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.
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.
Abstract Koopman operator theory has been successfully applied to problems from various research areas such as fluid dynamics, molecular dynamics, climate science, engineering, and biology. Applications include detecting metastable or coherent sets, coarse-graining, system identification, and control. There is an intricate connection between dynamical systems driven by stochastic differential equations and quantum mechanics. In this paper, we compare the ground-state transformation and Nelson’s stochastic mechanics and demonstrate how data-driven methods developed for the approximation of the Koopman operator can be used to analyze quantum physics problems. Moreover, we exploit the relationship between Schrödinger operators and stochastic control problems to show that modern data-driven methods for stochastic control can be used to solve the stationary or imaginary-time Schrödinger equation. Our findings open up a new avenue toward solving Schrödinger’s equation using recently developed tools from data science.
Variational quantum time evolution is a fundamental issue in physics and engineering. It provides a strong framework for modeling the dynamics of quantum states, which is essential for efficiently running quantum networks. It presents an excellent toolkit that can improve the efficacy, reliability, resilience and performance of quantum networks, opening the door to more sophisticated distributed computer systems, quantum communication, and other quantum technologies. The imaginary time evolution of quantum systems is one useful tool that enables the generation of ground or thermal states. In this work, we address standard variational principles, such as the Dirac and Frenkel variational principles, the McLachlan variational principle and the time-dependent variational principle to model real-time dynamics, as well as the Rayleigh–Ritz approach to address static challenges. We also present the variational quantum time evolution VarQITE and VarQRTE formalism in the density matrix method, which is not influenced by the global phase of the quantum state. We calculate the ground-state energy of our quantum Ising spin chain, using VarQITE with different interaction energies J , where three cases are considered. In the non-interacting ( NI ) case ( J = 0 ), we show that the energy of the ground state accomplished with VarQITE converges to an energy of − 2 . 00 Hartree at t = 1 . 1 seconds. In the Ferromagnetic ( FM ) interaction case ( J > 0 ), four scenarios are used to obtain the energy of the Hamiltonian. We show that the convergence and the energy 〈 H 〉 are sensitive to the energy of interaction J . Furthermore, the influence of the negative values of J < 0 , which correspond to the antiferromagnetic ( AFM ) case, on the energy of the ground state is also studied. We find that VarQITE and the exact classical solution are matched. We also investigated the expectation values for magnetization 〈 m z 〉 , employing VarQRTE and the exact classical solution. The NI, FM, and AFM cases are accomplished here. We find that when the spins are not interacting, 〈 m z 〉 = 0 , while it oscillates over time t when J > 0 and J < 0 . We believe that our outcomes will improve our knowledge of the variational quantum time evolution and lead to a better understanding of real and imaginary time evolution of quantum systems (which form a quantum network) and therefore, to more information on the time in networks.
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.
Abstract Data-driven methods for establishing quantum optimal control (QOC) using time-dependent control pulses tailored to specific quantum dynamical systems and desired control objectives are critical for many emerging quantum technologies. We develop a data-driven regression procedure, bilinear dynamic mode decomposition (biDMD), that leverages time-series measurements to establish quantum system identification for QOC. The biDMD optimization framework is a physics-informed regression that makes use of the known underlying Hamiltonian structure. Further, the biDMD can be modified to model both fast and slow sampling of control signals, the latter by way of stroboscopic sampling strategies. The biDMD method provides a flexible, interpretable, and adaptive regression framework for real-time, online implementation in quantum systems. Further, the method has strong theoretical connections to Koopman theory, which approximates nonlinear dynamics with linear operators. In comparison with many machine learning paradigms minimal data is needed to construct a biDMD model, and the model is easily updated as new data is collected. We demonstrate the efficacy and performance of the approach on a number of representative quantum systems, showing that it also matches experimental results.
Quantum algorithms offer an exponential advantage with respect to the number of dependent variables for solving certain nonlinear ordinary differential equations (ODEs). These algorithms typically begin by transforming the original nonlinear ODE into a higher-dimensional linear ODE using a linearization technique, most commonly Carleman linearization. Existing works restrict their analysis to ODEs where the nonlinearities are polynomial functions of the dependent variables, significantly limiting their applicability. In this work we construct an efficient quantum algorithm for solving ODEs with `Fourier' nonlinear terms expressible as du/dt=G0+G1eiu, where u denotes a vector of n complex variables evolving with t, G0 is an n-dimensional complex vector, G1 is an n×n complex matrix and eiu denotes the vector with entries {eiuj}. To tackle the Fourier nonlinear term, which is not expressible as a finite sum of polynomials of u, our algorithm employs a generalization of the Carleman linearization technique known as Koopman linearization. We also make other methodological advances towards relaxing the stringent dissipativity condition required for efficient solution extraction and towards integrated readout of classical quantities from the solution state. Our results open avenues to the development of efficient quantum algorithms for a significantly wider class of high-dimensional nonlinear ODEs, thereby broadening the scope of their applications.
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.