Low-Dimensional Structure in Quantum Variability: Evidence from State-Space Reconstruction in Monitored Quantum Systems This study investigates whether quantum variability, typically interpreted as intrinsically stochastic, exhibits underlying dynamical structure. Using publicly available data from monitored quantum circuits, we analyze time-ordered observables across circuit depth and evaluate their predictability using state-space reconstruction techniques. By constructing trajectories from measurement outcomes and comparing first-order and second-order embeddings, we test whether future states can be inferred from prior states. Both linear and nonlinear models are employed, and results are evaluated using root mean squared error (RMSE). To ensure that observed structure is not a statistical artifact, all results are compared against shuffled controls that preserve marginal distributions while removing ordered dependence. We find that prediction error for real data is significantly lower than for shuffled controls, demonstrating the presence of genuine temporal structure. Incorporating prior state information improves predictive performance consistently, indicating measurable memory or path dependence. However, the improvement beyond first-order embedding is modest, and nonlinear models provide only limited additional gains. These results show that quantum variability in the monitored system is not fully characterized by featureless stochastic behavior. Instead, it exhibits low-dimensional, weakly nonlinear, and memory-dependent structure, with most predictive information contained in the immediately preceding state. This work provides empirical evidence that quantum variability can be described as a constrained dynamical process governed by a small number of effective degrees of freedom. While the probabilistic framework of quantum mechanics remains operationally valid, the findings suggest that observed randomness may reflect structured underlying dynamics rather than purely intrinsic indeterminacy.
Low-Dimensional Structure in Quantum Variability: Evidence from State-Space Reconstruction in Monitored Quantum Systems This study investigates whether quantum variability, typically interpreted as intrinsically stochastic, exhibits underlying dynamical structure. Using publicly available data from monitored quantum circuits, we analyze time-ordered observables across circuit depth and evaluate their predictability using state-space reconstruction techniques. By constructing trajectories from measurement outcomes and comparing first-order and second-order embeddings, we test whether future states can be inferred from prior states. Both linear and nonlinear models are employed, and results are evaluated using root mean squared error (RMSE). To ensure that observed structure is not a statistical artifact, all results are compared against shuffled controls that preserve marginal distributions while removing ordered dependence. We find that prediction error for real data is significantly lower than for shuffled controls, demonstrating the presence of genuine temporal structure. Incorporating prior state information improves predictive performance consistently, indicating measurable memory or path dependence. However, the improvement beyond first-order embedding is modest, and nonlinear models provide only limited additional gains. These results show that quantum variability in the monitored system is not fully characterized by featureless stochastic behavior. Instead, it exhibits low-dimensional, weakly nonlinear, and memory-dependent structure, with most predictive information contained in the immediately preceding state. This work provides empirical evidence that quantum variability can be described as a constrained dynamical process governed by a small number of effective degrees of freedom. While the probabilistic framework of quantum mechanics remains operationally valid, the findings suggest that observed randomness may reflect structured underlying dynamics rather than purely intrinsic indeterminacy.
Quantum entanglement is traditionally described as a statistical correlation encoded in a many-body wavefunction. While this framework successfully predicts measurement outcomes, it leaves unresolved whether entanglement reflects purely probabilistic structure or an intrinsic physical organization of quantum states. Recent experiments have demonstrated that high-dimensional entangled states possess thousands of distinct, experimentally accessible topological invariants that persist under basis changes, noise, and local perturbations. These features are difficult to reconcile with a purely statistical interpretation of the wavefunction, but arise naturally if entanglement corresponds to intrinsic geometric structure within state space. This work analyzes experimental results reported in Nature Communications (2025) showing robust, high-dimensional topological structure embedded in conventional entangled states. We show that the multiplicity, stability, and basis-independence of these invariants support an interpretation of entanglement as intrinsic geometry rather than as observer-dependent probability. Under this interpretation, nonlocal correlations reflect shared geometric structure rather than information transfer, and robustness arises from topological protection rather than statistical averaging. The paper does not modify the formal machinery of quantum mechanics, but reframes the physical meaning of the quantum state. Entanglement is interpreted as a geometric configuration of collective modes whose topology constrains allowed correlations. This perspective provides a natural physical explanation for robustness, nonlocality, and measurement independence, and places strong constraints on interpretations in which the wavefunction is purely epistemic. Together with independent results in condensed matter and gravitational physics, these findings support a geometry-first interpretation of quantum phenomena and contribute to a growing body of experimental evidence that quantum states possess intrinsic physical structure beyond probabilistic description.
This project introduces a computational framework for quantum trajectory reconstruction using weak measurements and probabilistic inference. In standard quantum mechanics, physical systems are described by a wavefunction, and observable quantities are inherently probabilistic. As a result, particle trajectories are not directly accessible. This work proposes an operational approach in which trajectories are treated as latent variables inferred from noisy measurement data. The framework combines three core components: A numerical solver for the time-dependent Schrödinger equation (using FFT-based split-operator methods) A weak measurement model that simulates continuous, noisy observations of the system Inference engines, including Sequential Monte Carlo (particle filtering) and recurrent neural networks, to reconstruct the most probable trajectory The central idea is to reformulate quantum dynamics as an inference problem over hidden states, where the goal is to estimate the trajectory that best explains the observed measurements. This approach does not violate the principles of quantum mechanics, including the Principe d'incertitude de Heisenberg, but instead provides a new computational lens through which quantum systems can be analyzed. Potential applications include: Quantum sensing and metrology Quantum control and error correction Hybrid AI-quantum systems Advanced simulation of open quantum systems The project is fully reproducible and open-source, with modular architecture designed for research extensibility.
At the heart of physics lies the human fascination with natural phenomena and the wish to understand the underlying principles. A well-known example is the butterfly effect. It describes how small changes in the initial state can cause significantly different measurement outcomes at later times. Such sensitivity to initial conditions is a key signature of chaotic systems and reflects their instability under small perturbations. The principle underlying this phenomenon is information spreading: with time, information about the perturbation propagates through the system, changing measurement results even at distant locations. In quantum systems, information not only spreads but can also scramble, meaning that the precise details about the perturbation are hidden in highly nonlocal correlations. Information scrambling has emerged as a universal feature of quantum chaos and, together with other universal features, serves as an organizing principle for quantum systems. This motivates imposing structure like locality on quantum systems in order to probe the capabilities and limits of such organizing principles. There have been significant advances in both theoretical and experimental directions in recent years. The newly gained flexibility in designing experiments and simulations also inspires the analytical study of quantum systems from the bottom up, where tractable models offer insights into the generic features of quantum mechanics. In this thesis we build upon these advances to explore the implications of adding structure to chaotic quantum systems for the behavior of such features. In particular, we consider the measures entanglement entropy and Krylov complexity as probes of structure. These quantities are closely connected to information spreading. This motivates us to study the influence of structure on information spreading in the second part. In the first part, we begin with a detailed discussion of the behavior of entanglement in chaotic systems. We impose a minimal amount of local structure by separating the system into two parts, each modeled as a single spin. We validate the predicted entanglement for typical states in such systems and show that it saturates the Page limit as the Hilbert space dimension increases. We then study how different initially unentangled product states approach the Page limit under time evolution. We consider three types of product states, each representing a distinct case. Random product states represent generic states only restricted by the absence of entanglement. Random dipole and quadrupolar states represent two distinct types of semi-classical structure. The results on the entanglement generation in the three different types of states refine our earlier conclusion, demonstrating that the semi-classical limit can suppress the generation of extensive entanglement. This suppression does not occur for random product or quadrupolar states; where the imposed structure is insufficient. We conclude the second chapter by presenting two protocols that can stabilize the dynamics at finite spin sizes, similar to the semi-classical limit. In the following chapter, we move from measuring the complexity of time-evolved states using entanglement entropy to investigating time-evolved operators through the lens of Krylov complexity. After a short introduction to Krylov complexity for Hamiltonian dynamics, we generalize the framework for discrete time evolution. We identify the linear growth of Krylov complexity as a universal feature of generic systems and analyze its behavior as we transition from chaotic to increasingly structured systems. Dual-unitary circuits converge toward this universal behavior at rates that can be as fast as exponential, consistent with their classification as maximally chaotic. We then approximately restore energy conservation by considering trotterized systems, ranging from unstructured and chaotic to highly structured and integrable ones. We find these systems to thermalize on a scale set by the trotterization step, slowly approaching the universal linear Krylov complexity growth regime. Approaching the linear growth regime is a fundamental property of trotterized systems with an unbounded spectrum, while systems with a bounded energy spectrum can evade thermalization as probed by Krylov complexity. Instead, they have a nonanalytic transition as the Trotter step size increases. Our study of Krylov complexity establishes its universal properties for discrete time evolution. The presented results deepen our understanding of the thermalizing behavior of different classes of systems in relation to their spectral properties by introducing an intuitive notion of a bath with chaotic characteristics. We link this thermalization behavior to information spreading, which we explore in the second part of this thesis. Information scrambling, along with the associated spreading, is a key phenomenon in chaotic systems with local structure. In the fourth chapter, we investigate information spreading in proximity to an integrable system exhibiting spontaneous symmetry breaking, the transverse field Ising model (TFIM). As expected, integrability prevents the scrambling of information. At the same time, information spreading probed by quenched out-of-time ordered correlators (OTOCs) exhibits features reminiscent of the quantum phase transition present in the TFIM. We unravel the intricate behavior of these OTOCs and identify the underlying mechanism that stabilizes the long-time values of OTOCs: dynamical relocalization. We provide two variations of the TFIM in which the relocalization phenomenon is not directly related to the presence of a quantum phase transition. This shows that no direct, universal relation exists between the two. Finally, we discuss the effects of breaking the integrable structure of the TFIM. Our results show that for short times, chaotic systems in proximity of integrability can retain signatures of integrability. In contrast, we observe significant changes at large times. This suggests that information scrambling, as a universal feature of chaotic systems, is restored at large times—even in systems near integrability that exhibit order. Following the analysis of information spreading close to integrable regimes, we study the stability of information spreading in more generic systems. First, we consider charge-conserving random circuits. In these systems, charge-charge correlators exhibit large fluctuations across different circuit instances with a slow polynomial decay in time. At the same time, they remain stable against fine-tuned perturbations, which can be interpreted as dephasing noise in the eigenbasis of the charge density operators. Equivalently, charge-charge correlators are mostly insensitive to decoherence, with interference effects contributing only subleading corrections. In this sense, the charge information spreading probed by charge-charge correlators is effectively classical. We perform a detailed analysis and quantify the dephasing insensitivity of these correlators. This suggests that circuit-to-circuit fluctuations can be captured by considering small regions where “quantum” effects are included, embedded within otherwise “classical” dynamics. This observation motivates the development of an algorithm to compute charge-charge correlators for individual circuits on classical computers with high accuracy. We quantitatively relate the algorithm’s performance to the dephasing analysis. We demonstrate the algorithm’s accuracy numerically for charge-charge correlators. Then, we argue that in generic circuits without conservation laws, the information spreading, as probed by OTOCs, follows similar principles. We conclude by comparing OTOC values measured on quantum devices with up to 95 superconducting qubits and 1000 gates to predictions from the approximate algorithm. This cross-validates experimental measurements and theoretical predictions. In this thesis, we explore the implications of adding structure to chaotic quantum systems, contributing to a deeper understanding of the organizing principles governing quantum chaos. We establish that entanglement can be suppressed in the semiclassical limit and that Krylov complexity behaves differently in thermalizing or integrable time-dependent systems. We show that the phenomenon information scrambling is also present in chaotic quantum systems even in the vicinity of a quantum phase transition. Conversely, when charge conservation is added, information spreading is stabilized and becomes reminiscent of that in classical systems. This work opens up further research directions towards a better understanding of the quantum nature of chaotic quantum systems and the development of probes of chaos in time dependent systems with different types of structure.
Recently we have presented a hidden-variable model of measurements for a qubit where the hidden-variable state-space dimension is one-half the quantum-state manifold dimension. The absence of a short memory (Markov) dynamics is the price paid for this dimensional reduction. The conflict between having the Markov property and achieving the dimensional reduction was proved by Montina [A. Montina, Phys. Rev. A 77, 022104 (2008)] using an additional hypothesis of trajectory relaxation. Here we analyze in more detail this hypothesis introducing the concept of invertible process and report a proof that makes clearer the role played by the topology of the hidden-variable space. This is accomplished by requiring suitable properties of regularity of the conditional probability governing the dynamics. In the case of minimal dimension the set of continuous hidden variables is identified with an object living an N-dimensional Hilbert space whose dynamics is described by the Schr\"odinger equation. A method for generating the economical non-Markovian model for the qubit is also presented.
In this work, we present a relational reinterpretation of the Schrödinger equation in which quantum evolution is understood as arising from the balance between competing tendencies toward propagation and localization. Within this framework, the Hamiltonian is re-expressed as an effective generator of relational tension, and quantum measurement is interpreted as a transition into a localization-dominated regime rather than a fundamental collapse. To test this interpretation, we analyze publicly available experimental data from monitored quantum circuits exhibiting measurement-induced transitions (Feng et al., Communications Physics, 2026). Using the merged dataset (k4-z-data-merged.npy), we examine the structure of variability in the experimentally defined order parameter across measurement strength and circuit depth. The analysis shows that variability is not adequately described by a single control parameter. Instead, it exhibits clear multi-variable structure, with significant dependence on circuit depth. Furthermore, incorporating prior-state information—specifically, the observable value at the previous depth—improves predictive accuracy, demonstrating that the system exhibits measurable path dependence. At the same time, a structured residual component remains, indicating that the system is only partially predictable within the available variable set. These findings show that quantum variability in this system is not featureless or purely stochastic, but instead reflects organized, multi-variable, and history-dependent structure. This result does not alter the formal predictions of quantum mechanics. Rather, it provides a data-supported interpretation in which quantum uncertainty is understood as the projection of deeper relational dynamics into a reduced descriptive framework. The presence of path dependence and partial predictability suggests that variability arises from partially resolved structure rather than irreducible randomness.
Quantum state tomography (QST) is a fundamental task in quantum information science that aims to reconstruct unknown quantum states from measurement data. However, the exponential growth of Hilbert-space dimension with system size makes full tomography of general quantum states statistically and computationally prohibitive. This challenge has motivated extensive research on structured quantum state tomography, where prior structure, such as low-rankness, tensor-network representations, shallow quantum circuits, and neural quantum states, can substantially reduce the effective degrees of freedom and enable scalable recovery. In this review, we provide a unified perspective on QST for structured quantum states through three closely related themes: compact state representations, measurement design, and computational algorithms. After reviewing common models for structured quantum states, we survey existing work on geometric preservation properties of measurement frameworks, ranging from informationally complete POVMs to randomized measurements, and their implications for sample complexity. On the algorithmic side, we review optimization methods for reconstructing structured quantum states from empirical measurements. By connecting QST with broader principles from compressive sensing, matrix sensing, and structured inverse problems, this survey highlights common theoretical foundations underlying sample complexity, measurement efficiency, and scalable recovery.