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Quantum open system identification via global optimization: Optimally accurate Markovian models of open systems from time-series data

Zakhar Popovych, Kurt Jacobs, Georgios Korpas, Jakub Marecek, Denys I. Bondar

TL;DR

The paper tackles learning physically valid open-quantum-system dynamics from time-series data, addressing the inability of standard SID to enforce positivity and complete positivity. It formulates SID as a polynomial optimization problem over the GKSL/GKS parameters $(H,C)$, using the moment–SOS hierarchy to guarantee a globally optimal solution with $H=H^\dagger$ and $C\succ 0$, and demonstrates this on the spin-boson model with an Ohmic bath. The results show that the global-optimization approach yields highly accurate Markovian models across damping regimes and outperforms traditional linear SID methods, while also providing an operational measure of non-Markovianity via model error; a simpler Lindblad ansatz suffices at high $Q$ but may lose accuracy at stronger coupling. The work offers a robust, physically grounded SID framework for quantum devices and suggests extensions to non-Markovian modeling, with potential impact on quantum control, sensing, and error mitigation.

Abstract

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.

Quantum open system identification via global optimization: Optimally accurate Markovian models of open systems from time-series data

TL;DR

The paper tackles learning physically valid open-quantum-system dynamics from time-series data, addressing the inability of standard SID to enforce positivity and complete positivity. It formulates SID as a polynomial optimization problem over the GKSL/GKS parameters , using the moment–SOS hierarchy to guarantee a globally optimal solution with and , and demonstrates this on the spin-boson model with an Ohmic bath. The results show that the global-optimization approach yields highly accurate Markovian models across damping regimes and outperforms traditional linear SID methods, while also providing an operational measure of non-Markovianity via model error; a simpler Lindblad ansatz suffices at high but may lose accuracy at stronger coupling. The work offers a robust, physically grounded SID framework for quantum devices and suggests extensions to non-Markovian modeling, with potential impact on quantum control, sensing, and error mitigation.

Abstract

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.
Paper Structure (15 sections, 41 equations, 3 figures)

This paper contains 15 sections, 41 equations, 3 figures.

Table of Contents

  1. Introduction
  2. System Identification using Markovian models
  3. Unique representation of Markovian evolution
  4. Formulating system identification as polynomial optimization
  5. Application to the spin-boson model
  6. Note: the cut-off frequency and the non-Markovian "initial slip"
  7. Employing as an ansatz the most general Markovian master equation
  8. Results from employing the most general Markovian master equation
  9. Employing a simpler Lindblad-form ansatz for higher $Q$
  10. Comparison with traditional linear system identification
  11. Two standard Linear system identification methods
  12. Applying linear system identification to the spin-boson model
  13. CONCLUSIONS
  14. Breuer's measure of non-Markovianity
  15. Linear system identification: mapping the model representation to the original state vector

Figures (3)

  • Figure 1: Performance of Markovian models obtained with polynomial (global) optimization ((a) and (b)) compared with a benchmark traditional optimization method from NLOPT (c), and the Breuer measure of non-Markovianity, all for the spin-boson system with a range of $Q$-factors. Left axis: Infidelity between the optimized identified model and the true evolution, $1-F$, where the fidelity $F(\rho^{M}, \rho^{SB})$ is defined in Eq.\ref{['eq:fidelity']}. The "violin plots" give the distribution of infidelities over all data points on the 10 distinct example trajectories (see text) for each of the models obtained with the three different methods. It is an important feature that the polynomial optimization always provide a model at least as good as the traditional search method. The former is essentially guaranteed to obtain the best possible model. (a) Polynomial optimization (POP) using the Kossakowski master equation, Eq. \ref{['eq:pop_kossak2']}, see Ref. popovych_kossak_pop_2024 regarding a Julia code used; (b) POP using the selected Lindblad ansatz, Eq.\ref{['eq:pop_lindblad2']}, see Ref. popovych_lindblad_pop_2024 regarding a Julia code used; (c) The best performing NLOPT optimiser (LD_SLSQP) for comparison, using the corresponding ansatz, and optimizing the Fidelity explicitly, see Ref. popovych_kossak_benchmark_2024popovych_lindblad_benchmark_2024regarding a Julia code used. Right axis: Brauer's non-Markovianity measure $\mathcal{N}$breuer_measure_2009, approximated over all available pairs $\{\rho^{(1)}, \rho^{(2)}\}$ of spin-boson system trajectories using Eq.\ref{['eq:nonmark_breuer']}, see Ref. popovych_non-mark-estims_2024 regarding a Julia code used. See Ref. popovych_pop_plot_2024 regarding a python code used to generate this figure.
  • Figure 2: The performance of Markovian master equation models obtained using polynomial (global) optimization as a function of the length of the time series employed, and for eight quality factors, $Q$. We see from this that different quality factors require different durations to obtain the best model. For all the quality factors we investigated, for the simulation times we used, all have reached their optimal models except for $Q=3.16$ which appears to be close. See Ref. popovych_pop_plot_by_train_2024 regarding a python code used to generate this figure.
  • Figure 3: Performance of linear system identification methods for the spin-boson system for different quality factors $Q$. For each value of $Q$ the performance of the model(s) is shown using a "violin" plot that shows the distribution of the fidelity of the model over all the data points of the time-series for our 10 distinct initial conditions (half the points of a dodecahedron on the Bloch sphere). The infidelity is shown on the left-hand axis, defined as $1-F$ where $F$ is the fidelity (Eq.(\ref{['eq:fidelity']})) between the density matrix predicted by the model and that of the exact evolution of the spin-boson two-level system. For the 4-dimensional models (shown in blue) we also plot in red the largest real eigenvalue of the dynamical matrix of the model. The small this value the more stable the model. This value is shown on the right-hand axis.