Stable matrix product state simulations with BUGs

Tensor networks have become a standard computational tool for simulating quantum many-body dynamics. Their time evolution requires the efficient and accurate update of low-rank tensor factors. In recent years, several new numerical schemes have been proposed for this task. All these methods can be understood within the unified framework of Basis Update and Galerkin (BUG) integrators, in which the basis matrices are evolved by low-dimensional matrix differential equations, while the core tensors are updated through a Galerkin projection step.

We begin with an introduction to matrix product states, also known as tensor trains in the mathematical literature. Further, it will introduce the basic ideas behind the BUG integrator. Particular emphasis will be placed on their remarkable properties like rank-adaptivity, parallelism and the preservation of physical invariants such as norm and energy. Numerical experiments from quantum physics will demonstrate the theoretical results and explore the potential of BUG integrators for large-scale many-body quantum systems.