Accurate identification of Hamiltonian parameters is essential for modeling and controlling open quantum systems. In this work, we demonstrate that the multichannel Hankel alternative view of Koopman (mHAVOK) algorithm is a robust and reliable spectral data-driven method for retrieving Hamiltonian parameters from the evolution of first-moment observables in open quantum systems. The method relies on the discrete spectrum of the Koopman operator to obtain these parameters, which are computed using the mHAVOK algorithm; a theoretical connection to this affirmation is presented. The method is tested on noiseless quadratures of an open two-dimensional quantum harmonic oscillator and shown to retrieve oscillation frequencies, damping rates, nonlinear Kerr shifts, the qubit-photon coupling strength of a Jaynes-Cummings interaction, and the modulated frequency of a time-dependent Hamiltonian. The majority of the recovered parameters remained within 5% of their actual values. Compared with Fourier and matrix-pencil estimators, our approach yields lower errors for dynamics with strong dissipation. Overall, these findings suggest that Koopman operator theory provides a practical framework for studying quantum dynamical systems.
Accurate identification of Hamiltonian parameters is essential for modeling and controlling open quantum systems. In this work, we demonstrate that the multichannel Hankel alternative view of Koopman (mHAVOK) algorithm is a robust and reliable spectral data-driven method for retrieving Hamiltonian parameters from the evolution of first-moment observables in open quantum systems. The method relies on the discrete spectrum of the Koopman operator to obtain these parameters, which are computed using the mHAVOK algorithm; a theoretical connection to this affirmation is presented. The method is tested on noiseless quadratures of an open two-dimensional quantum harmonic oscillator and shown to retrieve oscillation frequencies, damping rates, nonlinear Kerr shifts, the qubit-photon coupling strength of a Jaynes-Cummings interaction, and the modulated frequency of a time-dependent Hamiltonian. The majority of the recovered parameters remained within 5% of their actual values. Compared with Fourier and matrix-pencil estimators, our approach yields lower errors for dynamics with strong dissipation. Overall, these findings suggest that Koopman operator theory provides a practical framework for studying quantum dynamical systems.
Machine learning algorithms designed to learn dynamical systems from data can be used to forecast, control and interpret the observed dynamics. In this work we exemplify the use of one of such algorithms, namely Koopman operator learning, in the context of open quantum system dynamics. We will study the dynamics of a small spin chain coupled with dephasing gates and show how Koopman operator learning is an approach to efficiently learn not only the evolution of the density matrix, but also of every physical observable associated to the system. Finally, leveraging the spectral decomposition of the learned Koopman operator, we show how symmetries obeyed by the underlying dynamics can be inferred directly from data.
In recent years, the application of machine learning to physics has been actively explored. In this paper, we study a method for estimating the ground-state energy of quantum Hamiltonians by applying data-driven Koopman analysis within the framework of variational wave functions. Koopman theory is a framework for analyzing the nonlinear dynamics of vectors, in which the dynamics are linearized by lifting the vectors to functions defined over the original vector space. We focus on the fact that the imaginary-time Schrödinger equation, when restricted to a variational wave function, is described by a nonlinear time evolution of the variational parameter vector. We collect sample points of this nonlinear dynamics at parameter configurations where the discrepancy between the true imaginary-time dynamics and the dynamics on the variational manifold is small, and perform data-driven continuous Koopman analysis. Within our formulation, the ground-state energy is reduced to the leading eigenvalue of a differential operator known as the Koopman generator. As a concrete example, we generate samples for the four-site transverse-field Ising model and estimate the ground-state energy using extended dynamic mode decomposition (EDMD). Furthermore, as an extension of this framework, we formulate the method for the case where the variational wave function is given by a uniform matrix product state on an infinite chain. By employing computational techniques developed within the framework of the time-dependent variational principle, all the quantities required for our analysis, including error estimation, can be computed efficiently in such systems. Since our approach provides predictions for the ground-state energy even when the true ground state lies outside the variational manifold, it is expected to complement conventional variational methods.
Solving high-dimensional dynamical systems by encoding their state in the amplitudes of a quantum state is expected to be a core application of fault-tolerant quantum computers. However, the conditions under which this paradigm yields quantum advantage can be restrictive, motivating complementary approaches. Here, we introduce Quantum Koopman Algorithms, where quantum amplitudes encode approximately closed sets of observables for a linear or nonlinear dynamics. These amplitudes evolve under the Koopman operator, and we establish conditions for efficient quantum algorithms addressing both dynamical and spectral problems. With this approach, we show that classes of open-system dynamics involving exponentially large fermionic systems can be efficiently simulated on a quantum computer. For nonlinear dynamics, we introduce a novel nonlinear interaction picture that enables quantum simulation beyond existing weak-nonlinearity conditions, and extract late-time spectral data where quantum phase estimation is unavailable.
Dynamic mode decomposition (DMD) is a data-driven method for the estimation, prediction, and control of complex dynamical systems, which has gained much attention in the fields of nonlinear dynamics and fluid mechanics. A DMD method for quantum spin systems, described by a linear dynamical system of a finite-dimensional set of observables, has been proposed recently. In this study, we propose two DMD methods applicable to infinite-dimensional open quantum systems, which use time-series data obtained by quantum state tomography. First, we propose a kernel DMD method for a data-driven spectral analysis of the Liouville superoperator. Second, we propose a method for the parameter estimation of the Liouville superoperator, which incorporates prior knowledge of the model structure into DMD. The proposed methods can accurately reconstruct the system dynamics and show that DMD frameworks can be applicable to infinite-dimensional open quantum systems.
Nonlinear dynamics is ubiquitous in nature, ranging from chemical pattern formation to ocean circulation, yet its simulation on quantum computers is fundamentally limited by the unitary nature of quantum evolution. We propose the quantum Koopman method, a data-driven framework that embeds nonlinear dynamics into a learned linear representation and implements the resulting evolution using shallow quantum circuits. This method learns Koopman observables from trajectory data, projects the lifted dynamics onto a finite-dimensional subspace, and decomposes the corresponding non-unitary propagator into parallel spectral channels. We utilize the Koopman method on a superconducting processor to simulate three distinct nonlinear systems, comprising reaction-diffusion dynamics, fluid motion on a sphere, and satellite-derived observations of Gulf Stream currents, employing up to 32 parallel circuits of 10 qubits. These quantum simulations capture the dominant multiscale patterns and statistical signatures of the underlying dynamics, and reveal a transition from performance limited by hardware noise in weakly nonlinear systems to performance limited by finite-dimensional Koopman representations as nonlinear scale interactions increase. This transition identifies a practical boundary for quantum-amenable nonlinear dynamics, establishing a hardware-validated route for simulating moderately nonlinear dynamics on near-term quantum hardware.
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.