Donaldson-Witten theory and indefinite theta functions

We consider partition functions with insertions of surface operators of topologically twisted N=2, SU(2) supersymmetric Yang-Mills theory, or Donaldson-Witten theory for short, on a four-manifold. If the metric of the compact four-manifold has positive scalar curvature, Moore and Witten have shown that the partition function is completely determined by the integral over the Coulomb branch parameter $a$, while more generally the Coulomb branch integral captures the wall-crossing behavior of both Donaldson polynomials and Seiberg-Witten invariants. We show that after addition of a Q-exact surface operator to the Moore-Witten integrand, the integrand can be written as a total derivative to the anti-holomorphic coordinate $\bar a$ using Zwegers' indefinite theta functions. In this way, we reproduce G\"ottsche's expressions for Donaldson invariants of rational surfaces in terms of indefinite theta functions for any choice of metric.Comment: 23 pages + appendices, comments welcome. v2: published versio

Predicting Ising Model Performance on Quantum Annealers

By analyzing the characteristics of hardware-native Ising Models and their performance on current and next generation quantum annealers, we provide a framework for determining the prospect of advantage utilizing adiabatic evolution compared to classical heuristics like simulated annealing. We conduct Ising Model experiments with coefficients drawn from a variety of different distributions and provide a range for the necessary moments of the distributions that lead to frustration in classical heuristics. By identifying the relationships between the linear and quadratic terms of the models, analysis can be done a priori to determine problem instance suitability on annealers. We then extend these experiments to a prototype of D-Wave's next generation device, showing further performance improvements compared to the current Advantage annealers

Iteration Complexity of Variational Quantum Algorithms

There has been much recent interest in near-term applications of quantum computers. Variational quantum algorithms (VQA), wherein an optimization algorithm implemented on a classical computer evaluates a parametrized quantum circuit as an objective function, are a leading framework in this space. In this paper, we analyze the iteration complexity of VQA, that is, the number of steps VQA required until the iterates satisfy a surrogate measure of optimality. We argue that although VQA procedures incorporate algorithms that can, in the idealized case, be modeled as classic procedures in the optimization literature, the particular nature of noise in near-term devices invalidates the claim of applicability of off-the-shelf analyses of these algorithms. Specifically, the form of the noise makes the evaluations of the objective function via circuits biased, necessitating the perspective of convergence analysis of variants of these classical optimization procedures, wherein the evaluations exhibit systematic bias. We apply our reasoning to the most often used procedures, including SPSA the parameter shift rule, which can be seen as zeroth-order, or derivative-free, optimization algorithms with biased function evaluations. We show that the asymptotic rate of convergence is unaffected by the bias, but the level of bias contributes unfavorably to both the constant therein, and the asymptotic distance to stationarity.Comment: 39 pages, 11 figure

Renormalization and BRST symmetry in Donaldson-Witten theory

The presence of a BRST symmetry in topologically twisted gauge theories makes a precise analysis of these theories feasible. While the global BRST symmetry suggests that correlation functions of BRST exact observables vanish, this decoupling might be obstructed due to a contribution from the boundary of field space. Motivated by divergent BRST exact observables on the Coulomb branch of Donaldson-Witten theory, we put forward a new prescription for the renormalization of correlation functions on the Coulomb branch. This renormalization is based on the relation between Coulomb branch integrals and integrals over a modular fundamental domain, and establishes that BRST exact observables indeed decouple in Donaldson-Witten theory.Comment: 32 pages + appendices, 2 figures; v2: minor change

Globally Optimal Quantum Control

Optimization methods for constrained quantum control problems power quantum technologies. Such control problems are notoriously difficult because they are non-convex and plagued with local extrema. Current optimization methods must be repeated many times to find good solutions, each time requiring many simulations of the system. Here we present Quantum Control via Polynomial Optimization (QCPOp), a method that eliminates this problem by directly finding globally optimal solutions. This remarkable ability is due to global optimization methods recently developed for polynomial functions. We demonstrate the tremendous improvement over current state-of-the-art methods using a number of non-trivial examples. Global optimization also allows QCPOp to find the simplest control solutions. Since QCPOp is able to reveal the optimum performance of quantum control with high confidence, we expect that it will not only enhance the utility of such control but provide a key tool for determining the limits of quantum technologies.Comment: Significantly updated content with many new cool examples (10 pages and 6 figures

Modern aspects of topological gauge theories - Polynomial invariants and mock modular forms

In this dissertation we present new results in the field of topologically twisted gauge theories evaluated on compact four-manifolds without boundary. We focus on the Donaldson-Witten theory, that is the N = 2 topologically twisted super Yang-Mills theory, We revisit and study the contribution on the Coulomb branch of the path integral of the low energy effective theory which is non-vanishing only for four-manifolds with b+2 ? 1. We establish new ways to evaluate path integrals of this theory using mock modular forms. Furthermore we propose a new regularization and renormalization of the path integral required for conservation of the BRST symmetry of the topological theory. We conclude by generalizing these considerations to the Donaldson-Witten theory in the presence of supersymmetric surface defects