Understanding and harnessing the power of quantum systems has the potential to transform many domains in science and technology. However, before we can achieve these aspirations, we must first build a better understanding of how quantum systems fundamentally behave. In this thesis, we approach this question through the lens of learning theory to develop new paradigms for learning about quantum systems and understanding their structural properties. We deliver several surprising results, upending previous beliefs about even fundamental laws and giving provably efficient algorithms for learning about quantum systems in settings previously conjectured to be intractable. Typically in quantum many-body systems, the particles in the system interact locally with respect to some geometry as described by a local Hamiltonian. Two key questions are first, understanding equilibrium properties of a system with a given Hamiltonian and second, recovering the Hamiltonian from measurements of the properties of the system. For the first, we prove a universal law that there is a sudden death of entanglement, at a critical temperature depending only on the geometry but not on the system size. For the second, we give the first efficient algorithm for recovering the Hamiltonian at any temperature, breaking a conjectured barrier at low temperatures. Beyond systems with local interactions, we also consider learning and testing properties of general quantum states, focusing on the interplay between statistical complexity and near-term quantum device constraints, only allowing for entangled measurements over a limited number of copies of the state. We characterize the optimal rates for learning and testing with single-copy measurements and for multi-copy measurements in many relevant near-term regimes.
Understanding and harnessing the power of quantum systems has the potential to transform many domains in science and technology. However, before we can achieve these aspirations, we must first build a better understanding of how quantum systems fundamentally behave. In this thesis, we approach this question through the lens of learning theory to develop new paradigms for learning about quantum systems and understanding their structural properties. We deliver several surprising results, upending previous beliefs about even fundamental laws and giving provably efficient algorithms for learning about quantum systems in settings previously conjectured to be intractable. Typically in quantum many-body systems, the particles in the system interact locally with respect to some geometry as described by a local Hamiltonian. Two key questions are first, understanding equilibrium properties of a system with a given Hamiltonian and second, recovering the Hamiltonian from measurements of the properties of the system. For the first, we prove a universal law that there is a sudden death of entanglement, at a critical temperature depending only on the geometry but not on the system size. For the second, we give the first efficient algorithm for recovering the Hamiltonian at any temperature, breaking a conjectured barrier at low temperatures. Beyond systems with local interactions, we also consider learning and testing properties of general quantum states, focusing on the interplay between statistical complexity and near-term quantum device constraints, only allowing for entangled measurements over a limited number of copies of the state. We characterize the optimal rates for learning and testing with single-copy measurements and for multi-copy measurements in many relevant near-term regimes.
Learning the Hamiltonian of a many-body system from its dynamics is a central task in quantum science, yet the algorithms with the strongest provable guarantees assume some level of quantum control--fast, arbitrary single-qubit gates interleaved with time evolution, and measurements in arbitrary bases--that is beyond the capabilities of near-term analog quantum simulators. Motivated by analog atom- and ion-based platforms, we study Hamiltonian learning under minimal access models.
Uniform state preparation and measurements: We first consider the setting where in every experiment, one can rotate each qubit to the same state, perform short-time evolution, and measure every qubit in the same basis. Surprisingly, we show that for generic 2-local Hamiltonians on any interaction graph, all of the parameters can be reconstructed from such experiments.
Computational basis state preparation and measurements: We then consider a similarly constrained setting, but where state preparation and measurement are restricted to the computational basis. For nearest-neighbor Hamiltonians with only Pauli X/Z interactions, a class which captures contemporary Rydberg atom platforms, we show that over 1D and 2D rectangular lattices, all of the parameters can be reconstructed from such experiments up to unavoidable gauges.
Our protocols introduce new techniques for solving structured polynomial systems over an extensive number of parameters. Taken together, our results suggest that one can learn a great deal from the dynamics of quantum many-body systems even under the most stringent experimental constraints.
Out-of-equilibrium quantum many-body systems stand at the forefront of modern theoretical physics, addressing fundamental questions on thermalization, transport, and universal phenomena. These fields, already well established in condensed matter, statistical physics, quantum optics, and quantum information theory, are progressively gaining even greater relevance with the rapid development of quantum technologies and simulation, which inherently operate in dynamical regimes. In recent years, the traditional paradigm of unitary quantum evolution has been expanded to include measurements, opening new directions in out-of-equilibrium physics. At the core of these advances lie measurement-induced phase transitions (MIPTs), which have emerged as a new class of dynamical critical phenomena characterizing the general behavior of monitored quantum dynamics. When external monitoring intertwines with unitary evolution, many-body quantum correlations change their structure, giving rise to distinct entanglement phases of matter. This discovery has sparked enormous interest in MIPTs, leading to significant advances in open quantum systems, entanglement theory, and more broadly quantum complexity. Despite much progress, a full understanding of monitored many-body dynamics is far from complete, leaving several open questions on the nature of MIPTs, their experimental observability, and the possibilities offered by measurements to enhance control over synthetic quantum matter. These issues persist due to the intrinsic complexity of the problem and the lack of efficient tools to study it, mainly caused by the stochastic character of monitored evolution. This thesis addresses these challenges by expanding the investigation of measurement-induced phenomena in new settings and introducing innovative probes of entanglement and many-body quantum complexity for MIPTs. A core question we investigate is the role of symmetries, non-ergodicity, and especially integrability in measurement-induced criticality, which dramatically affect the non-equilibrium phases. We further explore how these phenomena extend beyond bipartite quantum correlations to multipartite entanglement and quantum non-stabilizerness, highlighting the non-trivial interplay between measurements and complexity notions rooted in quantum information theory. Finally, we focus on the compelling problem of decoherence, modeling how noise spoils entanglement structures. These findings, supported by advanced numerical simulations and theoretical analysis, deepen the current understanding of entanglement, complexity, and integrability in monitored quantum many-body systems, offering new perspectives on their rich behavior. In parallel, we address the experimental problem of dissipation in MIPTs, which is of key relevance for practical implementations. We anticipate the present investigation to foster future research on the nature of monitored dynamical critical phenomena and, more broadly, the applications of measurements in quantum state evolution.
Quantum mechanics led us to reconsider the scope of physics and its building principles, such as the notions of realism and locality. More recently, quantum theory has changed in an equally dramatic manner our understanding of information processing and computation. On one hand, the fundamental properties of quantum systems can be harnessed to transmit, store, and manipulate information in a more efficient and secure way than possible in the realm of classical physics. On the other hand, the development of systematic procedures to manipulate systems of a large number of particles in the quantum regime, crucial to the implementation of quantum-based information processing, has triggered new possibilities in the exploration of quantum many-body physics and related areas. In this thesis, we present new results relevant to two important problems in quantum information science: the development of a theory of entanglement, intrinsically quantum correlations, and the exploration of the use of controlled quantum systems to the computation and simulation of quantum many-body phenomena. In the first part we introduce a new approach to the study of entanglement
Recently, the possibility of investigating single molecule, or single spin observables, as well as the necessity of a better understanding of the mechanisms underlying quantum dynamics in order to obtain nanoscale devices and nanostructered materials suitable for quantum computing tasks, have revived the interest in foundational aspects of quantum statistical mechanics. This thesis aims to give a contribution to this field by re-considering the statistical characterization of a quantum system at the light of some paradigmatic changes in our understanding of quantum theory which have taken place in the last two decades. In particular the impressive development of quantum information theory has changed the perceptions of quantum entanglement: for a long time it has been considered a somewhat paradoxical property of the matter at the atomic scale, but now it is regarded as an essential and ubiquitous phenomenon whose consequences are affecting the very macroscopic world that we experience. Still the decoherence program has brought out the importance of considering a quantum system together with its environment in order to clarify some key aspects of quantum dynamics. Thus, we start from the idea that quantum correlations are ubiquitous and somewhat uncontrollable in systems with many degrees of freedom which are typically considered in statistical mechanics. As a consequence we assume the standpoint that quantum statistical mechanics has not to be based on the underlying idea of a collection of many, independent quantum systems but rather it has to emerge at the level of a global wavefunction (pure state) which describes the system as well as its environment as a whole. In order to investigate the consequences of these assumptions we study the equilibrium distribution of an isolated quantum system. This is defined, in analogy with the ergodic foundations of classical statistical mechanics, on the basis of the time evolution of the quantum state. Then, we study the emergence of thermodynamic properties in a quantum system by studying the probability distribution of some function of interests, as the entropy and the equilibrium state of a subsystem, on Ensembles of Pure States. Such a probability distribution is derived from the geometry of the Hilbert space, and the theoretical tools suitable for its characterization are developed. On the one hand we perform a numerical sampling of the ensemble distributions by employing Monte Carlo techniques, on the other hand simpler analytical approximation of the geometrical distributions are derived by means of a maximum entropy principle. Model systems composed of an ensemble of spins are chosen to illustrate the salient features which emerge from the developed theoretical framework: the main point is that the Ensemble Distributions of “thermodynamic observables” (entropy or equilibrium state of a subsystem) are sharply peaked around a typical value. From the analysis it emerges that each of the overwhelming majority of the wavefunctions which has appreciable weight in the considered ensemble, is characterized by the same value of the “macroscopic” functions. This is a striking evidence of the “typicality” of these properties. In the essence, our impossibility to know the state of the system in detail does not matter, just for the remarkable fact that almost all quantum states behave essentially in the same way. By virtue of this typicality the study of the behaviour of the typical values of the thermodynamic function become meaningful. Notably, under certain conditions, one recovers the results of standard statistical mechanics, that is, the equilibrium average of the state of a subsystem can be cast in the Boltzmann canonical form at the temperature given by the usual thermodynamical relation . In the second part of the thesis we consider the dynamical aspects of the equilibrium state of a subsystem interacting with its environment. The fluctuations around the equilibrium average critically depends on the entanglement between the system and the environment and on the form of the interaction Hamiltonian. The connection between the dynamics of the fluctuations of an observable at the equilibrium and the relaxation toward the equilibrium from a “non typical” initial value is also investigated with the aid of simple model systems. The study presented in this thesis was partly motivated by a critical analysis of the statistical methods available for the theoretical modelling of magnetic resonance experiments. One of these, the Stochastic Liouville Equation, has been employed in a work completed during the first year of my Ph.D. program in order to interpret some feature of a two dimensional electron spin resonance experiment, [Fresch B., Frezzato D., Moro G. J., Kothe G., Freed J. H.; J. Phys. Chem. B., 110, 24238, (2006)].
The task of learning the Hamiltonian interactions governing a quantum system, given samples of its thermal (or 'Gibbs') states, is a foundational question at the intersection of quantum learning theory and many-body physics. In this paper, we draw connections to the quantum Gibbs sampling literature to introduce a natural learning algorithm we call the stationarity test: which simply "guesses" the Hamiltonian, and measures the rate-of-change of local observables, under the associated detailed-balanced quantum Markov chain [CKG23]. We leverage the stationarity test to address the following applications:
1. To give the first learning algorithm for the underlying interaction graph, i.e. structure learning, of lattice Hamiltonians at all temperatures, given copies of their Gibbs states.
2. To give the first learning algorithm for the coefficients of a lattice Hamiltonian, given only copies of its thermal metastable states, modeled as the "local minima" of the free energy.
3. In addition, we present a refinement to the recent area law for thermal metastable states [BCV25], which holds in the thermodynamic limit.
Our learning algorithms are rigorous, time-efficient, and nearly sample-optimal in system size and accuracy. At a technical level, our arguments are based on new approximate locality and convexity properties for the quantum Fisher information of these Gibbs sampling algorithms.
This dissertation contributes to a better understanding of concepts, theories, and applications of classical and quantum information in various large-scale systems. The dissertation is structured in two parts: \n \nThe first half is concerned with classical systems: \n \nFirst, we study a basic-yet-never-fully-appreciated property of many real-world complex networks---the scale-free (SF) property---which was often considered to be only represented by the degree distribution. We define a new fundamental quantity, however, the degree--degree distance, which can better represent the SF property by showing statistically more significant power laws and better explain the evolution of real-world networks, e.g., Wikipedia Webpages. \n \nSecond, we study brain tractography of a healthy subject by diffusion-weighted magnetic resonance imaging (dMRI) data and find that the dependence of dMRI signal on the interpulse time can decode the smaller-than-resolution brain structure and might unravel how information transmits. This finding is confirmed by Monte-Carlo simulation of water-molecule diffusion which let us understand how to optimally measure the thickness of axon sheets in the brain. \n \nThe second half is concerned with quantum systems: \n \nOur first work is to understand how to establish long-distance entanglement transmission in a quantum network where each link has non-zero concurrence---a measure of bipartite entanglement. We introduce a fundamental statistical theory, concurrence percolation theory (ConPT), and find the existence of an entanglement transmission threshold predicted by ConPT which is lower than the known classical-percolation-based results---a “quantum advantage” that is more general and efficient than expected. ConPT also shows a percolation-like universal critical behavior derived by finite-size analysis. \n \nOur second work is to study continuous-time quantum walk as an open system that strongly interacts with the environment where non-Markovianity may significantly speed up the dynamics. We confirm this speed-up by first introducing a general multi-scale perturbation method that works on integro-differential equations and then building the Hamiltonian on regular networks, e.g., star or complete graphs, which can be mapped to an error correction algorithm scheme of practical significance. \n \nOur third work explores the possible use of entanglement entropy (EE) in machine-learning fields. We introduce a new long-short-term-memory-based recurrent neural network architecture using tensorization techniques to forecast chaotic time series, the learnability of which is determined not only by the number of free parameters but also the tensorization complexity---recognized as how EE scales.