Riemannian optimization of isometric tensor networks
Abstract
Several tensor networks are built of isometric tensors, i.e. tensors acting as linear operators $W$ satisfying $W^\dagger W = \mathrm{I}$. Prominent examples include matrix product states (MPS) in canonical form and the multiscale entanglement renormalization ansatz (MERA). Such tensor networks can also represent quantum circuits and are thus of interest for quantum computing tasks, such as state preparation and quantum variational eigensolvers. We show how wellknown methods of gradientbased optimization 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 stateoftheart optimization methods like nonlinear conjugate gradient and quasiNewton algorithms can be implemented in this context. We demonstrate how these methods can be applied in the context of infinite MPS and MERA, and show benchmark results that indicate that they can outperform current optimization methods, which are tailormade for those specific variational classes. We also provide opensource implementations of our algorithms.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 arXiv:
 arXiv:2007.03638
 Bibcode:
 2020arXiv200703638H
 Keywords:

 Quantum Physics;
 Condensed Matter  Strongly Correlated Electrons
 EPrint:
 13 pages, 2 figures