Michael S Albergo

Sampling using SU(N) gauge equivariant flows

We develop a flow-based sampling algorithm for SU(N) lattice gauge theories that is gauge-invariant by construction. Our key contribution is constructing a class of flows on an SU(N) variable (or on a U(N) variable by a simple alternative) that respect matrix conjugation symmetry. We apply this technique to sample distributions of single SU(N) variables and to construct flow-based samplers for SU(2) and SU(3) lattice gauge theory in two dimensions.

Preprint: ArXiv

Equivariant flow-based sampling for lattice gauge theory

We define a class of machine-learned flow-based sampling algorithms for lattice gauge theories that are gauge invariant by construction. We demonstrate the application of this framework to U(1) gauge theory in two spacetime dimensions, and find that, at small bare coupling, the approach is orders of magnitude more efficient at sampling topological quantities than more traditional sampling procedures such as hybrid Monte Carlo and heat bath.

Published: Physical Review Letters

Normalizing Flows on Tori and Spheres

Normalizing flows are a powerful tool for building expressive distributions in high dimensions. So far, most of the literature has concentrated on learning flows on Euclidean spaces. Some problems however, such as those involving angles, are defined on spaces with more complex geometries, such as tori or spheres. In this paper, we propose and compare expressive and numerically stable flows on such spaces. Our flows are built recursively on the dimension of the space, starting from flows on circles, closed intervals or spheres.

Published: ICML 2020

Learnability scaling of quantum states: Restricted Boltzmann machines

Generative modeling with machine learning has provided a new perspective on the data-driven task of reconstructing quantum states from a set of qubit measurements. As increasingly large experimental quantum devices are built in laboratories, the question of how these machine learning techniques scale with the number of qubits is becoming crucial. We empirically study the scaling of restricted Boltzmann machines (RBMs) applied to reconstruct ground-state wavefunctions of the one-dimensional transverse-field Ising model from projective measurement data. We define a learning criterion via a threshold on the relative error in the energy estimator of the machine. With this criterion, we observe that the number of RBM weight parameters required for accurate representation of the ground state in the worst case - near criticality - scales quadratically with the number of qubits. By pruning small parameters of the trained model, we find that the number of weights can be significantly reduced while still retaining an accurate reconstruction. This provides evidence that over-parametrization of the RBM is required to facilitate the learning process.

Published: Physical Review B -- Editor's Suggestions

Preprint: ArXiv

Flow-based generative models for Markov chain Monte Carlo in lattice field theory

A Markov chain update scheme using a machine-learned flow-based generative model is proposed for Monte Carlo sampling in lattice field theories. The generative model may be optimized (trained) to produce samples from a distribution approximating the desired Boltzmann distribution determined by the lattice action of the theory being studied. Training the model systematically improves autocorrelation times in the Markov chain, even in regions of parameter space where standard Markov chain Monte Carlo algorithms exhibit critical slowing down in producing decorrelated updates. Moreover, the model may be trained without existing samples from the desired distribution. The algorithm is compared with HMC and local Metropolis sampling for ϕ⁴ theory in two dimensions.

Published: Physical Review D