Riemannian optimization of isometric tensor networks

TL;DR

This paper introduces Riemannian optimization techniques for isometric tensor networks, improving the efficiency of ground state approximations in quantum many-body systems.

Contribution

It applies Riemannian manifold optimization methods to tensor networks of isometries, outperforming existing specialized algorithms for infinite MPS and MERA.

Findings

Riemannian optimization outperforms previous methods

Open-source implementations are provided

Effective for ground state calculations in 1D quantum systems

Abstract

Several tensor networks are built of isometric tensors, i.e. tensors satisfying $W^\dagger W = \mathrm{I}$. Prominent examples include matrix product states (MPS) in canonical form, the multiscale entanglement renormalization ansatz (MERA), and quantum circuits in general, such as those needed in state preparation and quantum variational eigensolvers. We show how gradient-based optimization methods on Riemannian manifolds can be used to optimize tensor networks of isometries to represent e.g. ground states of 1D quantum Hamiltonians. We discuss the geometry of Grassmann and Stiefel manifolds, the Riemannian manifolds of isometric tensors, and review how state-of-the-art optimization methods like nonlinear conjugate gradient and quasi-Newton algorithms can be implemented in this context. We apply these methods in the context of infinite MPS and MERA, and show benchmark results in which they outperform the best previously-known optimization methods, which are tailor-made for those specific variational classes. We also provide open-source implementations of our algorithms.