Graphon Neural Networks and the Transferability of Graph Neural Networks
Luiz Chamon (UC Berkeley)
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Adaptive Wavelet Distillation from DNNs and Dictionary Learning
Bin Yu (UC Berkeley)Uniform Convergence of Interpolators: Gaussian Width, Norm Bounds, and Benign Overfitting
Frederic Koehler (Simons Institute)Realizing a dynamical topological phase without symmetry protection in trapped ions
Andrew Potter University of British Columbia
Probing topological invariants from a ground state wave function
Ze-Pei Cian University of New Mexico
Equivariant Machine Learning Structured Like Classical Physics
Soledad Villar (Johns Hopkins)UltraLight Dark Matter Dynamics in the Language of Eigenstates
Luna Zagorac Perimeter Institute for Theoretical Physics
The Entropic Dynamics approach to Quantum Mechanics
Ariel Caticha State University of New York (SUNY)
Algebraically closed higher categories
Theo Johnson-Freyd Dalhousie University
Symmetry Adapted Gram Spectrahedra
Serkan Hosten (San Francisco State University)Holographic entanglement in spin network states: bulk-to-boundary isometries and horizon-like regions from volume correlations
Eugenia Colafranceschi University of California, Santa Barbara
Graphon Neural Networks and the Transferability of Graph Neural Networks
Luiz Chamon (UC Berkeley)Graph neural networks (GNNs) generalize convolutional neural networks (CNNs) by using graph convolutions that enable information extraction from non-Euclidian domains, e.g., network data. These graph convolutions combine information from adjacent nodes using coefficients that are shared across all nodes. Since these coefficients do not depend on the graph, one can envision using the same coefficients to define a GNN on a different graph. In this talk, I will tackle this problem by introducing graphon NNs as limit objects of sequences of GNNs and characterizing the difference between the output of a GNN and its limit graphon-NN. This bound vanishes as the number of nodes grows as long as the graph convolutional filters are bandlimited in the graph spectral domain. This establishes a tradeoff between discriminability and transferability of GNNs and sheds light on the effect of training using smaller (possibly corrupted) graph convolutions.Uniform Convergence of Interpolators: Gaussian Width, Norm Bounds, and Benign Overfitting
Frederic Koehler (Simons Institute)We consider interpolation learning in high-dimensional linear regression with Gaussian data, and prove a generic uniform convergence guarantee on the generalization error of interpolators in an arbitrary hypothesis class in terms of the class's Gaussian width. Applying the generic bound to Euclidean norm balls recovers the consistency result of Bartlett et al. (2020) for minimum-norm interpolators, and confirms a prediction of Zhou et al. (2020) for near-minimal-norm interpolators in the special case of Gaussian data. We demonstrate the generality of the bound by applying it to the simplex, obtaining a novel consistency result for minimum l1-norm interpolators (basis pursuit). Our results show how norm-based generalization bounds can explain and be used to analyze benign overfitting, at least in some settings. Joint work with Lijia Zhou, Danica Sutherland, and Nathan Srebro.Realizing a dynamical topological phase without symmetry protection in trapped ions
Andrew Potter University of British Columbia
In thermal equilibrium, 1d bosonic systems (e.g. spin- or qubit- chains) cannot support intrinsically topological phases without symmetry protection. For example, the edge states of the Haldane spin chain are fragile to magnetic fields, in contrast to the absolutely stable Majorana edge states of a topological superconducting wire of fermionic electrons. This fragility is a serious drawback to harnessing topological edge states as protected quantum memories in existing AMO and qubit platforms for quantum simulation and information processing. In this talk, I will present evidence for a non-equilibrium topological phase of quasiperiodically-driven trapped ion chains, that exhibits topological edge states that are protected purely by emergent dynamical symmetries that cannot be broken by microscopic perturbations. This represents both the first experimental realization of a non-equilibrium quantum phase, and the first example of a 1d bosonic topological phase that does not rely on symmetry-protection.
Probing topological invariants from a ground state wave function
Ze-Pei Cian University of New Mexico
With the rapid development of programmable quantum simulators, the quantum states can be controlled with unprecedented precision. Thus, it opens a new opportunity to explore the strongly correlated phase of matter with new quantum technology platforms. In quantum simulators, one can engineer interactions between the microscopic degree of freedom and create exotic phases of matter that presumably are beyond the reach of natural materials. Moreover, quantum states can be directly measured instead of probing physical properties indirectly via optical and electrical responses of material as done in traditional condensed matter. Therefore, it is pressing to develop new approaches to efficiently prepare and characterize desired quantum states in the novel quantum technology platforms.
In this talk, I will introduce our recent works on the characterization of the topological invariants from a ground state wave function of the topological order phase and the implementation in noisy intermediate quantum devices. First, using topological field theory and tensor network simulations, we demonstrate how to extract the many-body Chern number (MBCN) given a bulk of a fractional quantum Hall wave function [1]. We then propose an ancilla-free experimental scheme for measuring the MBCN without requiring any knowledge of the Hamiltonian. Specifically, we use the statistical correlations of randomized measurements to infer the MBCN of a wave function [2]. Finally, I will present an unbiased numerical optimization scheme to systematically find the Wilson loop operators given a ground state wave function of a gapped, translationally invariant Hamiltonian on a disk. We then show how these Wilson loop operators can be cut and glued through further optimization to give operators that can create, move, and annihilate anyon excitations. We then use these operators to determine the braiding statistics and topological twists of the anyons, yielding a way to fully characterize topological order from the bulk of a ground state wave function [3].
[1] H. Dehghani, Z.P. Cian, M. Hafezi, and M. Barkeshl, Phys. Rev. B 103, 075102
[2] Z.P. Cian, H. Dehghani, A. Elben, B. Vermersch, G. Zhu, M. Barkeshli, P. Zoller, and M. Hafezi, Phys. Rev. Lett. 126, 050501
[3] Z.P. Cian, M. Hafezi, and M. Barkeshl, Manuscript in preparation.Equivariant Machine Learning Structured Like Classical Physics
Soledad Villar (Johns Hopkins)There has been enormous progress in the last few years in designing neural networks that respect the fundamental symmetries and coordinate freedoms of physical law. Some of these frameworks make use of irreducible representations, some make use of high-order tensor objects, and some apply symmetry-enforcing constraints. Different physical laws obey different combinations of fundamental symmetries, but a large fraction (possibly all) of classical physics is equivariant to translation, rotation, reflection (parity), boost (relativity), and permutations. Here we show that it is simple to parameterize universally approximating polynomial functions that are equivariant under these symmetries, or under the Euclidean, Lorentz, and Poincaré groups, at any dimensionality d. The key observation is that nonlinear O(d)-equivariant (and related-group-equivariant) functions can be universally expressed in terms of a lightweight collection of scalars -- scalar products and scalar contractions of the scalar, vector, and tensor inputs. We complement our theory with numerical examples that show that the scalar-based method is simple, efficient, and scalable.UltraLight Dark Matter Dynamics in the Language of Eigenstates
Luna Zagorac Perimeter Institute for Theoretical Physics
Self-gravitating quantum matter may exist in a wide range of cosmological and astrophysical settings: from the very early universe through to present-day boson stars. Such quantum matter arises in UltraLight Dark Matter (ULDM): an exciting axion-like particle candidate which keeps the successes of CDM on large scales but alleviates tensions on small scales. This small scale behavior is due to characteristic cores in ULDM called solitons, which also correspond to the ground state of the self-gravitating quantum system governing ULDM. We calculate the full spectrum of eigenstates and decompose simulations of ULDM into these states, allowing us to precisely track the evolution of the tell-tale soliton cores and the surrounding halo “skirt”. Using this formalism, we investigate formation of halos through binary soliton collisions and the dependence of the final halo product on initial parameters. We further link characteristic ULDM halo behavior—such as the soliton “breathing mode” and random walk of the center of mass—to the presence of certain modes. Finally, we comment on the relationship between eigenenergies and oscillatory timescales present in the system, as well as future directions for understanding ULDM through the language of its eigenstates.
The Entropic Dynamics approach to Quantum Mechanics
Ariel Caticha State University of New York (SUNY)
Entropic Dynamics (ED) is a framework in which Quantum Mechanics is derived as an application of entropic methods of inference. In ED the dynamics of the probability distribution is driven by entropy subject to constraints that are codified into a quantity later identified as the phase of the wave function. The challenge is to specify how those constraints are themselves updated.
The important ingredients are two: the cotangent bundle associated to the probability simplex inherits (1) a natural symplectic structure from ED, and (2) a natural metric structure from information geometry.
The requirement that the dynamics preserves both the symplectic structure (a Hamilton flow) and the metric structure (a Killing flow) leads to a Hamiltonian dynamics of probabilities in which the linearity of the Schrödinger equation, the emergence of a complex structure, Hilbert spaces, and the Born rule, are derived rather than postulated.Algebraically closed higher categories
Theo Johnson-Freyd Dalhousie University
I will report on my progress, joint with David Reutter, to construct and analyze the algebraic closure of nVec – in other words, the universal n-category of framed nD TQFTs. The invertibles are Pontryagin dual to the stable homotopy groups of spheres. The Galois group is almost, but not quite, the stable PL group. An invertible TQFT can be condensed from the vacuum if and only if it trivializes on (possibly-exceptional) spheres.
Equivariant RL
Max Welling (University of Amsterdam)Symmetries play a unifying role in physics and many other sciences. In deep learning, symmetries have been incorporated into neural networks through the concept of equivariance. One of the major benefits is that it will reduce the number parameters through parameter sharing and as such can learn with less data. In this talk I will ask the question, can equivariance also help in RL? Besides the obvious idea of using equivariant value functions, we explore the idea of deep equivariant policies. We make a connection between equivariance and MDP homomorphisms, and generalize to distributed multi-agent settings. Joint work with Elise van der Pol (main contributor), Herke van Hoof and Frans Oliehoek.Symmetry Adapted Gram Spectrahedra
Serkan Hosten (San Francisco State University)We report on the geometric structure of symmetry adapted PSD cones and symmetry adapted Gram spectrahedra of symmetric polynomials. We determine the dimension of symmetry adapted PSD cones, describe its extreme rays, and discuss the structure of its matrix representations. We also focus on symmetry adapted Gram spectrahedra of symmetric binary forms, quadrics, ternary quartics and sextics. In particular, we characterize extreme points of these spectrahedra for symmetric binary forms that are of rank two, and we report what we know about the facial structure of the same spectrahedra. The talk will be based on two collaborations, one with Alex Heaton and Isabelle Shankar, and another one with Matthew Heid.Holographic entanglement in spin network states: bulk-to-boundary isometries and horizon-like regions from volume correlations
Eugenia Colafranceschi University of California, Santa Barbara
For quantum gravity states associated to open spin network graphs, we study how the boundary (the set of open edges, which carries spin degrees of freedom) is affected by the bulk, specifically by its combinatorial structure and by the quantum correlations among the intertwiners. In particular, we determine under which conditions certain classes of quantum gravity states map bulk degrees of freedom into boundary ones isometrically (which is a necessary condition for holography). We then look at the entanglement entropy of the boundary and recover, for slightly entangled intertwiners, the Ryu-Takayanagi formula with corrections induced by the entanglement entropy of the bulk state. We also show that the presence of a region with highly entangled intertwiners deforms the minimal-area surface, which is prevented from entering that region when the entanglement entropy of the latter exceeds a certain bound, a mechanism which thus leads to the rise of a black hole-like region in the bulk.
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