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.
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.
We study nonequilibrium quantum dynamics of spin chains by employing principal component analysis on data sets of wave function snapshots and examine how information propagates within these data sets. The quantities we employ are derived from the spectrum of the sample second moment matrix, built directly from data sets. Our investigations on several interacting spin chains featuring distinct spin or energy transport reveal that the growth of data information spreading follows the same dynamical exponents as that of the underlying quantum transport of spin or energy. Specifically, our approach enables an easy, data-driven, and, importantly, interpretable diagnostic to track energy transport with a limited number of samples, which is usually challenging without any assumption on the Hamiltonian form. These observations are obtained at a modest finite-size and evolution time, which aligns with experimental and numerical constraints. Our framework directly applies to experimental quantum simulator data sets of dynamics in higher-dimensional systems, where classical simulation methods usually face significant limitations and apply equally to both near- and far-from-equilibrium quenches.
Analog quantum simulators natively return large collections of single-shot projective measurements, the wave-function snapshots. These are high-dimensional configurations drawn from the Born distribution of the state they prepare. These collections carry far more information than is retained when they are simply compressed into low-order observables fixed in advance. Moreover, this procedure requires knowing beforehand what one is looking for. This thesis chooses to take a different route, asking what structure a dataset of snapshots possesses as a set of points in configuration space, and what kind of physics that structure reflects. The tools used here are drawn from un supervised learning and network theory, namely principal component analysis (PCA), intrinsic-dimension estimators (TWO-NN and BID), and the wave-function networks obtained by promoting snapshots to nodes and activating links according to the met ric structure of configuration space. They are chosen under two constraints, namely interpretability at every step, and the smallest possible amount of physical input: no order parameter, no symmetry, and no knowledge of the Hamiltonian. At equilibrium, wave-function networks are used to build a classification of one dimensional quantum phases based on snapshots alone. The scheme first selects, in the spirit of Occam’s razor, the measurement basis carrying the minimal information content, and then analyses the networks constructed in that basis. Paramagnetic and symmetry-broken phases display low complexity and “probability” networks, symmetry protected topological phases high complexity and Erdős–Rényi networks, and critical points high complexity and scale-free networks. Interestingly, gapped and gapless mat ter are treated on the same footing, and real-space decimation enters the scheme to resolve massive phases rather than flowing towards criticality. Out of equilibrium, the dynamical exponent governing transport of a conserved quan tity is shown to be imprinted on the singular-value spectrum of the snapshot matrix. In particular, it is shown that the PCA entropy, and in fact the leading eigenvalue alone, grows with the same exponent as the underlying spin or energy transport, at modest system sizes, evolution times and sample numbers. Ergodicity breaking is then ad dressed through the binary intrinsic dimension (BID) of the snapshots, in the quantum sun model: the BID grows and saturates at the Thouless time, reaching its maximum in the ergodic phase, remaining system-size independent in the non-ergodic one, and growing sub-maximally at criticality, where the data occupy an extensive but lower dimensional manifold. The associated networks reproduce the equilibrium dichotomy, being random in the ergodic phase and scale-free at the transition, whose location can be identified by the BID within a few percent of spectral estimates. Preliminary experimental observations, not reported here, show analogous features. Taken together, these results indicate that datasets orders of magnitude smaller than the Hilbert-space dimension suffice to identify collective properties of quantum matter through the study of the structure of the data points in configuration space. Moreover, the same structural classes recur across ground states and quenched dynamics, with no order parameter supplied and under realistic experimental imperfections. The final chapter elaborates on another aspect of quantum simulator capabilities, that is, that to realize gauge theory dynamics. There, one-dimensional U(1) lattice gauge theories with a topological θ term are mapped exactly onto kinetically con strained fermions in an emergent Stark potential. The mapping accounts for the slow relaxation reported in recent Rydberg experiments: prethermalization near an inte grable point at θ = 0, and Stark localization with Hilbert-space fragmentation at θ ̸ =0.
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.
Principal component analysis (PCA) has been widely adopted to reduce the dimension of data while preserving the information. The quantum version of PCA (qPCA) can be used to analyze an unknown low-rank density matrix by rapidly revealing the principal components of it, i.e., the eigenvectors of the density matrix with the largest eigenvalues. However, because of the substantial resource requirement, its experimental implementation remains challenging. Here, we develop a resonant analysis algorithm with minimal resource for ancillary qubits, in which only one frequency-scanning probe qubit is required to extract the principal components. In the experiment, we demonstrate the distillation of the first principal component of a 4 × 4 density matrix, with an efficiency of 86.0% and a fidelity of 0.90. This work shows the speedup ability of quantum algorithm in dimension reduction of data and thus could be used as part of quantum artificial intelligence algorithms in the future.
Abstract Kernel principal component analysis (kernel PCA) is a nonlinear dimensionality reduction technique that employs kernel functions to map data into a high-dimensional feature space, thereby extending the applicability of linear PCA to nonlinear data and facilitating the extraction of informative principal components. However, kernel PCA necessitates the manipulation of large-scale matrices, leading to high computational complexity and posing challenges for efficient implementation in big data environments. Quantum computing has recently been integrated with kernel methods in machine learning, enabling effective analysis of input data within intractable feature spaces. Although existing quantum kernel PCA proposals promise exponential speedups, they impose stringent requirements on quantum hardware that are challenging to fulfill. In this work, we propose a quantum algorithm for kernel PCA by establishing a connection between quantum kernel methods and block encoding, thereby diagonalizing the centralized kernel matrix on a quantum computer. The query complexity is logarithmic with respect to the size of the data vector, D , and linear with respect to the size of the dataset. An exponential speedup could be achieved when the dataset consists of a few high-dimensional vectors, wherein the dataset size is polynomial in log ( D ) , with D being significantly large. In contrast to existing work, our algorithm enhances the efficiency of quantum kernel PCA and reduces the requirements for quantum hardware. Furthermore, we have also demonstrated that the algorithm based on block encoding matches the lower bound of query complexity, indicating that our algorithm is nearly optimal. Our research has laid down new pathways for developing quantum machine learning algorithms aimed at addressing tangible real-world problems and demonstrating quantum advantages within machine learning.
This thesis investigates universality and complexity in classical and quantum statistical physics, with a particular emphasis on nonequilibrium phenomena. While the emergence of universal behavior in equilibrium systems stands as one of the cornerstones of statistical physics, the extension of these ideas to nonequilibrium settings presents fundamental and still open challenges. The inherently diverse and system-specific nature of nonequilibrium dynamics calls for a broad spectrum of conceptual and analytical tools. Furthermore, nonequilibrium dynamics is nowadays accessible to experiments in both classical and quantum regimes. These modern experiments produce a large amount of experimental data, which calls for analysis of physics through data-driven methods, complexity theory and high-dimensional statistics. The possibility of probing and accessing correlated states of matter out of equilibrium in modern experiments raises several questions. Can we understand the output of experiments in terms of complexity theory and dimensional reduction? Is it possible to extend concepts of universality to data structures arising in simulations? Can we describe phase transitions (either classical or quantum) concepts of modern data science? How to control entanglement in quantum dynamics? Which nonequilibrium phenomena can we hope to observe in quantum simulators? The research work presented in this thesis is an attempt to answer to these questions. In this thesis, we argue that order parameters and diagnostics of criticality in nonequilibrium classical systems can be found by unsupervised learning and dimensional reduction, which allow to reduce the complexity of datasets by projections on lower-dimensional spaces, while still preserving essential universal features. We then move to the study of models of quantum dynamics relevant for quantum simulators. In particular, we show that the same concepts of dimensional reduction apply also to quantum systems, allowing for the extraction of information from outputs of quantum simulators. We then study a protocol to prepare complex, entangled states via unitary evolution and enhance entanglement via measurements in a fermionic model. Finally, we study the emergence of anomalous transport in one-dimensional chiral systems with non-Abelian symmetries.