Arithmetic Tensor Networks and Integration - IPAM at UCLA

Explore a 32-minute lecture on arithmetic tensor networks and integration presented by Garnet Chan from the California Institute of Technology at IPAM's Quantum Numerical Linear Algebra Workshop. Delve into the intricacies of performing arithmetic with tensor networks and its implications for function integration. Examine topics such as tensor network contraction, approximate contraction techniques, and their applications in multivariable function integration. Discover how arithmetic circuit representations and tensor network circuits can be utilized for polynomial integration, and investigate the relationship between exact and approximate contractions. Gain insights into the dependence of integration on various factors, including the number of variables and the nature of the integrand. Conclude with a discussion on Gaussian integration in a hypercube, providing a comprehensive overview of this advanced mathematical concept.

Intro
Tensor networks and quantum circuits
Tensor network contraction encodes many problems
Exact tensor network contraction
Approximate tensor network contraction
Approximate contraction often works well
Multivariable function integration
From quadrature to tensor networks
Arithmetic circuit representations
Arithmetic tensor network circuits
Examples
Polynomial integration
Exact contraction
Approximate contraction and integration
Simple case
Discovering identities
General case
Away from exact compressibility
dependence on number of variables N
Dependence on integrand
Integrand dependence cont'd
Gaussian integration in a hypercube
Summary