Provably efficient machine learning for quantum many-body problems

Classical machine learning (ML) provides a potentially powerful approach to solving challenging quantum many-body problems in physics and chemistry. However, the advantages of ML over more traditional methods have not been firmly established. In this work, we prove that classical ML algorithms can efficiently predict ground state properties of gapped Hamiltonians in finite spatial dimensions, after learning from data obtained by measuring other Hamiltonians in the same quantum phase of matter. In contrast, under widely accepted complexity theory assumptions, classical algorithms that do not learn from data cannot achieve the same guarantee. We also prove that classical ML algorithms can efficiently classify a wide range of quantum phases of matter. Our arguments are based on the concept of a classical shadow, a succinct classical description of a many-body quantum state that can be constructed in feasible quantum experiments and be used to predict many properties of the state. Extensive numerical experiments corroborate our theoretical results in a variety of scenarios, including Rydberg atom systems, 2D random Heisenberg models, symmetry-protected topological phases, and topologically ordered phases.Comment: 10 pages, 12 figures + 57 page appendi

Brittleness of Bayesian inference and new Selberg formulas

The incorporation of priors in the Optimal Uncertainty Quantification (OUQ) framework \cite{OSSMO:2011} reveals brittleness in Bayesian inference; a model may share an arbitrarily large number of finite-dimensional marginals with, or be arbitrarily close (in Prokhorov or total variation metrics) to, the data-generating distribution and still make the largest possible prediction error after conditioning on an arbitrarily large number of samples. The initial purpose of this paper is to unwrap this brittleness mechanism by providing (i) a quantitative version of the Brittleness Theorem of \cite{BayesOUQ} and (ii) a detailed and comprehensive analysis of its application to the revealing example of estimating the mean of a random variable on the unit interval $[0,1]$ using priors that exactly capture the distribution of an arbitrarily large number of Hausdorff moments. However, in doing so, we discovered that the free parameter associated with Markov and Kre\u{\i}n's canonical representations of truncated Hausdorff moments generates reproducing kernel identities corresponding to reproducing kernel Hilbert spaces of polynomials. Furthermore, these reproducing identities lead to biorthogonal systems of Selberg integral formulas. This process of discovery appears to be generic: whereas Karlin and Shapley used Selberg's integral formula to first compute the volume of the Hausdorff moment space (the polytope defined by the first $n$ moments of a probability measure on the interval $[0,1]$), we observe that the computation of that volume along with higher order moments of the uniform measure on the moment space, using different finite-dimensional representations of subsets of the infinite-dimensional set of probability measures on $[0,1]$ representing the first $n$ moments, leads to families of equalities corresponding to classical and new Selberg identities.Comment: 73 pages. Keywords: Bayesian inference, misspecification, robustness, uncertainty quantification, optimal uncertainty quantification, reproducing kernel Hilbert spaces (RKHS), Selberg integral formula

The Conformal Bootstrap: Theory, Numerical Techniques, and Applications

Conformal field theories have been long known to describe the fascinating universal physics of scale invariant critical points. They describe continuous phase transitions in fluids, magnets, and numerous other materials, while at the same time sit at the heart of our modern understanding of quantum field theory. For decades it has been a dream to study these intricate strongly coupled theories nonperturbatively using symmetries and other consistency conditions. This idea, called the conformal bootstrap, saw some successes in two dimensions but it is only in the last ten years that it has been fully realized in three, four, and other dimensions of interest. This renaissance has been possible both due to significant analytical progress in understanding how to set up the bootstrap equations and the development of numerical techniques for finding or constraining their solutions. These developments have led to a number of groundbreaking results, including world record determinations of critical exponents and correlation function coefficients in the Ising and $O(N)$ models in three dimensions. This article will review these exciting developments for newcomers to the bootstrap, giving an introduction to conformal field theories and the theory of conformal blocks, describing numerical techniques for the bootstrap based on convex optimization, and summarizing in detail their applications to fixed points in three and four dimensions with no or minimal supersymmetry.Comment: 81 pages, double column, 58 figures; v3: updated references, minor typos correcte

Multi-Modal Multi-Scale Deep Learning for Large-Scale Image Annotation

Image annotation aims to annotate a given image with a variable number of class labels corresponding to diverse visual concepts. In this paper, we address two main issues in large-scale image annotation: 1) how to learn a rich feature representation suitable for predicting a diverse set of visual concepts ranging from object, scene to abstract concept; 2) how to annotate an image with the optimal number of class labels. To address the first issue, we propose a novel multi-scale deep model for extracting rich and discriminative features capable of representing a wide range of visual concepts. Specifically, a novel two-branch deep neural network architecture is proposed which comprises a very deep main network branch and a companion feature fusion network branch designed for fusing the multi-scale features computed from the main branch. The deep model is also made multi-modal by taking noisy user-provided tags as model input to complement the image input. For tackling the second issue, we introduce a label quantity prediction auxiliary task to the main label prediction task to explicitly estimate the optimal label number for a given image. Extensive experiments are carried out on two large-scale image annotation benchmark datasets and the results show that our method significantly outperforms the state-of-the-art.Comment: Submited to IEEE TI