Abstract Dynamical probes are popular experimental tools for studying quantum systems nowadays. Though a well-defined eigen-mode will manifest itself as a single frequency oscillation in the dynamical probe, the real experimental data can be quite messy because a probe usually excites multiple modes which are also mixed together with random white noise. Moreover, the oscillations usually quickly damp out due to finite decoherence time. These make it difficult to extract the frequencies of these oscillation measurement data, that is, the eigen-energies of these eigen-modes. Here we develop an unsupervised machine learning algorithm to solve this problem. Our method is inspired by the independent component analysis method and its application to the ‘cocktail party problem’, where the goal is to recover each voice from detectors that detect signals of many mixed voices. We demonstrate the advantage of our method by an example of analyzing the collective mode of a Bose–Einstein condensate of atomic gases. We believe that this method can find broad applications in analyzing data of dynamical experiments in quantum systems of different fields.
Abstract Dynamical probes are popular experimental tools for studying quantum systems nowadays. Though a well-defined eigen-mode will manifest itself as a single frequency oscillation in the dynamical probe, the real experimental data can be quite messy because a probe usually excites multiple modes which are also mixed together with random white noise. Moreover, the oscillations usually quickly damp out due to finite decoherence time. These make it difficult to extract the frequencies of these oscillation measurement data, that is, the eigen-energies of these eigen-modes. Here we develop an unsupervised machine learning algorithm to solve this problem. Our method is inspired by the independent component analysis method and its application to the ‘cocktail party problem’, where the goal is to recover each voice from detectors that detect signals of many mixed voices. We demonstrate the advantage of our method by an example of analyzing the collective mode of a Bose–Einstein condensate of atomic gases. We believe that this method can find broad applications in analyzing data of dynamical experiments in quantum systems of different fields.
In many dynamical probes of a quantum system, quite often multiple eigenmodes are excited. Therefore, the experimental data can be quite messy due to the mixing of different modes, as well as the background noise, despite that each mode manifests itself as a single frequency oscillation. Here we develop an unsupervised machine learning algorithm to extract the frequencies of these oscillations from such measurement data, that is, the eigenenergies of these modes. This method is particularly useful when the measurement time is not long enough to perform the Fourier transformation. Our method is inspired by the independent component analysis method and its application to the cocktail party problem. In that problem, the goal is to recover each voice from detectors that detect signals of many mixed voices, and the principle is to find out signals that possess features and are away from a Gaussian distribution. Instead, our generalization is to find out signals that are close to a single frequency oscillation. We demonstrate the advantage of our method by an example of analyzing the collective mode of cold atoms. We believe this method can find broad applications in analyzing data from dynamical experiments in quantum systems.
To extract useful information about quantum effects in cold atom experiments, one central task is to identify the intrinsic fluctuations from extrinsic system noises of various kinds. As a data processing method, principal component analysis can decompose fluctuations in experimental data into eigenmodes, and give a chance to separate noises originated from different physical sources. In this paper, we demonstrate for Bose-Einstein condensates in one-dimensional optical lattices that the principal component analysis can be applied to time-of-flight images to successfully separate and identify noises from different origins of leading contribution, and can help to reduce or even eliminate noises via corresponding data processing procedures. The attribution of noise modes to their physical origins is also confirmed by numerical analysis within a mean-field theory. As the method does not rely on any a priori knowledge of the system properties, it is potentially applicable to the study of other quantum states and quantum critical regions.
For a linear system, the response to a stimulus is often superposed by its responses to other decomposed stimuli. In quantum mechanics, a state is the superposition of multiple eigenstates. Here, by taking advantage of the phase difference, a common feature as we identified in data sets, we propose eigen component analysis (ECA), an interpretable linear learning model that incorporates the principle of quantum mechanics into the design of algorithm design for feature extraction, classification, dictionary and deep learning, and adversarial generation, etc. The simulation of ECA, possessing a measurable class-labelH, on a classical computer outperforms the existing classical linear models. Eigen component analysis network (ECAN), a network of concatenated ECA models, enhances ECA and gains the potential to be not only integrated with nonlinear models, but also an interface for deep neural networks to implement on a quantum computer, by analogizing a data set as recordings of quantum states. Therefore, ECA and ECAN promise to expand the feasibility of linear learning models, by adopting the strategy of quantum machine learning to replace heavy nonlinear models with succinct linear operations in tackling complexity.
In modern machine learning circuits and systems, the need for centralized, standardized data poses significant challenges, especially in privacy-sensitive, resource-constrained, real-time, or asynchronous environments. We introduce Eigen-Component Analysis (ECA), a quantum-inspired linear model that circumvents these requirements by leveraging intrinsic data structures. ECA uses an orthogonal transformation to extract meaningful features from decentralized, non-standardized data, ensuring high classification accuracy without extensive preprocessing. Unlike traditional linear models, ECA adapts to complex data distributions and shows superior performance in feature extraction and classification across various datasets. Experiments indicate that ECA achieves robust accuracy with fewer parameters, highlighting its potential where data centralization and standardization are impractical. Our work extends the feasibility of linear models in diverse scenarios, enhancing interpretability and efficiency in challenging data environments.
Advanced data analysis techniques have proved to be crucial for extracting information from noisy images. Here we show that principal component analysis can be successfully applied to ultracold gases to unveil their collective excitations. By analyzing the correlations in a series of images we are able to identify the collective modes which are excited, determine their population, image their eigenfunction, and measure their frequency. Our method allows us to discriminate the relevant modes from other noise components and is robust with respect to the data sampling procedure. It can be extended to other dynamical systems, including cavity polariton quantum gases and trapped ions.
Adaptive data collection and analysis, where data are being fed back to update the measurement settings, can greatly increase speed, precision, and reliability of the characterization of quantum systems. However, decoherence tends to make adaptive characterization difficult. As an example, we consider two coupled discrete quantum systems. When one of the systems can be controlled and measured, the standard method to characterize another, with an unknown frequency ${\ensuremath{\omega}}_{\mathrm{r}}$, is swap spectroscopy. Here, adapting measurements can provide estimates whose error decreases exponentially in the number of measurement shots rather than as a power law in conventional swap spectroscopy. However, when the decoherence time is so short that an excitation oscillating between the two systems can only undergo less than a few tens of vacuum Rabi oscillations, this approach can be marred by a severe limit on accuracy unless carefully designed. We adopt machine learning techniques to search for efficient policies for the characterization of decohering quantum systems. We find, for instance, that when the system undergoes more than 2 Rabi oscillations during its relaxation time T1,O(103) measurement shots are sufficient to reduce the squared error of the Bayesian initial prior of the unknown frequency ${\ensuremath{\omega}}_{\mathrm{r}}$ by a factor O(104) or larger. We also develop policies optimized for extreme initial parameter uncertainty and for the presence of imperfections in the readout.