The intersection of physics and information theory has advanced rapidly in the past few decades. Information-theoretic concepts have been employed to formulate physical laws, while quantum information processing, such as quantum computation and communication, has been developed both theoretically and experimentally. Simultaneously, understanding the universal or specific behaviors of quantum many-body systems has remained a central challenge since shortly after the inception of quantum mechanics. This is studied in part by statistical mechanics, which models complex many-body systems using simple ensembles. In this thesis, we investigate information-theoretic problems in quantum many-body dynamics that lie at the intersection of these avenues. To begin with, we investigate the extent of randomness, captured by unitary designs, in random quantum circuits. Quantum circuits serve as discrete models for random many-body quantum dynamics. Specifically, we study how the internal structure of random gates influences the generation of randomness in quantum circuits. We demonstrate that the spectral gap of structured random circuits can be larger than that of unstructured ones, implying that the former can converge to designs faster than the latter. In this analysis, free-fermionic techniques play a crucial role in obtaining exact results on the spectrum of second-moment operators. We then investigate the effect of quantum measurements on the quantum complexity in random circuits. Here, the quantum complexity of a state is defined as the minimum number of local gates required for its exact preparation in quantum circuits. We prove that in monitored random quantum circuits, the complexity undergoes a measurement-induced phase transition by leveraging techniques from percolation theory. Above the critical measurement rate, the complexity saturates quickly. Below the threshold, it grows at least linearly for an exponentially long time. Finally, we discuss quantum communication scenarios in noisy quantum dynamics, where we simulate the identity channel by combining a given noisy quantum dynamics and our quantum operational strategies. Specifically, we consider multi-time noisy processes and optimize our strategies to simulate near-noiseless channels, both of which are mathematically formulated using quantum combs. We introduce formal concepts of simulation and capacity for quantum combs, and we derive lower-bounds on the minimum simulation error in terms of semidefinite programming (SDP). We impose certain conditions on our strategies, such as the positive partial transpose (PPT) preserving constraint. We then analyze these SDPs for simple models, such as the depolarizing process, to determine their fundamental communication limits.
The intersection of physics and information theory has advanced rapidly in the past few decades. Information-theoretic concepts have been employed to formulate physical laws, while quantum information processing, such as quantum computation and communication, has been developed both theoretically and experimentally. Simultaneously, understanding the universal or specific behaviors of quantum many-body systems has remained a central challenge since shortly after the inception of quantum mechanics. This is studied in part by statistical mechanics, which models complex many-body systems using simple ensembles. In this thesis, we investigate information-theoretic problems in quantum many-body dynamics that lie at the intersection of these avenues. To begin with, we investigate the extent of randomness, captured by unitary designs, in random quantum circuits. Quantum circuits serve as discrete models for random many-body quantum dynamics. Specifically, we study how the internal structure of random gates influences the generation of randomness in quantum circuits. We demonstrate that the spectral gap of structured random circuits can be larger than that of unstructured ones, implying that the former can converge to designs faster than the latter. In this analysis, free-fermionic techniques play a crucial role in obtaining exact results on the spectrum of second-moment operators. We then investigate the effect of quantum measurements on the quantum complexity in random circuits. Here, the quantum complexity of a state is defined as the minimum number of local gates required for its exact preparation in quantum circuits. We prove that in monitored random quantum circuits, the complexity undergoes a measurement-induced phase transition by leveraging techniques from percolation theory. Above the critical measurement rate, the complexity saturates quickly. Below the threshold, it grows at least linearly for an exponentially long time. Finally, we discuss quantum communication scenarios in noisy quantum dynamics, where we simulate the identity channel by combining a given noisy quantum dynamics and our quantum operational strategies. Specifically, we consider multi-time noisy processes and optimize our strategies to simulate near-noiseless channels, both of which are mathematically formulated using quantum combs. We introduce formal concepts of simulation and capacity for quantum combs, and we derive lower-bounds on the minimum simulation error in terms of semidefinite programming (SDP). We impose certain conditions on our strategies, such as the positive partial transpose (PPT) preserving constraint. We then analyze these SDPs for simple models, such as the depolarizing process, to determine their fundamental communication limits.
Recently, the dynamics of quantum systems that involve both unitary evolution and quantum measurements have attracted attention due to the exotic phenomenon of measurement-induced phase transitions. The latter refers to a sudden change in a property of a state of n qubits, such as its entanglement entropy, depending on the rate at which individual qubits are measured. At the same time, quantum complexity emerged as a key quantity for the identification of complex behaviour in quantum many-body dynamics. In this work, we investigate the dynamics of the quantum state complexity in monitored random circuits, where n qubits evolve according to a random unitary circuit and are individually measured with a fixed probability at each time step. We find that the evolution of the exact quantum state complexity undergoes a phase transition when changing the measurement rate. Below a critical measurement rate, the complexity grows at least linearly in time until {saturating to a value e Ω ( n ) .} Above, the complexity does not exceed poly ( n ) . In our proof, we make use of percolation theory to find paths along which an exponentially long quantum computation can be run below the critical rate, and to identify events where the state complexity is reset to zero above the critical rate. We lower bound the exact state complexity in the former regime using recently developed techniques from algebraic geometry. Our results combine quantum complexity growth, phase transitions, and computation with measurements to help understand the behavior of monitored random circuits and to make progress towards determining the computational power of measurements in many-body systems.
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.
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.
Quantum complexity is a measure of the minimal number of elementary operations required to approximately prepare a given state or unitary channel. Recently, this concept has found applications beyond quantum computing—in studying the dynamics of quantum many-body systems and the long-time properties of anti–de Sitter black holes. In this context, Brown and Susskind [] conjectured that the complexity of a chaotic quantum system grows linearly in time up to times exponential in the system size, saturating at a maximal value, and remaining maximally complex until undergoing recurrences at doubly exponential times. In this work, we prove the saturation and recurrence of complexity in two models of chaotic time evolutions based on (i) random local quantum circuits and (ii) stochastic local Hamiltonian evolution. Our results advance an understanding of the long-time behavior of chaotic quantum systems and could shed light on the physics of black-hole interiors. From a technical perspective, our results are based on establishing new quantitative connections between the Haar measure and high-degree approximate designs, as well as the fact that random quantum circuits of sufficiently high depth converge to approximate designs. Published by the American Physical Society 2024
The concept of quantum complexity has far-reaching implications spanning theoretical computer science, quantum many-body physics, and high-energy physics.The quantum complexity of a unitary transformation or quantum state is defined as the size of the shortest quantum computation that executes the unitary or prepares the state.It is reasonable to expect that the complexity of a quantum state governed by a chaotic many-body Hamiltonian grows linearly with time for a time that is exponential in the system size; however, because it is hard to rule out a shortcut that improves the efficiency of a computation, it is notoriously difficult to derive lower bounds on quantum complexity for particular unitaries or states without making additional assumptions.To go further, one may study more generic models of complexity growth.We provide a rigorous connection between complexity growth and unitary k-designs, ensembles that capture the randomness of the unitary group.This connection allows us to leverage existing results about design growth to draw conclusions about the growth of complexity.We prove that local random quantum circuits generate unitary transformations whose complexity grows linearly for a long time, mirroring the behavior one expects in chaotic quantum systems and verifying conjectures by Brown and Susskind.Moreover, our results apply under a strong definition of quantum complexity based on optimal distinguishing measurements.
According to one of the fundamental axioms of quantum mechanics, unitary operators rule the evolution of any quantum system; it is thus of prominent importance to investigate the corresponding group and discover its properties. Random processes over the unitary group have indeed a wide range of applications in the context of quantum information; in particular, they are involved in the construction of peculiar distributions called unitary designs which mimic the uniform Haar measure by matching its moments and are thus deeply connected to the description of phenomena such as quantum tomography, equilibration, thermalization, encryption and scrambling. Previous approaches have shown that unitary designs are efficiently approximated by random quantum circuits with local unitary operators. Moreover, these circuits are used to characterize the precision of experimental implementations of unitary gates via randomized benchmarking protocols and to rapidly decouple a system from the environment. The main novel contribution of this work is to extend these mixing properties to a continuous-time framework, namely, Brownian motion over the unitary group induced by stochastic local Hamiltonians. In order to achieve these new results, on the one hand we move to a representation theoretic formulation and make use of its tools to establish the gap of local generators linked to the moments of the distribution induced by the diffusion process; from this we then derive an expression for the length of time it takes to ensure convergence toward the moments of the Haar measure. On the other hand we project the stochastic evolution onto a random walk on Pauli matrices and tie this description to the analogous one for random quantum circuits to achieve decoupling in a run time scaling almost linearly with respect to the system size. In addition to providing a unifying framework for random processes over the unitary group, we hence aim at presenting new mathematical results and techniques for quantum information. We furthermore discuss applications to black holes dynamics in the perspective of the information paradox. As an additional novel result not related to Brownian motion, we propose a randomized benchmarking protocol exploiting the symmetry of the gate whose accuracy has to be estimated, in order to overcome current shortcomings afflicting the known schemes. We again rely on representation theory to reduce the computational effort and the amount of employed quantum resources.