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Journal of Physics A Mathematical and Theoretical

Robust observable control of open and closed quantum systems

Vaibhav Bhutoria, Andy Koswara, Raj Chakrabarti

Published Mar 29, 2022• 4 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8121/ac623f

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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.
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.
Recent advancements in quantum technologies have highlighted the importance of mitigating system imperfections, including parameter uncertainties and decoherence effects, to improve the performance of experimental platforms. However, most of the previous efforts in quantum control are devoted to the realization of arbitrary unitary operations in a closed quantum system. Here, we improve the algorithm that suppresses system imperfections and noises, providing notably enhanced scalability for robust and optimal control of open quantum systems. Through experimental validation in a superconducting quantum circuit, we demonstrate that our approach outperforms its conventional counterpart for closed quantum systems with an ultra-low infidelity of about 0.60%0.60\%0.60%, while the complexity of this algorithm exhibits the same scaling, with only a modest increase in the prefactor. This work represents a notable advancement in quantum optimal control techniques, paving the way for realizing quantum-enhanced technologies in practical applications.
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.
The impact of control field fluctuations on the optimal manipulation of quantum dynamics phenomena is investigated. The quantum system is driven by an optimal control field, with the physical focus on the evolving expectation value of an observable operator. A relationship is shown to exist between the system dynamics and the control field fluctuations, wherein the process of seeking optimal performance assures an inherent degree of system robustness to such fluctuations. The presence of significant field fluctuations breaks down the evolution of the observable expectation value into a sequence of partially coherent robust steps. Robustness occurs because the optimization process reduces sensitivity to noise-driven quantum system fluctuations by taking advantage of the observable expectation value being bilinear in the evolution operator and its adjoint. The consequences of this inherent robustness are discussed in the light of recent experiments and numerical simulations on the optimal control of quantum phenomena. The analysis in this paper bodes well for the future success of closed-loop quantum optimal control experiments, even in the presence of reasonable levels of field fluctuations.
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 closed-loop 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 ats=0s = 0s=0. Additional difficulties occur when symmetry gives rise to multiple open-loop poles, which under symmetry-breaking 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.
The impact of control field fluctuations on the optimal manipulation of quantum dynamics phenomena is investigated. The presence of significant field fluctuations is shown to break down the evolution into a sequence of partially coherent robust steps. Robustness occurs because the optimization process reduces sensitivity to noise-driven quantum system fluctuations. This process takes advantage of the observable expectation value being bilinear in the evolution operator and its adjoint. The consequences of this inherent robustness bodes well for the future success of closed loop quantum optimal control experiments.