# Riemannian optimization of isometric tensor networks

### Submission summary

 As Contributors: Jutho Haegeman · Markus Hauru Arxiv Link: https://arxiv.org/abs/2007.03638v4 (pdf) Date accepted: 2021-02-03 Date submitted: 2021-01-14 11:21 Submitted by: Hauru, Markus Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Condensed Matter Physics - Computational Quantum Physics Approaches: Theoretical, Computational

### 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.

Published as SciPost Phys. 10, 040 (2021)

### Submission & Refereeing History

Resubmission 2007.03638v4 on 14 January 2021
Submission 2007.03638v3 on 8 December 2020

## Reports on this Submission

### Anonymous Report 2 on 2021-1-29 Invited Report

• Cite as: Anonymous, Report on arXiv:2007.03638v4, delivered 2021-01-29, doi: 10.21468/SciPost.Report.2476

### Strengths

Elegant and authoritative summary of how to best use isometric structure in the optimisation of tensor networks.

### Report

The authors have made a number of revisions to the manuscript in response to the first round of review. The additional discussion about the role of preconditioning, in particular, is very helpful.

The authors declined make modifications in response to point 1. I still think that there are advantages to writing the transformation as right multiplication of [W,W_\perp] by the appropriate unitary. The unitary structure combines the MPS left orthogonalization and left tangent gauge choice, and parallel transport expressed as right multiplication by a unitary explicitly preserves these two gauge choices. The parallel transport is also then explicitly independent of the point on the manifold from which it is performed. Both of these points are hidden a little in the present form of (9) and (10). However, this was mainly an aesthetic suggestion and it is acceptable that the authors decide not to follow it.

This is an informative and useful paper and I recommend it for publication.

• validity: top
• significance: top
• originality: high
• clarity: top
• formatting: perfect
• grammar: perfect

### Report 1 by Edwin Miles Stoudenmire on 2021-1-23 Invited Report

• Cite as: Edwin Miles Stoudenmire, Report on arXiv:2007.03638v4, delivered 2021-01-23, doi: 10.21468/SciPost.Report.2459

### Strengths

Based on my previous review, and the authors' response and modifications to the paper, I now recommend it for publication. The new material explaining and motivating preconditioning is helpful and welcome, including the pedagogical illustration.

### Report

I recommend this paper for publication in SciPost Physics.

• validity: top
• significance: high
• originality: high
• clarity: high
• formatting: perfect
• grammar: perfect