RE: Yet another suggestion for Discussion: analytical techniques for quantum control [Sophie]

Motivated by Tuesday's talks I wanted to suggest another discussion on analytical techniques. Do we need more analytical techniques? Is it useful to develop analytical approaches even if we can only deal with a few parameters? E.g. explicit Hamilton-Jacobi equations or Lie group methods? What other techniques? What would be the most important questions to answer?

RE: Suggestion for further Discussionon state-of-the-art and challenges of quantum control [Sophie]

Motivated by Birgitta's very nice talk yesterday I wanted to suggest that maybe we could have further discussions to establish what's been demonstrated experimentally, what's been done theoretically, and what we think may be possible in the near future or shown to be/believed to be impossible. Many talks have touched on these issues but it would be nice to pull it all together and write a little summary for the wiki. Topics might include

- Quantum Information:

Robust Quantum Gates:

Quantum state preparation

- Quantum Metrology

- Magnetic Resonance Spectroscopy & Imaging
Molecular MRI?

- Photo-chemistry

- Quantum biology: light harvesting and beyond

- Theory / computation

fundamental issues (e.g. quantum-classical interface)

algorithms (optimal control), controllability,

system identification and parameter estimation

RE: Control paradigms for quantum systems etc. [Sophie]

I promised to write something for this wiki, so here we go. Let me start by saying that Jun's talk today gave a nice overview of classical control and system identification paradigms, I think.

What distinguishes control of quantum systems from its classical counterpart (for me) is not open vs closed-loop, although people may have got the impression from my talk. Both paradigms are clearly useful for classical and quantum systems. Quantum error correction clearly meets the definition of feedback control posted here (by Matt, I think) and it is very useful to deal with noise, disturbances and uncertainty, just as it is in the classical case. Some people have argued that quantum error correction is really feedforward control but Carlton's talk and diagrams have convinced me that this is definitively not the case for standard quantum error correction. We are not measuring the disturbance and correcting the open-loop control by anticipating the effect it will have on the system, but rather we really do perform measurements on the system to infer the effect of disturbances after they have acted on the system, and to mitigate it and restore the desired behaviour. At the same time open-loop control is very useful to construct controls that solve non-trivial problems from implementing quantum gates to putting systems into quantum states we are interested in (as opposed to what nature provides). But as Jun mentioned today, open-loop control is also used for classical systems, just so happens that it is generally a much simpler problem and therefore not given as much attention. This being said, the distinction between open and closed-loop control is really too simplistic. Some paradigms such as learning control do not seem to fit neatly into either category, at least if we go by the definitions posted below. Going by the definitions below, I would say that learning control fits more into the feedback paradigm (we certainly require feedback to learn) but it is quite a different setting from feedback control acting on the same system. But again, those paradigms have classical analogues too. The same can be said for other paradigms that have been mentioned such as distributed control -- distributed quantum computing anyone?

Where quantum and classical control seem to diverge in the system and control characteristics and accordingly the tools at our disposal. A lot of classical control (though not all) focuses on linear systems, for multiple reasons, I think. Clearly, linear systems are much easier to analyze and there are far more analytical tools available for linear systems, so linear models are nice, but the most important thing seems to be that they are also well suited to classical applications. As Jun said today, most non-linear classical control systems can be linearized quite successfully, and some people would say that good plant designs should have a (mostly) linear response and that non-linearity is undesirable (though one might argue about desirability of course). The problem with quantum systems is that the control usually enters into the Hamiltonian which acts on the state, and thus in the best of cases we end up with a bilinear control model. The bad news about this is that bilinear systems are much more complex from a control design point of view, but the good news is that the complexity also creates opportunities. A single control can be enough to generate very complex dynamics because the dynamical Lie algebra generated by the system (drift) and control Hamiltonian is usually very large. There are exceptions of quantum systems that can be modelled as linear systems such as the harmonic oscillators but I would argue that these are systems are not typical for quantum control and somehow not very powerful. The Hilbert space of a Harmonic oscillator may be infinite dimensional but the dynamical Lie algebra is equivalent to su(2), i.e., no bigger than that of a qubit and all we can really control is basically position and momentum. Of course, there is a difference here in that the quantum mechanical position and momentum operators do not commute, so unlike the classical case, we cannot hope to simultaneously control position and momentum because of uncertainty relations. It is a valid point to say that non-linearity is not specific to quantum systems as there are obviously classical bilinear control systems. Robotics comes to mind for example. In this sense non-commutativity does seem to be a more fundamental characteristic of quantum systems but the non-linearity strikes me as more relevant from an open-loop control design point of view as it makes the control design problem non-trivial. This effect is compounded by the fact that most quantum systems have non-trivial drift terms that are beyond our control and can't be switched off. This would seem to distinguish them from the typical situation in robotics(?).

Things do become a lot more interesting when measurements enter the picture, as here quantum systems clearly differ from classical systems because the measurements have a backaction. To my mind this makes the sensors co-actuators. This is annoying in some sense as it really makes things complicated. In the simplest example of direct Hamiltonian feedback this effect can be seen very nicely in the form of the Lindblad master equation -- the feedback Hamiltonian enters the reservoir operators while the measurement enters the Hamiltonian. So the measurement and the feedback both act as actuators changing the dynamics of the system. Now at first one might say, this is really ugly but it opens up new possibilities for control. For instance, let's take a single qubit with some free evolution Hamiltonian (WLOG Z) and some control (say X) subject to dissipation, let's say decay from the excited state to the ground state. You would like to stabilize the system in some kind of superposition state. Open-loop control cannot do much for you here. OK, one can show that the system always has a unique steady state which depends on both the strength of the decay and the open-loop control strength. But except for the case where we switch the control off and the steady state is the ground state, all other states we can hope to stabilize are mixed states forming an ellipsoid inside the Bloch ball. The was observed a long time ago by Howard and Gerard (and possibly by other people before then) [x]. It may not seem obvious but if you write down the Bloch equation (which I'm not going to do here) it is not hard too see. OK, it's something but not very exciting. Now what is interesting to me is that in the indirect feedback setting, you really don't seem to be able to do a great deal better. If you take H = H_0 + f(t) H_1 but now allow f(t) to be determined by the results of measuring an operator M, it has been shown that one can stabilize both of the eigenstates of H_0 provided measurement commutes with H_0. But what if you want to stabilize a superposition of the eigenstates of H_0 or measurement operator does not commute with H_0, which strikes me as the more interesting case? I know of at least one negative result in that if H_0 and M do not commute then there is at least one eigenstate of H_0 that cannot be stabilized (though I maybe paraphrasing the result badly). That's not even considering superposition states (of eigenstates of H_0). What is very interesting to me is that when you add a bit of direct Hamiltonian feedback stabilizing superposition states suddenly becomes becomes very feasible, as pointed out a long-time ago by Howard and Gerard [x]. This idea of stabilizing states using direct feedback can be generalized to much more complicated systems and allows us (in principle) to stabilize most states, as discussed in [x] for example. To me this is an interesting example of a feedback control paradigm that does not really have a classical analogue. Sure, you can have proportional direct feedback for classical systems but you don't have the backaction effect. Another thing that makes approach interesting to me is that it has a certain build-in robustness and simplicity. Since the design is such that the target state is asymptotically stable, it doesn't really matter what your initial state is (very useful for state preparation starting from an unknown initial state), and your feedback is simple direct feedback, you don't need any real-time state-estimation or other algorithm to determine the control to apply conditioned on the measurement record, ideally instantaneously or at least really fast. In fact you don't really need the measurement record at all if you can design the master equation so that the target state is the only steady state. Maybe this still doesn't sound exciting but it could be useful for all sorts of things if the states you can stabilize are interesting enough (say cluster states for example). This is closely related to reservoir engineering in that the feedback in this case changes not only the Hamiltonian but the reservoir operators, which is why I tend to include direct Hamiltonian feedback in the reservoir engineering category. Of course one could try to combine this is open-loop or indirect feedback control and ask how much better one might do. Say if one had some estimate of the initial state, one could design a fast open-loop control scheme to transform it into the target state but use the direct Hamiltonian feedback (and backaction!) to make it robust and stable and maybe faster.

I could go on about control paradigms forever here but let's move on. Another important question one might ask is why bother with this semi-classical approach of trying to control a quantum system with what is effectively a classical control. I'd say that the main reason for considering this case is that this seems to be the situation ultimately in a lot of applications where one usually has some classical input and wants classical output as Gerard nicely illustrated in his blackboard talk. So the interface of classical and quantum systems seems like the main challenge. This not withstanding, coherent control is an exciting possibility though I'm not sure it can really avoid the issue -- at some stage there will be some classical input to a quantum system. Some classical information to control the state of the control say. Can coherent control really complete side-step this issue?

RE: Embedding a non-Markov system into a Markov system (Jan. 9, 2013)

I think I may have found the paper on an embedding of system with non-Markovian coupling into a Markovian system by Imamoglu that was mentioned in today's discussion (information courtesy of Andrew Doherty). It was published PRA (1994) and you can retrieve it at the following link:

Quantum simulation is an important theme for our program. However, there continues to be a lack of consensus about what a quantum simulation is and, given a definition, what is necessary to achieve that goal. I am particularly interested in addressing the following questions:

- Is there the level of noise and imperfection, higher than that required for universal fault-tolerant quantum computing, such a quantum simulator we can teach us something that we could not simulate classically? If so, what is that level of noise? - Is the complexity class of problems we can solve with a quantum simulator different from that of a universal quantum computer? - Can analog quantum simulators achieve something different than a digital simulator?

Attached in a small opinion article I wrote for the newsletter of the APS Topical Group on Quantum Information back in 2010.

I'm keen to organize a discussion group around this subject. Please see me if interested. Jan. 14, 2013, GĂ¶ran Johansson

In circuit QED, the technology right now matured to the point where:
1) Coherence times are approaching 100 microseconds
2) Near quantum limited homodyne detection can be performed using parametric amplifiers
3) The long coherence times thus allows for measurements, some signal processing and feedback within the coherence time.
This enables closed loop measurement base quantum feedback. I find this very exciting.

However, last week I learnt that anything that can be done by measurement based closed loop feedback can be done in open loop coherentized control, so the measurements are not needed.

Did I misunderstand something?

Jan. 17, 2013, Josh Combes

Doing a coherentized version of a feedback control protocol may be appropriate in some physical scenarios and has some advantages (over measurement based feedback) such as bandwidth. But in a particular experimental senario it might be better to do measurement based control. This will depend on the relative quality / efficiency / losses of the devices one uses in a coherent vs a measurement based feedback protocol. February 6, 2013: Carlton Caves

Here are the notes I used for my discussion of quantum control and quantum circuits on Monday, February 4. They're semi-interpretable by someone who was at the discussion, perhaps less so for someone who wasn't.

See also the page on closed/open loop control here: Closed and open loop quantum feedback## RE: Yet another suggestion for Discussion: analytical techniques for quantum control [Sophie]

Motivated by Tuesday's talks I wanted to suggest another discussion on analytical techniques. Do we need more analytical techniques? Is it useful to develop analytical approaches even if we can only deal with a few parameters? E.g. explicit Hamilton-Jacobi equations or Lie group methods? What other techniques? What would be the most important questions to answer?

RE: Suggestion for further Discussiononstate-of-the-art and challenges of quantum control [Sophie]Motivated by Birgitta's very nice talk yesterday I wanted to suggest that maybe we could have further discussions to establish what's been demonstrated experimentally, what's been done theoretically, and what we think may be possible in the near future or shown to be/believed to be impossible. Many talks have touched on these issues but it would be nice to pull it all together and write a little summary for the wiki. Topics might include

- Quantum Information:

- Quantum Metrology

- Magnetic Resonance Spectroscopy & Imaging

Molecular MRI?

- Photo-chemistry

- Quantum biology: light harvesting and beyond

- Theory / computation

RE: Control paradigms for quantum systems etc. [Sophie]I promised to write something for this wiki, so here we go. Let me start by saying that Jun's talk today gave a nice overview of classical control and system identification paradigms, I think.

What distinguishes control of quantum systems from its classical counterpart (for me) is not open vs closed-loop, although people may have got the impression from my talk. Both paradigms are clearly useful for classical and quantum systems. Quantum error correction clearly meets the definition of feedback control posted here (by Matt, I think) and it is very useful to deal with noise, disturbances and uncertainty, just as it is in the classical case. Some people have argued that quantum error correction is really feedforward control but Carlton's talk and diagrams have convinced me that this is definitively not the case for standard quantum error correction. We are not measuring the disturbance and correcting the open-loop control by anticipating the effect it will have on the system, but rather we really do perform measurements on the system to infer the effect of disturbances after they have acted on the system, and to mitigate it and restore the desired behaviour. At the same time open-loop control is very useful to construct controls that solve non-trivial problems from implementing quantum gates to putting systems into quantum states we are interested in (as opposed to what nature provides). But as Jun mentioned today, open-loop control is also used for classical systems, just so happens that it is generally a much simpler problem and therefore not given as much attention. This being said, the distinction between open and closed-loop control is really too simplistic. Some paradigms such as learning control do not seem to fit neatly into either category, at least if we go by the definitions posted below. Going by the definitions below, I would say that learning control fits more into the feedback paradigm (we certainly require feedback to learn) but it is quite a different setting from feedback control acting on the same system. But again, those paradigms have classical analogues too. The same can be said for other paradigms that have been mentioned such as distributed control -- distributed quantum computing anyone?

Where quantum and classical control seem to diverge in the system and control characteristics and accordingly the tools at our disposal. A lot of classical control (though not all) focuses on linear systems, for multiple reasons, I think. Clearly, linear systems are much easier to analyze and there are far more analytical tools available for linear systems, so linear models are nice, but the most important thing seems to be that they are also well suited to classical applications. As Jun said today, most non-linear classical control systems can be linearized quite successfully, and some people would say that good plant designs should have a (mostly) linear response and that non-linearity is undesirable (though one might argue about desirability of course). The problem with quantum systems is that the control usually enters into the Hamiltonian which acts on the state, and thus in the best of cases we end up with a bilinear control model. The bad news about this is that bilinear systems are much more complex from a control design point of view, but the good news is that the complexity also creates opportunities. A single control can be enough to generate very complex dynamics because the dynamical Lie algebra generated by the system (drift) and control Hamiltonian is usually very large. There are exceptions of quantum systems that can be modelled as linear systems such as the harmonic oscillators but I would argue that these are systems are not typical for quantum control and somehow not very powerful. The Hilbert space of a Harmonic oscillator may be infinite dimensional but the dynamical Lie algebra is equivalent to su(2), i.e., no bigger than that of a qubit and all we can really control is basically position and momentum. Of course, there is a difference here in that the quantum mechanical position and momentum operators do not commute, so unlike the classical case, we cannot hope to simultaneously control position and momentum because of uncertainty relations. It is a valid point to say that non-linearity is not specific to quantum systems as there are obviously classical bilinear control systems. Robotics comes to mind for example. In this sense non-commutativity does seem to be a more fundamental characteristic of quantum systems but the non-linearity strikes me as more relevant from an open-loop control design point of view as it makes the control design problem non-trivial. This effect is compounded by the fact that most quantum systems have non-trivial drift terms that are beyond our control and can't be switched off. This would seem to distinguish them from the typical situation in robotics(?).

Things do become a lot more interesting when measurements enter the picture, as here quantum systems clearly differ from classical systems because the measurements have a backaction. To my mind this makes the sensors co-actuators. This is annoying in some sense as it really makes things complicated. In the simplest example of direct Hamiltonian feedback this effect can be seen very nicely in the form of the Lindblad master equation -- the feedback Hamiltonian enters the reservoir operators while the measurement enters the Hamiltonian. So the measurement and the feedback both act as actuators changing the dynamics of the system. Now at first one might say, this is really ugly but it opens up new possibilities for control. For instance, let's take a single qubit with some free evolution Hamiltonian (WLOG Z) and some control (say X) subject to dissipation, let's say decay from the excited state to the ground state. You would like to stabilize the system in some kind of superposition state. Open-loop control cannot do much for you here. OK, one can show that the system always has a unique steady state which depends on both the strength of the decay and the open-loop control strength. But except for the case where we switch the control off and the steady state is the ground state, all other states we can hope to stabilize are mixed states forming an ellipsoid inside the Bloch ball. The was observed a long time ago by Howard and Gerard (and possibly by other people before then) [x]. It may not seem obvious but if you write down the Bloch equation (which I'm not going to do here) it is not hard too see. OK, it's something but not very exciting. Now what is interesting to me is that in the indirect feedback setting, you really don't seem to be able to do a great deal better. If you take H = H_0 + f(t) H_1 but now allow f(t) to be determined by the results of measuring an operator M, it has been shown that one can stabilize both of the eigenstates of H_0 provided measurement commutes with H_0. But what if you want to stabilize a superposition of the eigenstates of H_0 or measurement operator does not commute with H_0, which strikes me as the more interesting case? I know of at least one negative result in that if H_0 and M do not commute then there is at least one eigenstate of H_0 that cannot be stabilized (though I maybe paraphrasing the result badly). That's not even considering superposition states (of eigenstates of H_0). What is very interesting to me is that when you add a bit of direct Hamiltonian feedback stabilizing superposition states suddenly becomes becomes very feasible, as pointed out a long-time ago by Howard and Gerard [x]. This idea of stabilizing states using direct feedback can be generalized to much more complicated systems and allows us (in principle) to stabilize most states, as discussed in [x] for example. To me this is an interesting example of a feedback control paradigm that does not really have a classical analogue. Sure, you can have proportional direct feedback for classical systems but you don't have the backaction effect. Another thing that makes approach interesting to me is that it has a certain build-in robustness and simplicity. Since the design is such that the target state is asymptotically stable, it doesn't really matter what your initial state is (very useful for state preparation starting from an unknown initial state), and your feedback is simple direct feedback, you don't need any real-time state-estimation or other algorithm to determine the control to apply conditioned on the measurement record, ideally instantaneously or at least really fast. In fact you don't really need the measurement record at all if you can design the master equation so that the target state is the only steady state. Maybe this still doesn't sound exciting but it could be useful for all sorts of things if the states you can stabilize are interesting enough (say cluster states for example). This is closely related to reservoir engineering in that the feedback in this case changes not only the Hamiltonian but the reservoir operators, which is why I tend to include direct Hamiltonian feedback in the reservoir engineering category. Of course one could try to combine this is open-loop or indirect feedback control and ask how much better one might do. Say if one had some estimate of the initial state, one could design a fast open-loop control scheme to transform it into the target state but use the direct Hamiltonian feedback (and backaction!) to make it robust and stable and maybe faster.

I could go on about control paradigms forever here but let's move on. Another important question one might ask is why bother with this semi-classical approach of trying to control a quantum system with what is effectively a classical control. I'd say that the main reason for considering this case is that this seems to be the situation ultimately in a lot of applications where one usually has some classical input and wants classical output as Gerard nicely illustrated in his blackboard talk. So the interface of classical and quantum systems seems like the main challenge. This not withstanding, coherent control is an exciting possibility though I'm not sure it can really avoid the issue -- at some stage there will be some classical input to a quantum system. Some classical information to control the state of the control say. Can coherent control really complete side-step this issue?

RE: Embedding a non-Markov system into a Markov system (Jan. 9, 2013)I think I may have found the paper on an embedding of system with non-Markovian coupling into a Markovian system by Imamoglu that was mentioned in today's discussion (information courtesy of Andrew Doherty). It was published PRA (1994) and you can retrieve it at the following link:

http://pra.aps.org/abstract/PRA/v50/i5/p3650_1

Would love to get your comments on this.

Thanks,

Hendra NurdinJan. 13, 2013, Ivan DeutschQuantum simulation is an important theme for our program. However, there continues to be a lack of consensus about what a quantum simulation is and, given a definition, what is necessary to achieve that goal. I am particularly interested in addressing the following questions:

- Is there the level of noise and imperfection, higher than that required for universal fault-tolerant quantum computing, such a quantum simulator we can teach us something that we could not simulate classically? If so, what is that level of noise?

- Is the complexity class of problems we can solve with a quantum simulator different from that of a universal quantum computer?

- Can analog quantum simulators achieve something different than a digital simulator?

Attached in a small opinion article I wrote for the newsletter of the APS Topical Group on Quantum Information back in 2010.

I have also co-authored a recent review:

I'm keen to organize a discussion group around this subject. Please see me if interested.

Jan. 14, 2013, GĂ¶ran JohanssonIn circuit QED, the technology right now matured to the point where:

1) Coherence times are approaching 100 microseconds

2) Near quantum limited homodyne detection can be performed using parametric amplifiers

3) The long coherence times thus allows for measurements, some signal processing and feedback within the coherence time.

This enables closed loop measurement base quantum feedback. I find this very exciting.

However, last week I learnt that anything that can be done by measurement based closed loop feedback can be done in open loop coherentized control, so the measurements are not needed.

Did I misunderstand something?

Jan. 17, 2013, Josh CombesDoing a coherentized version of a feedback control protocol may be appropriate in some physical scenarios and has some advantages (over measurement based feedback) such as bandwidth. But in a particular experimental senario it might be better to do measurement based control. This will depend on the relative quality / efficiency / losses of the devices one uses in a coherent vs a measurement based feedback protocol.

February 6, 2013: Carlton CavesHere are the notes I used for my discussion of quantum control and quantum circuits on Monday, February 4. They're semi-interpretable by someone who was at the discussion, perhaps less so for someone who wasn't.