Stability against perturbations is a defining property of quantum many-body phases of matter. However, most rigorous stabilities are only established for static perturbations; whether any system can remain stable against generic time-dependent perturbations is largely elusive. Here, we identify a universal phenomenon, where the evolving state driven by time-dependent q-local Hamiltonians can be exponentially localized in an energy window of instantaneous spectrum, and prove its survival under generic time-dependent perturbations. Applying such energy-space localization to classical and quantum LDPC codes whose codewords are separated by extensive energy barriers, we show that the system remains localized near the original codeword for an exponentially long time under generic time-dependent perturbations. For classical optimization problems with clustered solution spaces, the stability becomes an obstacle for quantum Hamiltonian-based algorithms to escape local minima. Our work provides a new lens for analyzing quantum non-equilibrium dynamics and tools for establishing stability and designing quantum algorithms.
Stability against perturbations is a defining property of quantum many-body phases of matter. However, most rigorous stabilities are only established for static perturbations; whether any system can remain stable against generic time-dependent perturbations is largely elusive. Here, we identify a universal phenomenon, where the evolving state driven by time-dependent q-local Hamiltonians can be exponentially localized in an energy window of instantaneous spectrum, and prove its survival under generic time-dependent perturbations. Applying such energy-space localization to classical and quantum LDPC codes whose codewords are separated by extensive energy barriers, we show that the system remains localized near the original codeword for an exponentially long time under generic time-dependent perturbations. For classical optimization problems with clustered solution spaces, the stability becomes an obstacle for quantum Hamiltonian-based algorithms to escape local minima. Our work provides a new lens for analyzing quantum non-equilibrium dynamics and tools for establishing stability and designing quantum algorithms.
In quantum many-body systems subjected to generic perturbations, eigenstates are typically fragile and prone to strong mixing, leading to rapid thermalization in dynamics to an equilibrium state described by statistical mechanics. However, there exist robust counterexamples that defy this typical expectation, exhibiting novel non-equilibrium phenomena and offering potential applications in (quantum) information processing. This dissertation explores such systems and establishes their stability against perturbations with mathematical rigor. First, by generalizing the exponentially slow tunneling effect in single-particle quantum mechanics, we develop a theory of metastability in quantum many-body systems, demonstrating their nonperturbatively long lifetimes. As a canonical application, we provide the first general and tight bounds on the slow decay of false vacua. Second, it is known that ground states in finite spatial dimensions can resist arbitrary local perturbations through topologically ordered phases of matter. We extend this notion of stability to quantum error-correcting codes in infinite dimensions, specifically quantum low-density parity-check (LDPC) codes. These codes are of significant interest due to their promise of lower fault-tolerance overhead compared to finite-dimensional codes. Lastly, we analyze classical counterparts of LDPC codes and prove that their eigenstates remain localized in the many-body Hilbert space under generic perturbations. This finding presents the first unambiguous example of a robust violation of the eigenstate thermalization hypothesis. To achieve these results, we introduce advanced techniques to control operator locality, leveraging and generalizing methods such as Lieb-Robinson bounds, cluster expansions, and Schrieffer-Wolff transformations. These tools not only underpin our specific findings but may also prove valuable for broader challenges in mathematical physics of quantum many-body systems.
We generalize the proof of stability of topological order, due to Bravyi, Hastings, and Michalakis, to stabilizer Hamiltonians corresponding to low-density parity-check (LDPC) codes without the restriction of geometric locality in Euclidean space. We consider Hamiltonians H 0 defined by ⟦ N , K , d ⟧ LDPC codes, which obey certain topological quantum order conditions: (i) code distance d ≥ c log ( N ) , implying local indistinguishability of ground states, and (ii) a mild condition on local and global compatibility of ground states—these include good quantum LDPC codes and the toric code on a hyperbolic lattice, among others. We consider stability under weak perturbations that are quasilocal on the interaction graph defined by H 0 and that can be represented as sums of bounded-norm terms. As long as the local perturbation strength is smaller than a finite constant, we show that the perturbed Hamiltonian has well-defined spectral bands originating from the O ( 1 ) smallest eigenvalues of H 0 . The band originating from the smallest eigenvalue has 2 K states, is separated from the rest of the spectrum by a finite energy gap, and has exponentially narrow bandwidth δ = C N e − Θ ( d ) , which is tighter than the best-known bounds even in the Euclidean case. We also obtain that the new ground-state subspace is related to the initial-code subspace by a quasilocal unitary, allowing one to relate their physical properties. Our proof uses an iterative procedure that performs successive rotations to eliminate non-frustration-free terms in the Hamiltonian. Our results extend to quantum Hamiltonians built from classical LDPC codes, which give rise to stable symmetry-breaking phases. These results show that LDPC codes very generally define stable gapped quantum phases, even in the non-Euclidean setting, initiating a systematic study of such phases of matter.
Quantum thermalization explains how an isolated many-body system naturally evolves towards a thermal state, rendering information about the initial conditions inaccessible to local measurements. This is precisely the mechanism utilized in quantum error correction, where information is protected by design through a nonlocal encoding. In this work, we leverage this connection to port the rigorous framework of (approximate) quantum error correction to the study of quantum thermalization. Treating typical late-time states as codewords, we characterize the error-correcting properties of generic thermalizing dynamics. We numerically uncover a universal relationship between the encoding rate, distance, and thermal entropy density of the emergent code. At infinite temperature, this universal curve saturates the quantum Singleton bound, achieving the same optimal limit as Haar-random codes. At finite temperature, we introduce a code family based on the Scrooge ensemble, the natural thermal analogue of the Haar ensemble, and prove it saturates the entropic quantum Singleton bound, establishing this family as optimal within entropic constraints. Our extracted universal curve independently saturates this same bound, revealing that finite-temperature thermalization is itself optimal. Finally, we show how conserved quantities limit the error-correcting behavior of thermalization: codewords with differing energies, or other conserved charges, leak only classical information, and correctability persists until the difference reaches the scale of thermal fluctuations. Our results reveal a universal optimal coding structure in thermalizing dynamics, while introducing new optimal codes that achieve fundamental limits of approximate quantum error correction.
The current theoretical framework for topological phases of matter is based on the thermodynamic limit of a system with geometrically local interactions. A natural question is to what extent the notion of a phase of matter remains well-defined if we relax the constraint of geometric locality, and replace it with a weaker graph-theoretic notion of k-locality. As a step towards answering this question, we analyze the stability of the energy gap to perturbations for Hamiltonians corresponding to general quantum low-density parity-check codes, extending work of Bravyi and Hastings [Commun. Math. Phys. 307, 609 (2011)]. A corollary of our main result is that if there exist constants ε1,ε2>0 such that the size Γ(r) of balls of radius r on the interaction graph satisfy Γ(r)=O(exp(r1−ε1)) and the local ground states of balls of radius r≤ρ∗=O(log(n)1+ε2) are locally indistinguishable, then the energy gap of the associated Hamiltonian is stable against local perturbations. This gives an almost exponential improvement over the D-dimensional Euclidean case, which requires Γ(r)=O(rD) and ρ∗=O(nα) for some α>0. The approach we follow falls just short of proving stability of finite-rate qLDPC codes, which have ε1=0; we discuss some strategies to extend the result to these cases. We discuss implications for the third law of thermodynamics, as k-local Hamiltonians can have extensive zero-temperature entropy.
Despite the exponentially large amount of information required in the quantum description of many-body systems, finite size numerical simulations have paved the way for recent progress in the understanding of quantum phases of matter both in and out-of-equilibrium. \n \nIn this spirit, using exact diagonalization for a small number of spins ~ 20, first, we investigate the fate of both ergodic quantum many-body systems that thermalize at long times and systems with quenched disorder that exhibit many-body localization (MBL) and fail to thermalize (due to an extensive set of local conserved degrees of freedom) under periodic driving. Ergodic systems always delocalize in energy space and heat up to infinite temperature, for both local driving, in which the time dependence of the Hamiltonian is restricted to a few contiguous degrees of freedom, and global driving. On the other hand, MBL systems remain localized at finite energy density for local driving, while in the case of global driving these systems can either heat up to infinite temperature or remain localized depending on the parameters of the drive. Underlying the latter is the emergence of an effective time independent MBL Hamiltonian describing the long time dynamics of the system. These numerical predictions have recently been verified experimentally in a one dimensional quasi-disordered optical lattice of interacting fermions wherein unitary time evolution can be experimentally probed. \n \nSubsequently, we study the stability of MBL systems with respect to another perturbation, in this case by thermal inclusions, which remains poorly understood. We consider a simple model to address this question: a two level system interacting with 𝑁 localized bits subject to random fields. In the large 𝑁 limit, the localized bits are thermalized for any nonzero ratio of interactions to disorder strength. Moreover, at finite 𝑁, in the transition region, the single-site eigenstate entanglement entropies exhibit bimodal distributions, so that localized bits are either "on" (strongly entangled) or "off" (weakly entangled) in eigenstates. The clusters of "on" bits vary significantly between eigenstates of the same sample, which provides evidence for a heterogenous discontinuous transition out of the localized phase in single site observables. These results also imply the MBL phase is unstable in systems with short-range interactions and quenched randomness in dimensions 𝑑 that are high but finite. \n \nThe second part of this thesis is motivated by the recent breakthroughs of artificial intelligence in domains such as computer vision involving highly dimensional data of complexity not unlike the one found in the description of many-body systems. Using Monte Carlo simulations to generate synthetic data, we find that kernel methods, a class of machine learning algorithms, can learn the mathematical \nform of physical discriminators associated with phase transitions, such as order parameters and Hamiltonian constraints, for a set of two dimensional spin models: the ferromagnetic Ising model, a conserved-order-parameter Ising model, and the Ising gauge theory. \n \nOur work shows that the combination of numerical simulations with the ability of machine learning algorithms for automatic feature detection hold promise to become standard tools in the exploration and characterization of different phases of matter in condensed matter and other many-body systems.
The quantum circuit complexity of an evolving many-body quantum system is believed to exhibit a sustained growth, maintained for timescales much longer than the onset of thermalization. Most previous works have focused on models which violate energy conservation, such as random unitary circuits. Here we study generic, local time-independent Hamiltonian dynamics. We unconditionally prove that for generic local Hamiltonians and typical initial product states at high effective temperature, the robust quantum circuit complexity must grow over a very long period of time, attaining an exponentially large value at late times. Our approach relies on two structural properties that we prove rigorously: (i) generic local Hamiltonians satisfy generalized spectral no-resonance conditions of arbitrary order, and (ii) typical high-temperature product states are effectively supported in exponentially many energy eigenstates. These properties have been widely assumed without proof in prior works. As corollaries of this result, we show the late-time state displays robust volume-law entanglement that is irremovable by polynomial-size circuits, and we establish a no fast-forwarding result for generic local Hamiltonians. We also lower bound the complexity of preparing sufficiently large subsystems with local quantum channels, in contrast with sufficiently small thermalizing regions which we show retain low complexity at late times.