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.
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.
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.
Quantum computers and simulators offer unparalleled capabilities of probing quantum many-body states, by obtaining snapshots of the many-body wave function via collective projective measurements. The probability distribution obtained by such snapshots (which are fundamentally limited to a negligible fraction of the Hilbert space) is of fundamental importance to determine the power of quantum computations. However, its relation to many-body collective properties is poorly understood. Here, we develop a theoretical framework to link quantum phases of matter to their snapshots, based on a combination of data complexity and network theory analyses. The first step in our scheme consists of applying Occam's razor principle to quantum sampling: given snapshots of a wave function, we identify a minimal-complexity measurement basis by analyzing the information compressibility of snapshots over different measurement bases. The second step consists of analyzing arbitrary correlations using network theory, building a wave-function network from the minimal-complexity basis data. This approach allows us to stochastically classify the output of quantum computers and simulations, with no assumptions on the underlying dynamics, and in a fully interpretable manner. We apply this method to quantum states of matter in one-dimensional translational invariant systems, where such classification is exhaustive, and where it reveals an interesting interplay between algorithmic and computational complexity for many-body states. Our framework is of immediate experimental relevance, and can be further extended both in terms of more advanced network mathematics, including discrete homology, as well as in terms of applications to physical phenomena, such as time-dependent dynamics and gauge theories.
Based on the references [P1–P6], we investigate in this cumulative PhD thesis the dynamics and transport in closed and open interacting spin chain models. We focus on (i) the connection between the open and the closed system dynamics [P1, P2], (ii) the influence of various system parameters on the transport behavior in closed and open systems [P3, P5], and in addtion (iii) the dynamics and transport in closed models [P4, P6]. As for (i), we explore the connection of the dynamics in open systems, modeled by the Lindblad equation, and the correlation functions in the closed system [P1]. Building on this connection for magnetization transport, we study whether the time evolution of the open quantum system can be described in terms of classical correlation functions generated by the Hamiltonian equations of motion for real vectors [P2]. Second, we study in (ii) the influence of different parameters on the diffusion constants in closed and open systems [P3, P5]. In Ref. [P3], we study the impact of integrability-breaking perturbations in both closed and open spin chain systems. While we find a continuous change of the diffusion constant with perturbation strength in the closed system, in the open system, in the weak-perturbation limit, we indicate the possibility of a diverging diffusion constant. Also, we investigate the influence of the system-bath coupling on the transport behavior in boundary-driven open systems [P5]. We observe that the diffusion constants depend on the system-bath coupling and that in the nonequilibrium steady state, they inevitably retain finite-size effects, limiting the computation of long-time transport coefficients. Additional for (iii), in Ref. [P4], we first examine the impact of perturbations on the dynamics in closed spin chain models. We employ projector-operator techniques to understand, on perturbative grounds, the nontrivial damping of the standard correlation function. Second, we also discuss a dynamical quantum typicality approach for studying the energy transport in bipartite systems at low temperatures [P6].
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.
Real-time dynamics of quantum observables provide direct access to excitation spectra and correlation functions in quantum many-body systems, but currently available quantum devices are limited to short evolution times due to decoherence. We propose a neural ordinary differential equation (Neural ODE) framework with a physics-driven neural network to reconstruct long-time operator dynamics from short-time quantum data. By expanding observables in the Pauli basis and exploiting locality and symmetry constraints, the operator evolution is reduced to a tractable set of coefficients whose dynamics are learned from the measurement data. Applied to the Ising spin chain with mixed fields, the operator learning method accurately extrapolates its long-time correlation function and resolves its excitation spectrum. We validate the framework by collecting short-time data from IBM's superconducting quantum processor and successfully reconstruct the Fourier spectrum of an N=8 Ising spin chain with 5 times enhanced precision. Our results demonstrate a scalable and data-efficient strategy for extracting dynamical and spectral information from practical quantum hardware.