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Smolyak algorithm assisted robust control for quantum systems with uncertainties

Zigui Zhang, Zibo Miao, Yu Chen, Xiu–Hao Deng

Published Oct 25, 2024• 0 citations
Publisher

Abstract

Abstract Efficient and systematic numerical methods for robust control design are crucial in quantum systems due to inevitable uncertainties or disturbances. We propose a novel approach that models uncertainties as random variables and quantifies robustness using the expectation of infidelity. By reformulating the robustness measure as a weighted tensor product quadrature, we employ the Smolyak sparse grid algorithm to develop a parametric robust quantum control scheme. This scheme significantly reduces computational cost while improving accuracy. We demonstrate the effectiveness of our Smolyak algorithm assisted gradient-based methods including \textbf{smGOAT} and \textbf{smGRAPE} in robust control problems regarding state transfer and quantum gate realization, with ultrahigh fidelity and strong robustness achieved. Our results contribute to improving the reliability and security of quantum computing and communication systems in the presence of real-world imperfections.

DOI
10.21203/rs.3.rs-5313186/v1

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Abstract Efficient and systematic numerical methods for robust control design are crucial in quantum systems due to inevitable uncertainties or disturbances. We propose a novel approach that models uncertainties as random variables and quantifies robustness using the expectation of infidelity. By reformulating the robustness measure as a weighted tensor product quadrature, we employ the Smolyak sparse grid algorithm to develop a parametric robust quantum control scheme. This scheme significantly reduces computational cost while improving accuracy. We demonstrate the effectiveness of our Smolyak algorithm assisted gradient-based methods including \textbf{smGOAT} and \textbf{smGRAPE} in robust control problems regarding state transfer and quantum gate realization, with ultrahigh fidelity and strong robustness achieved. Our results contribute to improving the reliability and security of quantum computing and communication systems in the presence of real-world imperfections.
Efficient and systematic numerical methods for robust control design are crucial in quantum systems due to inevitable uncertainties or disturbances. We propose a novel approach that models uncertainties as random variables and quantifies robustness using the expectation of infidelity by reformulating it as a weighted tensor product quadrature. We employ the Smolyak algorithm to develop a parametric robust quantum control scheme, which balances the reduction of computational cost with the enhancement of estimation accuracy. We demonstrate the effectiveness of our proposed algorithm by incorporating the Smolyak sparse grids into conventional gradient-based quantum optimal control methods such as GRAPE and GOAT. In robust control problems concerning quantum gate realization, low infidelity and strong robustness can be achieved. These results contribute to improving the reliability and security of quantum computing and communication systems in the presence of real-world imperfections.
Efficient and systematic numerical methods for robust control design are crucial in quantum systems due to inevitable uncertainties or disturbances. We propose an approach that models uncertainties as random variables and quantifies robustness using the expectation of infidelity by reformulating it as a weighted tensor product quadrature. We employ the Smolyak algorithm to develop a parametric robust quantum control scheme, which balances the reduction of computational cost with the enhancement of estimation accuracy. We demonstrate the effectiveness of our proposed algorithm by incorporating the Smolyak sparse grids into conventional gradient-based quantum optimal control methods such as grape and goat. In robust control problems concerning quantum gate realization, low infidelity and strong robustness can be achieved. These results contribute to improving the reliability and security of quantum computing and communication systems in the presence of real-world imperfections.
In view of inevitable uncertainties or disturbances in quantum systems, efficient and systematic numerical methods for robust control design are of significant importance. Although reinforcement learning has recently been applied in the field of quantum control, completing quantum control tasks in practice is always challenging due to the lack of data from real-world environments. In this paper, a generic and robust agent can be constructed catering to system imperfections by applying a curriculum meta-reinforcement learning quantum control (CMQC) approach. The proposed approach incorporates an objective module that can break down a complex quantum control objective into multiple sub-objectives using an exponential decay strategy, enabling the agent to gradually learn to solve the final goal through experience transfer and knowledge consolidation. Furthermore, a meta module is employed to achieve adaptation to environmental imperfections. This allows the agent to adjust its strategy in real time, obtaining strong robustness even in the presence of uncertainties. The superiority of CMQC is demonstrated in robust control problems regarding state transfer in a two-level quantum system and realization of theRx(π)R_{x}(\pi)Rx​(π)gate, with reduced computational cost and improved accuracy achieved. Our results pave the way towards more reliable and secure quantum information processing with uncertainties involved.
Dynamic control via optimized, piecewise-constant pulses is a common paradigm for open-loop control to implement quantum gates. While numerous methods exist for the synthesis of such controls, there are many open questions regarding the robustness of the resulting control schemes in the presence of model uncertainty; unlike in classical control, there are generally no analytical guarantees on the control performance with respect to inexact modeling of the system. In this paper, a new robustness measure based on the differential sensitivity of the gate fidelity error to parametric (structured) uncertainties is introduced, and bounds on the differential sensitivity to parametric uncertainties are used to establish performance guarantees for optimal controllers for a variety of quantum gate types, system sizes, and control implementations. Specifically, it is shown how a maximum allowable perturbation over a set of Hamiltonian uncertainties that guarantees a given fidelity error can be reliably computed. This measure of robustness is inversely proportional to the upper bound on the differential sensitivity of the fidelity error evaluated under nominal operating conditions. Finally, the results show that the nominal fidelity error and differential sensitivity upper bound are positively correlated across a wide range of problems and control implementations, suggesting that in the high-fidelity control regime, rather than there being a trade-off between fidelity and robustness, higher nominal gate fidelities are positively correlated with increased robustness of the controls in the presence of parametric uncertainties.
Dynamic control via optimized, piecewise-constant pulses is a common paradigm for open-loop control to implement quantum gates. While numerous methods exist for the synthesis of such controls, there are many open questions regarding the robustness of the resulting control schemes in the presence of model uncertainty; unlike in classical control, there are generally no analytical guarantees on the control performance with respect to inexact modeling of the system. In this paper a new robustness measure based on the differential sensitivity of the gate fidelity error to parametric (structured) uncertainties is introduced, and bounds on the differential sensitivity to parametric uncertainties are used to establish performance guarantees for optimal controllers for a variety of quantum gate types, system sizes, and control implementations. Specifically, it is shown how a maximum allowable perturbation over a set of Hamiltonian uncertainties that guarantees a given fidelity error, can be reliably computed. This measure of robustness is inversely proportional to the upper bound on the differential sensitivity of the fidelity error evaluated under nominal operating conditions. Finally, the results show that the nominal fidelity error and differential sensitivity upper bound are positively correlated across a wide range of problems and control implementations, suggesting that in the high-fidelity control regime, rather than there being a trade-off between fidelity and robustness, higher nominal gate fidelities are positively correlated with increased robustness of the controls in the presence of parametric uncertainties.
Robust control design in open quantum systems under parametric uncertainty is essential for practical quantum applications. In this work, we propose an efficient algorithm for identifying robust control solutions in high-dimensional quantum systems. The algorithm reduces the computational complexity by utilizing the Suzuki-Trotter expansion to accelerate the solution of system dynamics. Our simulation results demonstrate that the proposed algorithm can achieve control solutions of high fidelity and robustness with enhanced efficiency.