The characterization of collective behavior and nonequilibrium phase transitions in quantum systems is typically rooted in the analysis of suitable system observables, so-called order parameters. These observables might not be known a priori, but they may in principle be identified through analyzing the quantum state of the system. Experimentally, this can be particularly demanding as estimating quantum states and expectation values of quantum observables requires a large number of projective measurements. However, open quantum systems can be probed in situ by monitoring their output, e.g. via heterodyne-detection or photon-counting experiments, which provide space-time resolved information about their dynamics. Building on this, we present a machine-learning approach to detect nonequilibrium phase transitions from the measurement time-records of continuously-monitored quantum systems. We benchmark our method using the quantum contact process, a model featuring an absorbing-state phase transition, which constitutes a particularly challenging test case for the quantum simulation of nonequilibrium processes.
Introduction.-Interacting many-body quantum systems far from equilibrium can display complex collective behavior and phase transitions [1][2][3][4][5][6][7][8][9][10][11][12]. The analysis of these phenomena is typically based on the identification of a suitable observable, a so-called order parameter, which captures the abrupt changes to the system state occurring close to a critical point. Their characterization thus effectively relies on a dimensional reduction, whereby collective effects and universal properties are fully described by the behavior of a single quantity. From a theoretical viewpoint, order parameters can be obtained by calculating expectation values of system observables or even by considering more involved properties of the quantum state, such as entanglement and quantum correlations. However, determining these quantities experimentally may come with significant overhead, which is associated with repeated state preparation and projective measurements or, in the most extreme case, with a full quantum state tomography.
Continuously monitored open quantum systems release information into the environment which can be in situ monitored and processed. This space-time resolved information (see sketch in Fig. 1) can be obtained using heterodyne or photon-counting measurements [13][14][15][16], using digital quantum simulators [17][18][19][20][21][22][23][24], or via ancilla-based measurement schemes [25,26]. In contrast to projective measurements, continuous monitoring does not destroy the state of the system. It rather provides continuous dynamical information [15,27], encoded in socalled quantum trajectories (cf. Fig. 1), which is experimentally accessible without the need of repeated state preparation, ensemble averaging, or postselection. More-over, one may expect that the space-time information contained in quantum trajectories allows to distinguish dynamical phases and to pinpoint phase transitions.
In this work, we introduce a machine-learning approach that learns collective properties of open quantum systems directly from the structure of quantum trajectories. While this output signal need not be directly related to any system observable or order parameter, we show that it encodes crucial information for the neural network to be able to discriminate between different phases. We achieve this by feeding the high-dimensional space-time resolved quantum trajectories to an unsupervised learning framework, which operates by mapping them onto a low-dimensional latent space. Here, trajectories are clustered according to the dynamical phase they originate from (see an example in Fig. 2). In this way, we exploit the intrinsic capability of machine-learning methods to reduce the problem’s dimensionality and to extract effective order parameters from quantum trajectories. This allows us to detect signatures of criticality from complex datasets, without relying on predefined system observables [28][29][30][31].
We illustrate our method focusing on the quantum contact process [32][33][34][35] and synthetically generate heterodyne-measurement quantum trajectories using tensor networks. We select this model because it features a nonequilibrium phase transition -even in one dimension -between an absorbing phase and an active phase. The presence of the absorbing phase renders studying the model extremely challenging and a benchmark problem for quantum hardware [34,36]. We emphasize that machine-learning has been employed for the identification FIG. 1. Quantum trajectories across the phase diagram of the quantum contact process. The top part of the figure sketches an experimental setting in which spatially resolved emissions from a many-body system (S) are monitored continuously in time. The sketched quantum trajectories labeled by S show qualitatively the dynamics of the expectation of a local order parameter (here the local density of active sites). The latter display clearly distinct structures in the subcritical, critical, and supercritical regimes of the quantum contact process. In contrast, space-time records O of the output from continuous monitoring [here the real part of the complex heterodyne current, Eq. ( 4)] appear noisy and structureless throughout all dynamical regimes. While trajectories of the order parameter are usually inaccessible due to postselection overheads, trajectories of the output signal are directly available in experiments. As we show, they can provide useful information to detect nonequilibrium phase transitions.
and classification of many-body phases in Bose-Hubbard quantum simulators [37][38][39][40] and, at a theoretical level, also for the quantum contact process [35]. However, in these works system observables with a direct connection to order parameters (density or density-density correlations) were considered, or measured. In contrast, our approach exploits the entire space-time record of the measured output, which is not linked in an apparent way with the order pa
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