We present a general approach to the classical dynamical systems simulation. This approach is based on classical systems extension to quantum states. The proposed theory can be applied to analysis of multiple (including nonHamiltonian) dissipative dynamical systems. As examples, we consider the logistic model, the Van der Pol oscillator, dynamical systems of Lorenz, Rössler (including Rössler hyperchaos) and Rabinovich-Fabrikant. Developed methods and algorithms integrated in quantum simulators will allow us to solve a wide range of problems with scientific and practical significance.
We present a general approach to the classical dynamical systems simulation. This approach is based on classical systems extension to quantum states. The proposed theory can be applied to analysis of multiple (including nonHamiltonian) dissipative dynamical systems. As examples, we consider the logistic model, the Van der Pol oscillator, dynamical systems of Lorenz, Rössler (including Rössler hyperchaos) and Rabinovich-Fabrikant. Developed methods and algorithms integrated in quantum simulators will allow us to solve a wide range of problems with scientific and practical significance.
A theory of stochastic flows over the algebra of observables of a dynamical system is presented in which the main objective is to ensure that the overall canonical/symplectic structure on the algebra is preserved. We study both classical and quantum systems and the importance of physical interpretation in the Stratonovich interpretation is stressed. We find the natural formulation of quantum dissipative systems to be given in terms of quantum stochastic calculus. This treatment allows for a physically meaningful treatment of both constant and nonlinear dissipation. As an application, we quantize a mechanical system with the same nonlinear damping mechanism as the van der Pol oscillator.
The exact response theory based on the Dissipation Function applies to general dynamical systems and has yielded excellent results in various applications. In this article, we propose a method to apply it to quantum mechanics. In many quantum systems, it has not yet been possible to overcome the perturbative approach, and the most developed theory is the linear one. Extensions of the exact response theory developed in the field of nonequilibrium molecular dynamics could prove useful in quantum mechanics, as perturbations of small systems or far-from-equilibrium states cannot always be taken as small perturbations. Here, we introduce a quantum analogue of the classical Dissipation Function. We then derive a quantum expression for the exact calculation of time-dependent expectation values of observables, in a form analogous to that of the classical theory. We restrict our analysis to finite-dimensional Hilbert spaces, for the sake of simplicity, and we apply our method to specific examples, like qubit systems, for which exact results can be obtained by standard techniques. This way, we prove the consistency of our approach with the existing methods, where they apply. Although not required for open systems, we propose a self-adjoint version of our Dissipation Operator, obtaining a second equivalent expression of response, where the contribution of an anti-self-adjoint operator appears. We conclude by using new formalism to solve the Lindblad equations, obtaining exact results for a specific case of qubit decoherence, and suggesting possible future developments of this work.
In this study, we investigate Quantum Long Short-Term Memory and Quantum Gated Recurrent Unit integrated with Variational Quantum Circuits in modeling complex dynamical systems, including the Van der Pol oscillator, coupled oscillators, and the Lorenz system. We implement these advanced quantum machine learning techniques and compare their performance with traditional Long Short-Term Memory and Gated Recurrent Unit models. The results of our study reveal that the quantum-based models deliver superior precision and more stable loss metrics throughout 100 epochs for both the Van der Pol oscillator and coupled harmonic oscillators, and 20 epochs for the Lorenz system. The Quantum Gated Recurrent Unit outperforms competing models, showcasing notable performance metrics. For the Van der Pol oscillator, it reports MAE 0.0902 and RMSE 0.1031 for variable x and MAE 0.1500 and RMSE 0.1943 for y; for coupled oscillators, Oscillator 1 shows MAE 0.2411 and RMSE 0.2701 and Oscillator 2 MAE is 0.0482 and RMSE 0.0602; and for the Lorenz system, the results are MAE 0.4864 and RMSE 0.4971 for x, MAE 0.4723 and RMSE 0.4846 for y, and MAE 0.4555 and RMSE 0.4745 for z. These outcomes mark a significant advancement in the field of quantum machine learning.
The simulation of quantum systems has been a key aim of quantum technologies for decades, and generalization to open systems is necessary to include physically realistic systems. We introduce an approach for quantum simulations of open system dynamics in terms of an environment of minimal size and a time-dependent Hamiltonian. This enables the implementation of a continuous-time simulation with a finite environment, whereas state-of-the-art methods require an infinite environment or only match the simulation at discrete times. We find the necessary and sufficient conditions for this Hamiltonian to be well behaved, and when these are not met, we show that there exists an approximate Hamiltonian that is well behaved and look into its applications.
Quantum state evolution is a fundamental aspect of quantum mechanics, particularly concerning the dynamics of quantum systems over time. This paper explores the development of a novel model for quantum state evolution, integrating principles of quantum field theory and advanced computational methods. The core claim is to construct a model capable of accurately simulating the non-linear time evolution of quantum states, offering predictive capabilities for complex environments. We introduce a "Quantum Field Theory Extension" concept, linking quantum states to the non-linear dynamic changes inherent in field theories, and establish a tailored algorithm for precise prediction. This work represents a significant advancement in quantum simulation, aiming to address limitations of existing models by incorporating the core dynamics of quantum evolution.
The theory of Gaussian quantum fluctuations around classical steady states in nonlinear quantum-optical systems (also known as standard linearization) is a cornerstone for the analysis of such systems. Its simplicity, together with its accuracy far from critical points or situations where the nonlinearity reaches the strong coupling regime, has turned it into a widespread technique, being the first method of choice in most works on the subject. However, such a technique finds strong practical and conceptual complications when one tries to apply it to situations in which the classical long-time solution is time dependent, a most prominent example being spontaneous limit-cycle formation. Here, we introduce a linearization scheme adapted to such situations, using the driven Van der Pol oscillator as a test bed for the method, which allows us to compare it with full numerical simulations. On a conceptual level, the scheme relies on the connection between the emergence of limit cycles and the spontaneous breaking of the symmetry under temporal translations. On the practical side, the method keeps the simplicity and linear scaling with the size of the problem (number of modes) characteristic of standard linearization, making it applicable to large (many-body) systems.