Quantum multiobjective control


In most practical applications of open loop engineering control, it is necessary to simultaneously satisfy multiple design criteria. For example, it may be necessary to drive multiple physical quantities to maximum or minimum values under limited resource constraints. Only in rare cases is the control of a single physical quantity sufficient to meet the demands of a technological application. Until recently, research in quantum optimal control, being in its nascent stages of development, has focused on single objective control problems, in some cases the constrained optimization of single objectives. 


Specifically, quantum optimal control problems studied to date fall into two major classes: 1) control of the expectation values of single quantum observables or more commonly, pure states; 2) control of quantum dynamical transformations. The former have been implemented in both simulations and experiments, whereas the latter has been tackled predominantly through numerical demonstrations, due to the expense of quantum process tomography. An important third class of QC problems, which lies between the latter two problem types, is the control of arbitrary numbers of quantum observables. To date, few effective techniques - either experimental or numerical - have been reported for multiobservable quantum control. We aim to fill this gap.



Quantum multiobservable control

R. Chakrabarti, R. Wu, and H. Rabitz, Phys. Rev. A 77:063425 (2008)
Eprint arXiv:0805.1556 [quant-ph] (full paper)
Reprinted in: Virt. J. Quant. Inf. 8: 7, Virt. J. Ultrafast Sci. 7: 7, July 2008.
                       

We present deterministic algorithms for the simultaneous control of an arbitrary number of quantum observables. Unlike optimal control approaches based on cost function optimization, quantum multiobservable tracking control (MOTC) is capable of tracking predetermined homotopic trajectories to target expectation values in the space of multiobservables. The convergence of these algorithms is facilitated by the favorable critical topology of quantum control landscapes. Fundamental properties of quantum multiobservable control landscapes that underlie the efficiency of MOTC, including the multiobservable controllability Gramian, are introduced. The effects of multiple control objectives on the structure and complexity of optimal fields are examined.
With minor modifications, the techniques described herein can be applied to general
quantum multiobjective control problems.

Two MOTC applications of particular importance are A) the simultaneous maximization of the expectation values of a set of observables and B) the preparation of arbitrary mixed states.



Quantum Pareto optimal control

R. Chakrabarti, R. Wu, and H. Rabitz, Phys. Rev. A 78: 033414 (2008).
Eprint arXiv:0805.4026[quant-ph]
Reprinted in: Virt. J. Quant. Inf. 8: 9, September 2008.


We extend the theory of MOTC to complex multiobjective optimization problems such as multiobservable maximization, and propose experimental techniques for the implementation of MOTC in such complex multiobservable control scenarios.

We describe algorithms, and experimental strategies, for the Pareto optimal control problem of simultaneously driving an arbitrary number of quantum observable expectation values to their respective extrema. Conventional quantum optimal control strategies are less effective at sampling points on the Pareto frontier of multiobservable control landscapes than they are at locating optimal solutions to single observable control problems. The present algorithms facilitate multiobservable optimization by following direct paths to the Pareto front, and are capable of continuously tracing the front once it is found to explore families of viable solutions. The numerical and experimental methodologies introduced are also applicable to other problems that require the simultaneous control of large numbers of observables, such as quantum optimal mixed state preparation.


Next steps

A class of stochastic search algorithms called multiobjective evolutionary algorithms (MOEA) have been successfully applied to multiobjective problems within conventional engineering optimization and control. My next step in this area is to apply these algorithms to quantum multiobservable control problems, both in an open-loop experimental incarnation and numerically.

Target applications of both classes of algorithms include:
  • Control of product selectivity in coherently driven chemical reactions 
  • Optimal quantum dynamical discrimination of similar molecules,
within the domain of chemical physics, and
  • Quantum optimal mixed state preparation
for quantum computing applications. The latter requires methodologies for mixed state estimation, discussed here.

Finally, the MOTC and MOEA algorithms should be tested numerically on open quantum systems in order to determine how quantum decoherence affects multiobservable control fidelity.


Raj Chakrabarti
6/08