Sophie G. Schirmer

Analysis of Lyapunov Method for Control of Quantum States

The natural trajectory tracking problem is studied for generic quantum states represented by density operators. A control design based on the Hilbert-Schmidt distance as a Lyapunov function is considered. The control dynamics is redefined on an extended space where the LaSalle invariance principle can be correctly applied even for non-stationary target states. LaSalles invariance principle is used to derive a general characterization of the invariant set, which is shown to always contain the critical points of the Lyapunov function. Critical point analysis of the latter is used to show that, for generic states, it is a Morse function with n! isolated critical points, including one global minimum, one global maximum and n!-2 saddles. It is also shown, however, that the actual dynamics of the system is not a gradient flow, and therefore a full eigenvalue analysis of the linearized dynamics about the critical points of the dynamical system is necessary to ascertain stability of the critical points. This analysis shows that a generic target state is locally asymptotically stable if the linearized system is controllable and the invariant set is regular, and in fact convergence to the target state (trajectory) in this case is almost global in that the stable manifolds of all other critical points form a subset of measure zero of the state space. On the other hand, if either of these sufficient conditions is not satisfied, the target state ceases to be asymptotically stable, a center manifold emerges around the target state, and the control design ceases to be effective.

Second order gradient ascent pulse engineering

We report some improvements to the gradient ascent pulse engineering (GRAPE) algorithm for optimal control of spin ensembles and other quantum systems. These include more accurate gradients, convergence acceleration using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton algorithm as well as faster control derivative calculation algorithms. In all test systems, the wall clock time and the convergence rates show a considerable improvement over the approximate gradient ascent.

Training Schrödinger’s cat: quantum optimal control

It is control that turns scientific knowledge into useful technology: in physics and engineering it provides a systematic way for driving a dynamical system from a given initial state into a desired target state with minimized expenditure of energy and resources. As one of the cornerstones for enabling quantum technologies, optimal quantum control keeps evolving and expanding into areas as diverse as quantum-enhanced sensing, manipulation of single spins, photons, or atoms, optical spectroscopy, photochemistry, magnetic resonance (spectroscopy as well as medical imaging), quantum information processing and quantum simulation. In this communication, state-of-the-art quantum control techniques are reviewed and put into perspective by a consortium of experts in optimal control theory and applications to spectroscopy, imaging, as well as quantum dynamics of closed and open systems. We address key challenges and sketch a roadmap for future developments.

Orbits of quantum states and geometry of Bloch vectors for N-level systems

Physical constraints such as positivity endow the set of quantum states with a rich geometry if the system dimension is greater than 2. To shed some light on the complicated structure of the set of quantum states, we consider a stratification with strata given by unitary orbit manifolds, which can be identified with flag manifolds. The results are applied to study the geometry of the coherence vector for n-level quantum systems. It is shown that the unitary orbits can be naturally identified with spheres in only for n = 2. In higher dimensions the coherence vector only defines a non-surjective embedding into a closed ball. A detailed analysis of the three-level case is presented. Finally, a refined stratification in terms of symplectic orbits is considered.

Entanglement generation between distant atoms by Lyapunov control

We show how to apply Lyapunov control design to the problem of entanglement creation between two atoms in distant cavities connected by optical fibers. The Lyapunov control design is optimal in the sense that the distance from the target state decreases monotonically and exponentially, and the concurrence increases accordingly. This method is far more robust than simple geometric schemes.

Analysis of Effectiveness of Lyapunov Control for Non-Generic Quantum States

A Lyapunov-based control design for natural trajectory- tracking problems is analyzed for quantum states where the analysis in the generic case is not applicable. Using dynamical systems tools we show almost global asymptotic stability for stationary target states subject to certain conditions on the Hamiltonians, and discuss effectiveness of the design when these conditions are not satisfied. For pseudo-pure target states the effectiveness of the design is studied further for both stationary and non-stationary states using alternative tools.

Quantum Control Landscapes: A Closer Look

The control landscape for various canonical quantum control problems is considered. For the class of pure-state transfer problems, analysis of the fidelity as a functional over the unitary group reveals no suboptimal attractive critical points (traps). For the actual optimization problem over controls in

Robust entanglement in antiferromagnetic Heisenberg chains by single-spin optimal control

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Stabilizing Quantum States by Constructive Design of Open Quantum Dynamics

, however, there are critical points for which the fidelity can assume any value in (0,1), critical points for which the second order analysis is inconclusive, and traps. For the class of unitary operator optimization problems analysis of the fidelity over the unitary group shows that while there are no traps over U(N), traps already emerge when the domain is restricted to the special unitary group. The traps on the group can be eliminated by modifying the performance index, corresponding to optimization over the projective unitary group. However, again, the set of critical points for the actual optimization problem for controls in

A CLOSER LOOK AT QUANTUM CONTROL LANDSCAPES AND THEIR IMPLICATION FOR CONTROL OPTIMIZATION

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