Quantum Control Theory

The environmental thrust of this blog will rise to the surface again soon, I promise. I’m just going to a lot of talks on quantum technology, condensed matter physics, and the like. Ultimately the two threads should merge in a larger discourse that ranges from highly theoretical to highly practical. But right now you’re probably just confused about the purpose of this blog — it’s smeared out all across the intellectual landscape.

Anyway, to add to the confusion: I just got a nice email from Giampiero Campa, who in week294 had pointed me to the fascinating papers on control theory by Jan Willems. Control theory is the art of getting open systems — systems that interact with their environment — to behave in ways you want.

Since complex systems like ecosystems or the entire Earth are best understood as made of many interacting open systems, and/or being open systems themselves, I think ideas from control theory could become very important in understanding the Earth and how our actions affect it. But I’m also fascinated by control theory because of how it combines standard ideas in physics with new ideas that are best expressed using category theory — a branch of math I happen to know and like. (See week296 and subsequent issues for more on this.) And quantum control theory — the art of getting quantum systems to do what you want — is the sort of thing people here at the CQT may find interesting.

In short, control theory seems like a promising meeting-place for some of my disparate interests. Not necessarily the most important thing for ‘saving the planet’, by any means! But the kind of thing I can’t resist thinking about.

In his email, Campa pointed me to two new papers on this subject:

• Anthony M. Bloch, Roger W. Brockett, and Chitra Rangan, Finite controllability of infinite-dimensional quantum systems, IEEE Transactions on Automatic Control 55 (August 2010), 1797-1805.

• Matthew James and John E. Gough, Quantum dissipative systems and feedback control design by interconnection, IEEE Transactions on Automatic Control 55 (August 2010), 1806-1821.

The second one is related to the ideas of Jan Willems:

Abstract: The purpose of this paper is to extend J.C. Willems’ theory of dissipative systems to open quantum systems described by quantum noise models. This theory, which combines ideas from quantum physics and control theory, provides useful methods for analysis and design of dissipative quantum systems. We describe the interaction of the plant and a class of external systems, called exosystems, in terms of feedback networks of interconnected open quantum systems. Our results include an infinitesimal characterization of the dissipation property, which generalizes the well-known Positive Real and Bounded Real Lemmas, and is used to study some properties of quantum dissipative systems. We also show how to formulate control design problems using network models for open quantum systems, which implements Willems’ “control by interconnection” for open quantum systems. This control design formulation includes, for example, standard problems of stabilization, regulation, and robust control.

I don’t have anything intelligent to say about these papers yet. Does anyone out know if ideas from quantum control theory have been used to tackle the problems that decoherence causes in quantum computation? The second article makes me wonder about this:

In the physics literature, methods have been developed to model energy loss and decoherence (loss of quantum coherence) arising from the interaction of a system with an environment. These models may be expressed using tools which include completely positive maps, Lindblad generators, and master equations. In the 1980s it became apparent that a wide range of open quantum systems, such as those found in quantum optics, could be described within a new unitary framework of quantum stochastic differential equations, where quantum noise is used to represent the influence of large heat baths and boson fields (which includes optical and phonon fields). Completely positive maps, Lindblad generators, and master equations are obtained by taking expectations.

Quantum noise models cover a wide range of situations involving light and matter. In this paper, we use quantum noise models for boson fields, as occur in quantum optics, mesoscopic superconducting circuits, and nanomechanical systems, although many of the ideas could be extended to other contexts. Quantum noise models can be used to describe an optical cavity, which consists of a pair of mirrors (one of which is partially transmitting) supporting a trapped mode of light. This cavity mode may interact with a free external optical field through the partially transmitting mirror. The external field consists of two components: the input field, which is the field before it has interacted with the cavity mode, and the output field, being the field after interaction. The output field may carry away energy, and in this way the cavity system dissipates energy. This quantum system is in some ways analogous to the RLC circuit discussed above, which stores electromagnetic energy in the inductor and capacitor, but loses energy as heat through the resistor. The cavity also stores electromagnetic energy, quantized as photons, and these may be lost to the external field…

13 Responses to Quantum Control Theory

  1. “Does anyone out know if ideas from quantum control theory have been used to tackle the problems that decoherence causes in quantum computation?”

    Tons of work has been done on this, in particular using control theory to help with quantum error correction. A good starting point is this paper by Ahn, Doherty and Landahl, Continuous quantum error correction via quantum feedback control.

    There is also this cool paper by Verstraete, Wolf and Cirac, Quantum computation, quantum state engineering, and quantum phase transitions driven by dissipation. From the abstract:

    We investigate the computational power of creating steady-states of quantum dissipative systems whose evolution is governed by time-independent and local couplings to a memoryless environment. We show that such a model allows for efficient universal quantum computation with the result of the computation encoded in the steady state. Due to the purely dissipative nature of the process, this way of doing quantum computation exhibits some inherent robustness and defies some of the DiVincenzo criteria for quantum computation…

    Basically, they pick a Lindbladian and the resulting master equation has a fixed point which is the output of a quantum computation. Since the computation is driven by coupling to a bath, it seems like it should be robust to decoherence. It is also independent of the initialization of the computer, which is neat.

    • John Baez says:

      Steve wrote:

      Since the computation is driven by coupling to a bath, it seems like it should be robust to decoherence.

      Zounds! I get the logic of this sentence, but it sounds too good to be true: quantum computation driven by dissipation?

      If it’s not too good to be true, this could be truly good. What do experts in this field think?

      • JB: quantum computation driven by dissipation?

        Of course. Without dissipation no measurement
        (permanent enough recording to be registered by
        a macroscopic device), and hence no quantum computation.

        The problem is that there are good and bad sides to dissipation, and it is extremely difficult to get only the good sides….

  2. Rod Carvalho says:

    Prof. Baez,

    An interesting paper for the neophyte is Matthew James’ Control Theory: from Classical to Quantum Optimal, Stochastic, and Robust Control [pdf], which seems to have been written for the Quantum Control Summer School at Caltech in August 2005. Hideo Mabuchi‘s group did some impressive work on quantum control at Caltech. I once visited his lab and was mesmerized.

    • John Baez says:

      Hey, that paper by Matthew James is really good — a great way for me to start catching up on a huge body of work! Thanks, Rod!

      And when I saw that Matthew James was at Australian National University, I suddenly remembered some emails from his colleague Hendra Nurdin, who pointed me to the webpage of the quantum control group at ANU, and recommended these papers:

      • J. Gough and M. R. James, The series product and its application to quantum feedforward and feedback networks, IEEE Trans. Automat. Contr. 54 (2009), 2530-2544.

      • J. Gough and M. R. James, Quantum feedback networks: Hamiltonian formulation, Comm. Math. Phys. 287 (2009), 1109-1132.

      I had set these emails aside at the time, since I was busy finishing other projects. But now this is something I’d like to look into.

      Thanks for all those other links to papers, too!

  3. Rod Carvalho says:

    Also interesting is Navin Khaneja’s PhD thesis: Geometric Control in Classical and Quantum Systems [pdf].

  4. John Sidles says:

    Classical Hamiltonian mechanics and fluctuation-dissipation theorems, which are both natural and interesting on Euclidean state-spaces, are similarly natural and (arguably) even more geometrically interesting on symplectic state-spaces.

    Similarly, quantum Hamiltonian mechanics and Lindbladian stochastic processes, which are both natural and interesting on Hilbert state-spaces, are similarly natural and (arguably) even more geometrically interesting on tensor network manifolds (that is, multilinear Kählerian state-spaces).

    • John Baez says:

      Wow — thanks for pointing that out! Just recently I’ve been noticing how ‘tensor networks’ are catching on in quantum information theory, and how a tensor network is a special case of the general concept of ‘spin network‘: the special case where the gauge group is the trivial group. I’m talking to people here at the CQT about this… but I hadn’t known of the connection between tensor networks and dissipation and control theory, which are some other subjects I’m hoping to work on here!

  5. Josh Combes says:

    RE: “Does anyone out know if ideas from quantum control theory have been used to tackle the problems that decoherence causes in quantum computation?”

    You might be interested in the following papers related to feedback stabilization of quantum states. It is complementary to the continuous in time error correction literature that Steve pointed out.

    • Bayesian feedback versus Markovian feedback in a two-level atom. Wiseman, Mancini and Wang, Phys. Rev. A 66, 013807 (2002)

    • Global stability criterion for a quantum feedback control process on a single qubit and exponential stability in case of perfect detection efficiency. de Vries, Phys. Rev. A 75, 032101 (2007)

    • Feedback control of nonlinear quantum systems: a rule of thumb. Jacobs and Lund, Phys. Rev. Lett. 99, 020501 (2007)

    • Stabilizing feedback controls for quantum systems. Mirrahimi and van Handel, SIAM J. Control Optim. 46, Issue 2, pp. 445-467 (2007) [the published version of the above arxiv paper]

  6. polariton says:

    It’s been ages since this post has been published but I thought I’d post this for anyone who might be interested. It’s a relatively recent (October 2009) review paper on quantum control which may be found at http://arxiv.org/abs/0910.2350 .

  7. spotmanoz says:

    Hey guys, I saw your thread on quantum control theory. That is my stock-in-trade. An ebook on the topic is located here at http://quantumcontrol.wikidot.com/start if you have any further questions just let me know over social media or email.

    Most of the current work is being done on autonomous systems, but the real power is in time optimal quantum control. If it isn’t time optimal, it’s slow to me!!!

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