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New Journal of Physics

Quantum robust control theory for Hamiltonian and control field uncertainty*

Andy Koswara, Vaibhav Bhutoria, Raj Chakrabarti

Published May 24, 2021• 24 citations
Publisher

Abstract

Abstract Quantum robust control—which can employ fast leading order approximations, slower but more accurate asymptotic methods, or a combination thereof for quantification of robustness—enables control of moments of quantum observables and gates in the presence of Hamiltonian uncertainty or field noise. In this paper, we present a generalized quantum robust control theory that extends the previously described theory of quantum robust control in several important ways. We present robust control theory for control of any moment of arbitrary quantum control objectives, introducing moment-generating functions and transfer functions for quantum robust control that generalize the tools of frequency domain response theory to quantum systems, and extend the Pontryagin maximum principle for quantum control to control optimization in the presence of noise in the manipulated amplitudes or phases used to shape the control field. To provide guidelines as to the types of quantum control systems and control objectives for which asymptotic robustness analysis is important for accuracy, we introduce methods for assessing the Lie algebraic depth of quantum control systems, and illustrate through examples drawn from quantum information processing how such accurate methods for quantification of robustness to noise and uncertainty are more important for control strategies that exploit higher order quantum pathways. In addition, we define the relationship between leading order Taylor expansions and asymptotic estimates for quantum control moments in the presence of Hamiltonian uncertainty and field noise, and apply such leading order approximations to significant pathways analysis and dimensionality reduction of asymptotic quantum robust control calculations, describing numerical methods for implementation of these calculations.

DOI
10.1088/1367-2630/ac0479

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Abstract Quantum robust control—which can employ fast leading order approximations, slower but more accurate asymptotic methods, or a combination thereof for quantification of robustness—enables control of moments of quantum observables and gates in the presence of Hamiltonian uncertainty or field noise. In this paper, we present a generalized quantum robust control theory that extends the previously described theory of quantum robust control in several important ways. We present robust control theory for control of any moment of arbitrary quantum control objectives, introducing moment-generating functions and transfer functions for quantum robust control that generalize the tools of frequency domain response theory to quantum systems, and extend the Pontryagin maximum principle for quantum control to control optimization in the presence of noise in the manipulated amplitudes or phases used to shape the control field. To provide guidelines as to the types of quantum control systems and control objectives for which asymptotic robustness analysis is important for accuracy, we introduce methods for assessing the Lie algebraic depth of quantum control systems, and illustrate through examples drawn from quantum information processing how such accurate methods for quantification of robustness to noise and uncertainty are more important for control strategies that exploit higher order quantum pathways. In addition, we define the relationship between leading order Taylor expansions and asymptotic estimates for quantum control moments in the presence of Hamiltonian uncertainty and field noise, and apply such leading order approximations to significant pathways analysis and dimensionality reduction of asymptotic quantum robust control calculations, describing numerical methods for implementation of these calculations.
Despite significant progress in theoretical and laboratory quantum control, engineering quantum systems remains principally challenging due to manifestation of noise and uncertainties associated with the field and Hamiltonian parameters. In this paper, we extend and generalize the asymptotic quantum control robustness analysis method, which provides more accurate estimates of quantum control objective moments than standard leading-order techniques, to diverse quantum observables, gates, and moments thereof and also introduce the Pontryagin maximum principle for quantum robust control. In addition, we present a Pareto optimization framework for achieving robust control via evolutionary open-loop (model-based) and closed-loop (model-free) approaches with the mechanisms of robustness and convergence described using asymptotic quantum control robustness analysis. In the open-loop approach, a multiobjective genetic algorithm is used to obtain Pareto solutions in terms of the expectation and variance of the transition probability under Hamiltonian parameter uncertainty. The set of numerically determined solutions can then be used as a starting population for model-free learning control in a feedback loop. The closed-loop approach utilizes a real-coded genetic algorithm with adaptive exploration and exploitation operators in order to preserve solution diversity and dynamically optimize the transition probability in the presence of field noise. Together, these methods provide a foundation for high-fidelity adaptive feedback control of quantum systems wherein open-loop control predictions are iteratively improved based on data from closed-loop experiments.
Abstract In recent work, we introduced the asymptotic theory of quantum robust control, which enables control of moments of quantum observables and gates in the presence of Hamiltonian uncertainty or field noise. In this paper, we extend this theory of quantum robust control to encompass two of the most important generalizations: robust control of arbitrary quantum observables and robust control of quantum systems sustaining environmental decoherence. In addition, we present deterministic Pareto optimization algorithms that can be applied in conjunction with either asymptotic or leading order measures of robustness. This enables robust control of any observable in quantum systems with any initial density matrix, and for which the entropy can change arbitrarily during the time evolution. Methods for robust optimal control of open quantum systems are presented that maximize the expected value of a quantum control objective while minimizing the expected environmentally induced decoherence.
The control of multilevel quantum systems is sensitive to implementation errors in the control field and uncertainties associated with system Hamiltonian parameters. A small variation in the control field spectrum or the system Hamiltonian can cause an otherwise optimal field to deviate from controlling desired quantum state transitions and reaching a particular objective. An accurate analysis of robustness is thus essential in understanding and achieving model-based quantum control, such as in the control of chemical reactions based on ab initio or experimental estimates of the molecular Hamiltonian. In this paper, theoretical foundations for quantum control robustness analysis are presented from both a distributional perspective---in terms of moments of the transition amplitude, interferences, and transition probability---and a worst-case perspective. Based on this theory, analytical expressions and a computationally efficient method for determining the robustness of coherently controlled quantum dynamics are derived. The robustness analysis reveals that there generally exists a set of control pathways that are more resistant to destructive interferences in the presence of control field and system parameter uncertainty. These robust pathways interfere and combine to yield a relatively accurate transition amplitude and high transition probability when uncertainty is present.
Resisting diverse noise effects is crucial for the accurate manipulation of quantum systems. For static or slowly varying noises, many efficient noise mitigation strategies have been developed, such as composite pulses and dynamical decoupling. However, for fast fluctuating noise in the Markovian limit, whether and to what extent coherent quantum control can enhance quantum engineering tasks remains unclear and less explored. Here, we propose a robust quantum optimal control method to tackle Markovian noises. The basic idea is that, regarding the Markovian noise channel as perturbation, we quantitatively characterize the noise-induced error evolution by the perturbative expansion term, and then take them as the objective functions to be suppressed. During optimization, the optimal controls are obtained by maximizing the control target function and meanwhile minimizing the perturbative terms due to Markovian noise order by order. As demonstration examples, we first apply our method to quantum state transfer tasks on two-level and three-level $\mathrm{\ensuremath{\Lambda}}$ systems, then use it to design quantum gates in two-level and three-level ladder systems, all under Markovian noises. The simulation results illustrate that our method can notably enhance quantum state transfer fidelities and have very limited improvement on gate fidelities. The method presented here is versatile and can be extended to enhance the performance of various control tasks under Markovian noise in multilevel or multiqubit quantum systems.
Reliable quantum information technologies depend on precise actuation and techniques to mitigate the effects of un-desired disturbances such as environmental noise and imperfect calibration. In this work, we present a general framework based in geometric optimal control theory to synthesize smooth control pulses for implementing arbitrary noise-robust quantum gates. The methodology applies to generic unitary quantum dynamics with any number of qubits or energy levels, any number of control fields, and any number of disturbances, extending existing dynamical decoupling approaches that are only applicable for limited gate sets or small systems affected by one or two disturbances. The noise-suppressing controls are computed via indirect trajectory optimization based on Pontryagin's maximum principle, eliminating the need to make heuristic structural assumptions on parameterized pulse envelopes.
In this paper, we demonstrate an approach to quantum robust control based on the tools of geometric optimal control. The central objects of interest are the sensitivity functions defined as the coefficients in the Taylor expansion of the trajectory with respect to the (unknown, small) parameters which describe the deviation of the actual model from nominal one. In terms of these quantities, we formalize an optimal control problem where one searches for the optimal nominal trajectory which minimizes the size of the sensitivity while taking into account other aspects of the control design such as the energy of the control field. We consider in detail the case of a single qubit with a dephasing Hamiltonian term, and the optimal control problem of obtaining a state transfer by minimizing the weighted sum of the energy of the controlling field and the first order sensitivity. At the limit of a very large weight on the sensitivity, we obtain the optimal control which zeros the sensitivity and minimizes the control field energy. This problem has a rich mathematical structure which enables its solution in terms of elliptic integrals. For this problem, we obtain an explicit solution which is particularly simple and also smooth, avoiding discontinuities which are present in other approaches. We extend the results to the robust control of two quantum bits minimizing cross-talk contamination, as we show that such a problem decouples in two independent one qubit problems.