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Non-Markovian quantum dynamics quantification in the Jaynes-Cummings model using principal component analisys

Carlos A. Galindez, Arturo Argüelles

Published Sep 11, 2025• 0 citations
Publisher

Abstract

Abstract Recent advancements in quantum information science have led to an increased focus on the study of open quantum systems and their non-Markovian dynamics. In this paper, we present a novel approach to quantifying the non-Markovian dynamics of a n-dimensional quantum system using Principal Component Analysis (PCA). The motivation for this work stems from the growing interest in understanding the complex interplay between quantum systems and their environment, alongside the potential of machine learning techniques to provide new insights into such systems dynamics. We specifically apply our methodology to the Jaynes-Cummings (JC) model, which describes the interaction between a two-level atom and a quantized electromagnetic field mode, and a cornerstone in quantum non-Markovianity studies due to its characteristic collapse and revival phenomena. Our PCA-based method aims to capture the non-Markovian features of the system dynamics in a comprehensive and interpretable manner. This involves constructing a covariance matrix from the state vector of the system for a set of initial states and employing PCA to identify the dominant modes of variation. Subsequently, we analyze the temporal behavior of these principal components to quantify the non-Markovian characteristics of the system. Importantly, our method provides a more nuanced understanding of non-Markovian dynamics, revealing insights that may not be fully captured by conventional approaches.

DOI
10.21203/rs.3.rs-7375569/v1

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Abstract Recent advancements in quantum information science have led to an increased focus on the study of open quantum systems and their non-Markovian dynamics. In this paper, we present a novel approach to quantifying the non-Markovian dynamics of a n-dimensional quantum system using Principal Component Analysis (PCA). The motivation for this work stems from the growing interest in understanding the complex interplay between quantum systems and their environment, alongside the potential of machine learning techniques to provide new insights into such systems dynamics. We specifically apply our methodology to the Jaynes-Cummings (JC) model, which describes the interaction between a two-level atom and a quantized electromagnetic field mode, and a cornerstone in quantum non-Markovianity studies due to its characteristic collapse and revival phenomena. Our PCA-based method aims to capture the non-Markovian features of the system dynamics in a comprehensive and interpretable manner. This involves constructing a covariance matrix from the state vector of the system for a set of initial states and employing PCA to identify the dominant modes of variation. Subsequently, we analyze the temporal behavior of these principal components to quantify the non-Markovian characteristics of the system. Importantly, our method provides a more nuanced understanding of non-Markovian dynamics, revealing insights that may not be fully captured by conventional approaches.
A precise understanding of the influence of a quantum system's environment on its dynamics, which is at the heart of the theory of open quantum systems, is crucial for further progress in the development of controllable large-scale quantum systems. However, existing approaches to account for complex system-environment interaction in the presence of memory effects are either based on heuristic and oversimplified principles or give rise to computational difficulties. In practice, one can leverage on available experimental data and replace first-principles simulations with a data-driven analysis that is often much simpler. Inspired by recent advances in data analysis and machine learning, we propose a data-driven approach to the analysis of the non-Markovian dynamics of open quantum systems. Our method allows, on the one hand, capturing the most important characteristics of open quantum systems such as the effective dimension of the environment and the spectrum of the joint system-environment quantum dynamics, and, on the other hand, reconstructing a predictive model of non-Markovian quantum dynamics, and denoising the measured quantum trajectories. We demonstrate the performance of the proposed approach with various models of open quantum systems, including a qubit coupled with a finite environment, a spin-boson model, and the damped Jaynes-Cummings model.
We study non-equilibrium quantum dynamics by performing principal component analysis on the data sets of wavefunction snapshots. We show that a specific transformation of the data sets maximizes the information content in the largest principal component and further enables its connection to certain observables. This connection enables us to explain the dynamical features revealed by such a dimensionality-reduction scheme. We demonstrate this using quantum dynamics of the Heisenberg spin chain, starting from different initial states, and further extend the approach to extract higher-order correlations. Our framework should also be applicable to other unsupervised machine-learning methods based on dimensionality-reduction schemes and is highly relevant to experiments with quantum simulators, including those in higher dimensions.
Quantum Machine Learning has gained significant attention in recent years as a way to leverage the relationship between quantum information and machine learning. Principal Component Analysis (PCA) is a fundamental technique in machine learning, and the potential for its quantum acceleration has been extensively studied. However, an algorithmic end-to-end implementation remains challenging. This paper covers quantum PCA implementation up to extracting the principal components. We extend existing processes for quantum state tomography to extract the eigenvectors from the output state, addressing the challenges of dealing with complex amplitudes in the case of non-integer eigenvalues. Finally, we apply our implementation to a practical quantum finance use case related to interest rate risk, and present the results of our experiments.
With the rapid progress in quantum hardware, there has been an increased interest in new quantum algorithms to describe complex many-body systems searching for the still-elusive goal of "useful quantum advantage."Surprisingly, quantum algorithms for the treatment of open quantum systems (OQSs) have remained underexplored, in part due to the inherent challenges of mapping non-unitary evolution into the framework of unitary gates.Evolving an open system unitarily necessitates dilation into a new effective system to incorporate critical environmental degrees of freedom.In this context, we present and validate a new quantum algorithm to treat non-Markovian dynamics in OQSs built on the ensemble of Lindblad's trajectories approach, invoking the Sz.-Nagy dilation theorem.Here we demonstrate our algorithm on the Jaynes-Cummings model in the strong-coupling and detuned regimes, relevant in quantum optics and driven quantum system studies.This algorithm, a key step towards generalized modeling of non-Markovian dynamics on a noisy-quantum device, captures a broad class of dynamics and opens up a new direction in OQS problems.
With the rapid progress in quantum hardware, there has been an increased interest in new quantum algorithms to describe complex many-body systems searching for the still-elusive goal of "useful quantum advantage." Surprisingly, quantum algorithms for the treatment of open quantum systems (OQSs) have remained underexplored, in part due to the inherent challenges of mapping non-unitary evolution into the framework of unitary gates. Evolving an open system unitarily necessitates dilation into a new effective system to incorporate critical environmental degrees of freedom. In this context, we present and validate a new quantum algorithm to treat non-Markovian dynamics in OQSs built on the ensemble of Lindblad's trajectories approach, invoking the Sz.-Nagy dilation theorem. Here we demonstrate our algorithm on the Jaynes-Cummings model in the strong-coupling and detuned regimes, relevant in quantum optics and driven quantum system studies. This algorithm, a key step towards generalized modeling of non-Markovian dynamics on a noisy-quantum device, captures a broad class of dynamics and opens up a new direction in OQS problems.
The Jaynes-Cummings-Hubbard (JCH) model is a fundamental many-body model for light-matter interaction. As a leading platform for quantum simulation, the trapped ion system has realized the JCH model for two to three ions. Here, we report the quantum simulation of the JCH model using up to 32 ions. We verify the simulation results even for large ion numbers by engineering low excitations and thus low effective dimensions; then we extend to 32 excitations for an effective dimension of 2^{77}, which is difficult for classical computers. By regarding the phonon modes as baths, we explore Markovian or non-Markovian spin dynamics in different parameter regimes of the JCH model, similar to quantum emitters in a structured photonic environment. We further examine the dependence of the non-Markovian dynamics on the effective Hilbert space dimension. Our Letter demonstrates the trapped ion system as a powerful quantum simulator for many-body physics and open quantum systems.