Reproducing kernel Hilbert space compactification of unitary evolution groups

A framework for coherent pattern extraction and prediction of observables of measure-preserving, ergodic dynamical systems with both atomic and continuous spectral components is developed. This framework is based on an approximation of the generator of the system by a compact operator W-tau on a reproducing kernel Hilbert space (RKHS). The operator W-tau is skew-adjoint, and thus can be represented by a projection-valued measure, discrete by compactness, with an associated orthonormal basis of eigenfunctions. These eigenfunctions are ordered in terms of a Dirichlet energy, and provide a notion of coherent observables under the dynamics akin to the Koopman eigenfunctions associated with the atomic part of the spectrum. In addition, W-tau generates a unitary evolution group {e(tW tau)}t epsilon R on the RKHS, which approximates the unitary Koopman group of the system. We establish convergence results for the spectrum and Borel functional calculus of W-tau as tau -> 0(+), as well as an associated data-driven formulation utilizing time series data. Numerical applications to ergodic systems with atomic and continuous spectra, namely a torus rotation, the Lorenz 63 system, and the Rossler system, are presented. (C) 2021 The Author(s). Published by Elsevier Inc.Peer reviewe

Discrete scale invariance and complex dimensions

We discuss the concept of discrete scale invariance and how it leads to complex critical exponents (or dimensions), i.e. to the log-periodic corrections to scaling. After their initial suggestion as formal solutions of renormalization group equations in the seventies, complex exponents have been studied in the eighties in relation to various problems of physics embedded in hierarchical systems. Only recently has it been realized that discrete scale invariance and its associated complex exponents may appear ``spontaneously'' in euclidean systems, i.e. without the need for a pre-existing hierarchy. Examples are diffusion-limited-aggregation clusters, rupture in heterogeneous systems, earthquakes, animals (a generalization of percolation) among many other systems. We review the known mechanisms for the spontaneous generation of discrete scale invariance and provide an extensive list of situations where complex exponents have been found. This is done in order to provide a basis for a better fundamental understanding of discrete scale invariance. The main motivation to study discrete scale invariance and its signatures is that it provides new insights in the underlying mechanisms of scale invariance. It may also be very interesting for prediction purposes.Comment: significantly extended version (Oct. 27, 1998) with new examples in several domains of the review paper with the same title published in Physics Reports 297, 239-270 (1998

On the representation theory of the Bondi-Metzner-Sachs group and its variants in three space-time dimensions

The original Bondi-Metzner-Sachs (BMS) group B is the common asymptotic symmetry group of all asymptotically flat Lorentzian radiating 4-dim space-times. As such, B is the best candidate for the universal symmetry group of General Relativity (G.R.). In 1973, with this motivation, P. J. McCarthy classified all relativistic B-invariant-systems in terms of strongly continuous irreducible unitary repesentations (IRS) of B. Here we introduce the analogue B(2,1) of the BMS group B in 3 space-time dimensions. B(2,1) itself admits thirty-four analogues both real in all signatures and in complex space-times. In order to find the IRS of both B(2,1) and its analogues we need to extend Wigner-Mackey's theory of induced representations. The necessary extension is described and is reduced to the solution of three problems. These problems are solved in the case where B(2,1) and its analogues are equipped with the Hilbert topology. The extended theory is necessary in order to construct the IRS of both B and its analogues in any number d of space-time dimensions, d is greater or equal to 3, and also in order to construct the IRS of their supersymmetric counterparts. We use the extended theory to obtain the necessary data in order to construct the IRS of B(2,1): The main results of the representation theory are: The IRS are induced from little groups which are compact. The finite little groups are cyclic groups of even order. The inducing construction is exhaustive notwithstanding the fact that B(2,1) is not locally compact in the employed Hilbert topology.Comment: 39 page

Numerical studies of the critical behaviour of non-commutative field theories

We study the critical behaviour of matrix models with builtin SU(2) geometry by using Hybrid Monte Carlo (HMC) techniques. The first system under study is a matrix regularization of the φ4 theory defined on the sphere. We develop a HMC algorithm together with an SU(2) gauge-fixing procedure in order to study the model. We extract the phase diagram of the model and give an estimation for the triple point for a system constructed of matrices of size N = 7. Our numerical results also suggest the existence of stripe phases- phases in which modes with higher momentum l have non-negligible contribution. The second system under study is a matrix model realized via competing Yang-Mills and Myers terms. In its low-temperature phase the system has geometrical phase with SO(3) symmetry: the ground state is represented by the su(2) generators. This geometry disappears in the high-temperature phase the system. Our results suggest that there are three main types of fluctuations in the system close to the transition: fluctuations of the fuzzy sphere, fluctuations which drive the system between the two phases, and fluctuations of the high-temperature regime. The fluctuations of the fuzzy sphere show the properties of a second order phase transition. We establish the validity of the finite size scaling ansatz in that regime. The fluctuations which bring the system between the phases show the properties of a first order transition. In the Appendix we provide in some detail the idea behind the HMC approach. We give some practical guidelines if one is to implement such an algorithm to study matrix models. We comment on the main sources for the phenomenon of autocorrelation time. As a final topic we present the basics of the OpenCL language which we used to port some of our algorithms for parallel computing architectures such as GPU’s

Transmon-based quantum computers from a many-body perspective

The quest for quantum computers is in full swing. Over the past decade, the frontiers of quantum computing have broadened from exploring few-qubit devices to developing viable multi-qubit processors. One of the protagonists of the present era is the superconducting transmon qubit. By harmoniously combining applied engineering with fundamental research in computer science and physics, transmon-based quantum processors have matured to a remarkable level. Their applications include the study of topological and nonequilibrium states of matter, and it is argued that they have already ushered us into the era of quantum advantage. Nevertheless, building a quantum computer that can solve problems of practical relevance remains a massive challenge. As the field progresses with unbridled panache, the question of whether we have a comprehensive picture of the potential dangers lurking in the wings acquires increasing urgency. In particular, it needs to be thoroughly clarified whether, with viable quantum computers of O(50) qubits at hand, new and hitherto unconsidered obstacles associated with the multi-qubit nature can emerge. For example, the high accuracy of quantum gates in small-scale devices is hard to obtain in larger processors. On the hardware side, the unique requirements posed by large quantum computers have already spawned new approaches to qubit design, control, and readout. This thesis introduces a novel, less applied perspective on multi-qubit processors. Specifically, we fuse the field of quantum engineering and many-body physics by applying concepts from the theories of localization and quantum chaos to multi-transmon arrays. From a many-body per- spective, transmon architectures are synthetic systems of interacting and disordered nonlinear quantum oscillators. While a certain amount of coupling between the transmons is indispensable for performing elementary gate operations, a delicate balancing with disorder—site-to-site varia- tions in the qubit frequencies—is required to prevent locally injected information from dispersing in extended many-body states. Transmon research has established different modalities to cope with this dilemma between inefficiency (slow gates due to small coupling or large disorder) and information loss (large couplings or too small disorder). We analyze small instances of transmon quantum computers in exact diagonalization studies, using contemporary quantum processors as blueprints. Scrutinizing the spectrum, many-body wave functions, and qubit-qubit correlations for experimentally relevant parameter regimes reveals that some of the prevalent transmon de- sign schemes operate close to a region of uncontrollable chaotic fluctuations. Furthermore, we establish a close link between the advent of chaos in the classical limit and the emergence of quantum chaotic signatures. Our concepts complement the traditional few-qubit picture that is commonly exploited to optimize device configurations on small scales. Destabilizing mecha- nisms beyond this local scale can be detected from our fresh perspective. This suggests that techniques developed in the field of many-body localization should become an integral part of future transmon processor engineering