 1970s.  It's a well-known work protection of given states with projective measurements, although  other work since the 70s has extended that to more general types of measurements as well  as protecting subspaces instead of just given particular states.  So what we want to do is protect an arbitrary state or subspace using the quantum Xeno effect.  And we want to try and understand, try and quantify how well we can do this with finite  resources.  So the Xeno effect typically is looking at infinitely many measurements, but what happens  if we do finite number of non-projective measurements?  So just to review the Xeno effect is this idea where you can do many measurements, and by  doing that, you basically slow down, freeze the dynamics of the system so that the watch  pod never boils.  So we're going to do these M measurements in some finite time interval, capital T, and  I want both M and T to be finite.  So I mentioned we're going to do with non-projective measurements.  We want weak measurements.  So what that means is a weak interaction between the system and the measurement apparatus or  the probe.  And the way that we describe this is with sort of an interpolation parameter, this parameter  epsilon.  Oh, I had it for a second.  There it is, which parameterizes the strength of the measurement from epsilon equals zero,  which is basically no measurement at all, to epsilon equal infinity, which is a projective  measurement.  So how do we do this?  How do we set up a Xeno effect and do it with the weak measurements to protect arbitrary  states from this environment?  So the basic idea is we're going to incorporate some ideas from quantum error correction.  We're going to make a stabilizer quantum error correction code, encode the initial state.  We're going to then weakly measure the stabilizer elements from that group.  And then simultaneously to doing that, we can think about doing some degree of computation  and control by using the logical operators.  Since they will commute with our measurement operators, we can kind of slide those in there.  And I want to emphasize the point that we're not doing error correction.  We're not closing the loop and correcting the errors.  We're just doing frequent detection of those errors.  So okay.  Again, we're taking an unknown state.  And some known code for some finite time.  And again, this system Hamiltonian that we talked about.  These two parameters I want to emphasize, J0 and J1, are going to be parameters that  we're going to use in this formulation.  So we're going to grab the norm of the system and bath Hamiltonians together and the norm  of the interaction term.  And then we've got this super operator from the Hamiltonian.  And we make this assumption that the bath interaction is at most D minus 1 local.  So that ensures that our system bath interaction term is a linear combination of errors that  will be detected by this code.  So we encode into the stabilizer group G. And then the protocol is just this very simple  thing of free evolution punctuated by measurements.  And we do that m times over the course of the time interval T. And the projection, well,  P here is the measurement, sort of a weak projection, if you like.  It's the measurement of either all of the elements of the stabilizer group or just the  generators of that group.  So those are two different protocols that we can look at.  So in order to sort of evaluate how well we're doing, we describe this distance metric between  the final state, rather, having applied this Zeno protocol compared to what the dynamics  would have been if we had no system bath interaction.  So we just had the internal Hamiltonians of the system and bath, but no interaction term.  So that's row not in this notation.  So I mentioned that we want to try and quantify how well we can do with finite resources.  So we're going to try and decompose this distance, try to develop a bound for this distance  that we can easily compute with the parameters that we've got at hand.  So I can't really go into the details.  They're kind of painful and shocking and, well, tedious, to say the least.  But just to give it a lot of line here, we can decompose the dynamics using the Dyson  series.  We can use basically the algebraic structure of stabilizer codes, these error subspaces,  to decompose the terms of the Dyson series into those various subspaces, and then apply  a bunch of triangle inequalities and so forth.  And what we get is a triple sum in which the sum looks like a combinatorial type of a quantity.  So there are binomial coefficients and exponentials in there.  And so that series is what's known as a hypergeometric series.  And that brings us to the mathematical none here.  So this is sister Mary Selene Faizenmaier, who did some nice work back in the 40s where  she came up with some paper and pencil methods that were quite simple but really nice for  evaluating these types of hypergeometric series.  And what she came up with was a way of basically searching for linear recurrence relations  that are satisfied by these series.  So she was interested in special functions.  These series show up in a lot of places, including special functions.  So she was interested in evaluating that.  But these methods have been developed and generalized and optimized and turned into packages for  Mathematica and Maple and these computer algebra systems.  So for at least certain subclasses of these series, you can just basically plug them in  and get out an answer.  It turns out that our series is a little bit different.  It has non-standard boundary conditions essentially and that introduces sort of an inhomogeneity  into that thing.  But most of this comes out right from this work.  And if we use these types of methods, we can get what's going to look like probably an  ugly expression for this bound.  And it's not going to get any prettier when I show you what the notation actually is.  But the nice piece of this is that these are closed expressions and they involve only the  half dozen parameters that we talked about, the norms of the Hamiltonian terms, the number  of measurements, M, the final time, the size of the stabilizer group, and the strength  of the measurement.  So we can just plug those things right in.  So we'd like to try and analyze this a little bit further and understand, is it a good bound,  for example, and what is it really telling us about the behavior of this method?  Or are we actually realizing a Zeno effect using these weak measurements?  So what we can show is that actually for any measurement strength greater than zero, at  least, we can expand around M equal to infinity, M being the number of measurements again.  And we can see that we get this 1 over M behavior going on here.  So it is converging to zero as M grows.  We are realizing a Zeno effect no matter what the strength of the measurement is.  And I'll point out that basically the strength of the measurement shows up in this term here.  So if the measurements are projective, epsilon is infinity and this term becomes zero.  So that vanishes.  And so this term is the only one that shows up in the effective limit.  In the other limit, when epsilon is going to zero, this thing dominates.  This becomes this gross to infinity.  So the rate of convergence of this thing, even though it's 1 over M for any arbitrary  epsilon, the rate of convergence is decreasing as epsilon decreases, as the strength of the  measurement decreases.  So I mentioned these two variations on the protocol.  One where we are measuring the full stabilizer group and one where we are measuring just  the generators of that group.  So there are trade-offs between the two.  Obviously, if you're just measuring the generators, it's an exponentially fewer measurement that  you have to do at each iteration.  But with the caveat that the behavior, at least of this bound, shows slower convergence.  So this, the Q here, which is related to the size of the stabilizer code, is replaced by  a 1 in the case where you go to the generators only.  So it's a little bit slower convergence.  Come on.  So we just plot out what the bound looks like for a certain set of parameters here.  So we've chosen J naught, T, to be 1, and then we've generated to be 4, and the size  of the interaction to be 1 tenth of the size of the system in bath, the Hamiltonians themselves.  And so you can see the bottom surface here is just when we take the limit of epsilon  going to zero.  So this is the limit of strong measurements, this purple surface here.  And then the middle surface is the full stabilizer group.  It's converging very quickly, and when you do the generators only, it converges a little  bit less quickly.  So that's recap here.  We've got this method or this protocol for protecting systems from the effect of the  environment.  We've got this rigorous distance bound that we've been able to come up with that characterizes  how well we can do with finite resources in this scheme.  I didn't really show it, but what you can show is that it doesn't really matter whether  the Hamiltonian is time-dependent or time-independent, the Dyson expansion will work either way.  So you've got the potential for application to adiabatic quantum computing.  And I mentioned that we could potentially also use the logical operators of the code  to do computation while we're protecting these things.  And I'll just mention briefly that what I showed you here was a result for non-selective  measurements so that that final state that I talked about is basically the average, stable  average over all possible measurement outcomes.  But we'd like to know what happens if we consider all of the individual possible measurement  outcomes, all the possible outcome states.  Is it just the average that's converging in this nice way, or is it sort of a concentration  of measure that's happening so that with high probability you're getting near the ideal  state that we're looking for?  So thank you for your attention.  If you repeatedly do the same weak measurements, then that converges to a strong measurement.  So it's not surprising that if you keep the measurement strength constant and let the  number of measurements go to infinity that you get a Xeno effect.  But I have a question.  If you measure all of the elements of the stabilizer versus measuring only the generators,  from measuring the generators, you would, because that's, you know, that gives you the  full information about the whole stabilizer, you could deduce from, if it were a strong  measurement, you could deduce the outcome of a measurement of any element of the stabilizer  just from measuring the generators.  So is the effect of it converging more rapidly when you measure the whole stabilizer just  that you're doing more measurements?  In other words, would just measuring the generator, if the ratio between the generators and the  whole thing is n, if you just did n times as many measurements of the generators, would  that be just the same as measuring the whole stabilizer?  It's a little hard to tell.  Just because what we're looking at here is this bound that we've created.  Certainly in the bound it doesn't look quite that clear cut.  But I think there's some saying that the additional measurements are contributing in that sort  of way, that you're not gaining any new information really.  But yeah, I think that's pretty close.  Do measurements, if you are dealing with the same analysis?  Yes we did.  For the purpose of the analysis, yes we did.  So many body measurements are not always easy to come by.  I'm wondering if you thought about physical systems where this could be implemented.  For example, when I was watching your talk, Kadeo Mabuchi's proposal from this morning  came to mind.  But have you thought about other physical systems where this might work?  I have not.  I've easily adapted.  Largely because I'm not a physicist really.  We have considered other possibilities of extending this beyond the stabilizer code  formalism to potentially the bake and shore code and two body measurements in that case.  In complete opposition to the spirit of this conference, do you have an anti-zeno effect  here also?  Can you get an anti-zeno?  Again, it's hard to say because we're only looking at this bound.  It's certainly possible that in the introductory part of that protocol that something bad is  happening but it's kind of hard to see in this analysis.
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