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arXiv:2003.08617 · Physical Review A

Inverse-system method for identification of damping-rate functions in non-Markovian quantum systems

Shibei Xue, Lingyu Tan, Re-Bing Wu, Min Jiang, Ian R. Petersen

Published Oct 27, 2020• 14 citations
arXivHTMLPDF

Abstract

Identification of complicated quantum environments lies in the core of quantum engineering, which systematically constructs an environment model with the aim of accurate control of quantum systems. In this paper, we present an inverse-system method to identify damping rate functions, which describe non-Markovian environments in time-convolution-less master equations. To access information on the environment, we couple a finite-level quantum system to the environment and measure time traces of local observables of the system. By using sufficient measurement results, an algorithm is designed, which can simultaneously estimate multiple damping rate functions for different dissipative channels. Further, we show that identifiability for the damping rate functions corresponds to the invertibility of the system and a necessary condition for identifiability is also given. The effectiveness of our method is shown in examples of an atom and three-spin-chain non-Markovian systems.

DOI
10.1103/physreva.102.042227

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Identification of complicated quantum environments lies in the core of quantum engineering, which systematically constructs an environment model with the aim of accurate control of quantum systems. In this paper, we present an inverse-system method to identify damping rate functions, which describe non-Markovian environments in time-convolution-less master equations. To access information on the environment, we couple a finite-level quantum system to the environment and measure time traces of local observables of the system. By using sufficient measurement results, an algorithm is designed, which can simultaneously estimate multiple damping rate functions for different dissipative channels. Further, we show that identifiability for the damping rate functions corresponds to the invertibility of the system and a necessary condition for identifiability is also given. The effectiveness of our method is shown in examples of an atom and three-spin-chain non-Markovian systems.
Quantum identification is a basic step for many quantum technologies. In this paper, we propose an inverse-system method to identify the damping rate function of a non-Markovian single-qubit system. Based on the time-convolution-less master equation, and using measured time traces of observables, the inverse-system method can extract the unknown damping rate function. For a computational singularity problem, we avoid it by switching between the two measured observebles. The effectiveness of our method is demonstrated in a numerical simulation.
Correlations of an environment are crucial for the dynamics of non-Markovian quantum systems, which may not be known in advance. In this brief, we propose a gradient algorithm for identifying the correlations in terms of the time-varying damping rate functions in a time-convolution-less master equation for spin chains. By measuring time trace observables of the system, the identification procedure can be formulated as an optimization problem. The gradient algorithm is designed based on a calculation of the derivative of an objective function with respect to the damping rate functions, whose effectiveness is shown in a comparison with a differential approach for a two-qubit spin chain.
In this paper, we present a gradient algorithm to identify a damping rate function for a non-Markovian single qubit system. The dynamics of the single qubit system in a non-Markovian environment are assumed to obey a time convolutionless master equation, where all the non-Markovian effects of the environment are combined in the unknown damping rate function. To identify the damping rate function, we measure time trace observables of the qubit such that we can formulate the identification procedure as an optimization problem. Thus, we design a gradient algorithm to optimally reveal the damping rate function.
In this paper, we present a linear least squares method to identify a damping rate function for a non-Markovian single qubit system. The dynamics of the system is described by a time-convolutionless master equation where the unknown damping rate function contains all the information of the environment. By expressing the function as a polynomial of time with unknown coefficients, we convert the identification problem into a parameter estimation problem. We transform the master equation into a Bloch differential equation, and obtain a reduced system whose output is the time trace of an observable. Then we use a linear least squares method to estimate unknown coefficients in the polynomial by utilizing the measured outputs. Finally, the effectiveness of our method is shown in an example of a two-level atom non-Markovian system.
Accurate models of the dynamics of quantum circuits are essential for optimizing and advancing quantum devices. Since first-principles models of environmental noise and dissipation in real quantum systems are often unavailable, deriving accurate models from measured time-series data is critical. However, identifying open quantum systems poses significant challenges: powerful methods from systems engineering can perform poorly beyond weak damping (as we show) because they fail to incorporate essential constraints required for quantum evolution (e.g., positivity). Common methods that can include these constraints are typically multi-step, fitting linear models to physically grounded master equations, often resulting in non-convex functions in which local optimization algorithms get stuck in local extrema (as we show). In this work, we solve these problems by formulating quantum system identification directly from data as a polynomial optimization problem, enabling the use of recently developed global optimization methods. These methods are essentially guaranteed to reach global optima, allowing us for the first time to efficiently obtain the most accurate Markovian model for a given system. In addition to its practical importance, this allows us to take the error of these Markovian models as an alternative (operational) measure of the non-Markovianity of a system. We test our method with the spin-boson model -- a two-level system coupled to a bath of harmonic oscillators -- for which we obtain the exact evolution using matrix-product-state techniques. We show that polynomial optimization using moment/sum-of-squares approaches significantly outperforms traditional optimization algorithms, and we show that even for strong damping Lindblad-form master equations can provide accurate models of the spin-boson system.
We propose a systematic scheme to engineer quantum states of a quantum system governed by a time-convolutionless non-Markovian master equation. According to the idea of reverse engineering, the general algebraic equation to determine the control parameters, such as coherent and incoherent control fields, is presented. Without artificially engineering the time-dependent decay rates and retaining the environment-induced Lamb shifts, the quantum state can still be transferred into the target state in a finite period of time along an arbitrary designed trajectory strictly in Hilbert space. As an application, we apply our scheme to a driven two-level non-Markovian system and realize instantaneous-steady-state tracking and a complete population inversion with control parameters which are available in experimental settings.