# Quantum computing and quantum control

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In 2009, Bernard Chazelle published a famous algorithms paper, "Natural Algorithms," in which he applied computational complexity techniques to a control theory model of bird flocking. Control theory methods (like Lyapunov functions) had been able to show that the models eventually converged to equilibrium, but could say nothing about the rate of convergence. Chazelle obtained tight bounds on the rate of convergence.

Inspired by this, I had the idea of seeing how to apply quantum computing to quantum control, or vice versa. However, I got nowhere with this, because the situations did not seem to be analogous. Further, QComp seemed to have expressive power in finite dimensional Hilbert spaces, while QControl seemed fundamentally infinite-dimensional to me. So I could not see how to make progress.

So my question: does this seem like a reasonable research direction, and I just didn't understand the technicalities well enough? Or are control theory and quantum control fundamentally different fields, which just happen to have similar names in English? Or some third possibility?

EDIT: I have not thought about this for a few years, and I just started Googling. I found this quantum control survey paper, which, at a glance, seems to answer one of my questions: that quantum control and traditional control theory are indeed allied fields and use some similar techniques. Whether there is a research question like Chazelle's analysis of the BOIDS model that would be amenable to quantum computing techniques, I have no idea.

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retagged Mar 7, 2014
of course, note that the "tight bounds" involve towers of 2s :)

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Well, obviously I don't know exactly what you were trying, but it's not an unreasonable direction. There is certainly a bit of interplay between the two areas. Many of the open loop techniques which have become standard practice in spin resonance (for example decoupling pulses such as WAHUHA [Waugh, J.S., Huber, L.M., Haeberlen, U. (1968) Phys. Rev. Lett., 20, 180.] etc.) are based on exactly the same Suzuki-Trotter tricks that are now used in quantum simulation algorithms.

Additionally, there have been some nice results from Steffen Glaser and others on optimal control for the basic building blocks of quantum computation, including some pretty high level operations like generating cluster states (see arXiv:0903.4066) and state transfer (see arXiv:0705.0378).

There is also a huge literature on trying to make stuff into quantum computers even when you don't have all the control knobs you might wish for (see for example Seth Lloyd's papers on Universal quantum interfaces, all of the stuff on global control, and recent papers by Daniel Burgarth and Alistair Kay on Lie algebraic control techniques).

Lastly, the two areas are very closely linked via the adiabatic model, where the efficiency is directly linked to how quickly you can adiabticly transition between two Hamiltonians.

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answered Sep 15, 2011 by (3,555 points)
Hi Joe, I was trying to come up with a cool-and-original question, and then to answer it. I did neither! Perhaps now I will at least be able to accomplish the first half, thank you.

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@Aaron: No problem. Hope there is something useful in this for you. The adiabatic stuff is hot at the moment, but tricky to prove anything useful.

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Also, quantum feedback control is perhaps an interesting direction which highlights the hidden distinction between classical and quantum control: the "back-action" of the system on the controller. Ideas from complexity theory are also popping up in this direction [recent papers by H. Wiseman, K. Jacobs, and J. Combes on the arXiv].

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@Kaveh: I didn't understand your first comment at all :( Do you mean Turing universality, or some th-ph notion? Also, do you mean make this a community wiki along the lines of "Big list: What are papers that apply QCmp to QCntrl or vice versa?"

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I mean the following [paper](http://www.jstor.org/stable/52686) by Deutsch et al al and here is the [landscape paper](http://arxiv.org/abs/0710.0684) by Chakrabarti and Rabbitz on the "landscape". Yes, that was what I meant by wiki. Otherwise the paper you mentioned covers many aspects.

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@Kaveh_kh: Perhaps you should post an answer then. For me, I see the obvious connection between circuit complexity and optimal control, but thought this was essentially to obvious to mention explicitly, and I saw the adiabatic algorithm as the deepest connection, but perhaps there is a better answer. I make no claim that this is definitive.

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:) I asked for some advice before posting an answer but was warned against the complications. The problem is quantum control is not well defined.

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I don't think it's an unreasonable direction at all - and something that is, I believe, being done by a few people. I actually came across this subject on John Baez's blog http://johncarlosbaez.wordpress.com/2010/08/16/quantum-control-theory/ . If you're looking for lit recommendations, I expect you could do worse than look at the post the comments. Also try http://cam.qubit.org/node/224 which a masters level course in quantum control course from Cambridge.

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answered Sep 17, 2011 by (70 points)
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This paper by G. Chiribella goes into a unique situation with quantum computing and quantum control. It defines a swap gate that has an entangled state where elements of a quantum computer are in a superposition of different connected states.

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answered Sep 16, 2011 by (100 points)

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