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 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. Introduction. Quantum systems are never perfectly isolated which makes the study of open quantum dynamics important for various disciplines including solid state physics [1], quantum chemistry [2], quantum sensing [3], quantum information transmission [4], and quantum computing [5]. Open quantum dynamics is a result of interaction between the system of interest and its environment. It is usually assumed that the environment is an infinitely large reservoir in statistical equilibrium, which has a well-defined interaction with the system [6]. However, the environments of many physical systems are rather complex and structured [7][8][9][10][11][12][13][14][15][16][17][18]. A model of the systemenvironment interaction is often heuristic and oversimplified (e.g., a harmonic environment), but even in this case the analysis is rater complicated and requires some elaborated analytical and numerical methods [19][20][21]. A theoretical model may also neglect some additional sources of decoherence and relaxation. The experimental analysis of the environmental degrees of freedom is difficult because of their inaccessibility in practice. In fact, one can only get some information about the actual environment by probing the system [22]. Therefore, one faces an important problem to learn the unknown environment and its interaction with the quantum system by probing and affecting the system only.

The difficulty to simulate the dynamics of open quantum systems resides in their coupling to many-body reservoirs with exponentially large Hilbert space. Applying a tensor network approach in the time domain, we demonstrate that effective small reservoirs can be defined and used for modeling open quantum dynamics. The key element of our technique is the timeline reservoir network (TRN), which contains all the information on the reservoir's characteristics, in particular, the memory effects timescale. The TRN has a one-dimensional tensor network structure, which can be effectively approximated in full analogy with the matrix product approximation of spinchain states. We derive the sufficient bond dimension in the approximated TRN with a reduced set of physical parameters: coupling strength, reservoir correlation time, minimal timescale, and the system's number of degrees of freedom interacting with the environment. The bond dimension can be viewed as a measure of the open dynamics complexity. Simulation is based on the semigroup dynamics of the system and effective reservoir of finite dimension. We provide an illustrative example showing scope for new numerical and machine learning-based methods for open quantum systems.

We consider a quantum system dynamics caused by successive selective and non-selective measurements of the probe coupled to the system. For the finite measurement rate $\tau^{-1}$ and the system-probe interaction strength $\gamma$ we derive analytical evolution equations in the stroboscopic limit $\tau \rightarrow 0$ and $\gamma^2 \tau = {\rm const}$, which can be considered as a deviation from the Zeno subspace dynamics on a longer timescale $T \sim (\gamma^2 \tau)^{-1} \gg \gamma^{-1}$. Non-linear quantum dynamics is analyzed for selective stroboscopic projective measurements of an arbitrary rank. Non-selective measurements are shown to induce the semigroup dynamics of the system-probe aggregate. Both non-linear and decoherent effects become significant at the timescale $T \sim (\gamma^2 \tau)^{-1}$, which is illustrated by a number of examples.Comment: 9 pages, 7 figures, published versio