Robust Control Performance for Open Quantum Systems

Abstract

Robust performance of control schemes for open quantum systems is investigated under classical uncertainties in the generators of the dynamics and nonclassical uncertainties due to decoherence and initial state preparation errors. A formalism is developed to measure performance based on the transmission of a dynamic perturbation or initial state preparation error to the quantum state error. This makes it possible to apply tools from classical robust control such as structured singular value analysis. A difficulty arising from the singularity of the closedloop Bloch equations for the quantum state is overcome by introducing the #-inversion lemma, a specialized version of the matrix inversion lemma. Under some conditions, this guarantees continuity of the structured singular value at s = 0. Additional difficulties occur when symmetry gives rise to multiple open-loop poles, which under symmetrybreaking unfold into single eigenvalues. The concepts are applied to systems subject to pure decoherence and a general dissipative system example of two qubits in a leaky cavity under laser driving fields and spontaneous emission. A nonclassical performance index, steady-state entanglement quantified by the concurrence, a nonlinear function of the system state, is introduced. Simulations confirm a conflict between entanglement, its log-sensitivity and stability margin under decoherence.

Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe

“…Robustness against specified frequency bands of the noise power spectral density can be achieved based by including a filter function which can either be derived from reverse engineering [148] or parametrized and optimized [637]. The design of robust control protocols has been explored for different sources of imperfections [52,367,478,512,551] and the optimal control of an inhomogeneous spin ensemble coupled to a cavity has been studied [33].…”

GRAPE optimization for open quantum systems with time-dependent decoherence rates driven by coherent and incoherent controls

“…Some problems for the two-qubit case were analyzed numerically in [86]. An important problem is to analyze robustness of controls to various fluctuations and imperfections [87][88][89][90][91][92][93][94][95][96][97][98][99]. In this work, we perform a basic analysis of robustness of obtained controls for the two-level case, although a detailed study deserves a separate work.…”

Robustness of energy landscape control to dephasing

“…To understand this effect of decoherence on the controller design we will focus on the intermediate regime where coherent dynamics play a significant role but are modified by dephasing as a result of weak interactions with the environment. The strongly dissipative regime where asymptotic stability can be recovered (Schirmer and Wang, 2010) and exploited to design backaction-based stabilization schemes (Ticozzi et al, 2010;Motzoi et al, 2016) has been considered in other work (Schirmer et al, 2022).…”