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arXiv:2009.03946 · Physical Review A

Estimating the degree of non-Markovianity using machine learning

F. F. Fanchini, Göktuğ Karpat, Daniel Z. Rossatto, Ariel Norambuena, Raúl Coto

Published Feb 24, 2021• 27 citations
arXivHTMLPDF

Abstract

In the last few years, the application of machine learning methods has become increasingly relevant in different fields of physics. One of the most significant subjects in the theory of open quantum systems is the study of the characterization of non-Markovian memory effects that emerge dynamically throughout the time evolution of open systems as they interact with their surrounding environment. Here we consider two well-established quantifiers of the degree of memory effects, namely, the trace distance and the entanglement-based measures of non-Markovianity. We demonstrate that using machine learning techniques, in particular, support vector machine algorithms, it is possible to estimate the degree of non-Markovianity in two paradigmatic open system models with high precision. Our approach can be experimentally feasible to estimate the degree of non-Markovianity, since it requires a single or at most two rounds of state tomography.

DOI
10.1103/physreva.103.022425

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In the last few years, the application of machine learning methods has become increasingly relevant in different fields of physics. One of the most significant subjects in the theory of open quantum systems is the study of the characterization of non-Markovian memory effects that emerge dynamically throughout the time evolution of open systems as they interact with their surrounding environment. Here we consider two well-established quantifiers of the degree of memory effects, namely, the trace distance and the entanglement-based measures of non-Markovianity. We demonstrate that using machine learning techniques, in particular, support vector machine algorithms, it is possible to estimate the degree of non-Markovianity in two paradigmatic open system models with high precision. Our approach can be experimentally feasible to estimate the degree of non-Markovianity, since it requires a single or at most two rounds of state tomography.
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Machine learning methods have proved to be useful for the recognition of patterns in statistical data. The measurement outcomes are intrinsically random in quantum physics, however, they do have a pattern when the measurements are performed successively on an open quantum system. This pattern is due to the system-environment interaction and contains information about the relaxation rates as well as non-Markovian memory effects. Here we develop a method to extract the information about the unknown environment from a series of projective single-shot measurements on the system (without resorting to the process tomography). The method is based on embedding the non-Markovian system dynamics into a Markovian dynamics of the system and the effective reservoir of finite dimension. The generator of Markovian embedding is learned by the maximum likelihood estimation. We verify the method by comparing its prediction with an exactly solvable non-Markovian dynamics. The developed algorithm to learn unknown quantum environments enables one to efficiently control and manipulate quantum systems.
We construct a general measure for the degree of non-Markovian behavior in open quantum systems. This measure is based on the trace distance which quantifies the distinguishability of quantum states. It represents a functional of the dynamical map describing the time evolution of physical states, and can be interpreted in terms of the information flow between the open system and its environment. The measure takes on nonzero values whenever there is a flow of information from the environment back to the open system, which is the key feature of non-Markovian dynamics.
Abstract Several applications of quantum machine learning (QML) rely on a quantum measurement followed by training algorithms using the measurement outcomes. However, recently developed QML models, such as variational quantum circuits (VQCs), can be implemented directly on the state of the quantum system (quantum data). Here, we propose to use a qubit as a probe to estimate the degree of non-Markovianity of the environment. Using VQCs, we find an optimal sequence of qubit-environment interactions that yield accurate estimations of the degree of non-Markovianity for the amplitude damping, phase damping, and the combination of both models. This work contributes to practical quantum applications of VQCs and delivers a feasible experimental procedure to estimate the degree of non-Markovianity.
Non-Markovian effects due to quantum memory in the dynamics of open systems typically correspond to information backflows from the surrounding environment to the system. We propose a witness to quantify the non-Markovianity of quantum evolutions using the Bhattacharyya distance (BD), a specific quantum statistical distance. This witness has the advantage of not requiring the calculation of the evolved density matrix and only computes through the initial and final states of the system, therefore leading to the improvement of quantum metrology. It means that we calculate the quantum angle between two states to detect non-Markovian effects. This proposal is investigated by considering several instances of open quantum systems, such as two and three-level atoms interacting in single and two-mode fields, respectively, and two effective two-level atoms interacting locally with two independent environments. We demonstrate that the suggested BD-based non-Markovianity witness identifies memory effects, consistent with well-established witnesses based on Bures distance, quantum Fisher information, and Hilbert-Schmidt speed, showing sensitivity to information backflows.
The modeling and analysis of the dynamics of complex systems often requires to employ non-Markovian stochastic processes. While there is a clear and well-established mathematical definition for non-Markovianity in the case of classical systems, the extension to the quantum regime recently caused a vivid debate, leading to many different proposals for the characterization and quantification of memory effects in the dynamics of open quantum systems. Here, we derive a mathematical representation for the non-Markovianity measure based on the exchange of information between the open system and its environment, which reveals the locality and universality of non-Markovianity in the quantum state space and substantially simplifies its numerical and experimental determination. We further illustrate the application of this representation by means of an all-optical experiment which allows the measurement of the degree of memory effects in a photonic quantum process with high accuracy.