Matrix product state renormalization

TL;DR

This paper presents a renormalization group perspective on matrix product states (MPS), interpreting spectrum truncation as an impurity problem and introducing a hybrid variational approach to improve understanding of ground state approximations.

Contribution

It introduces a novel RG interpretation of MPS truncation as an impurity problem and develops a hybrid variational ansatz combining MPS and MERA ideas for better ground state analysis.

Findings

Verified impurity interpretation of MPS compression numerically.

Demonstrated usefulness of layered MPS structure for elementary excitations.

Connected MPS truncation to Wilson's numerical renormalization group.

Abstract

The truncation or compression of the spectrum of Schmidt values is inherent to the matrix product state (MPS) approximation of one-dimensional quantum ground states. We provide a renormalization group picture by interpreting this compression as an application of Wilson's numerical renormalization group along the imaginary time direction appearing in the path integral representation of the state. The location of the physical index is considered as an impurity in the transfer matrix and static MPS correlation functions are reinterpreted as dynamical impurity correlations. Coarse-graining the transfer matrix is performed using a hybrid variational ansatz based on matrix product operators, combining ideas of MPS and the multi-scale entanglement renormalization ansatz. Through numerical comparison with conventional MPS algorithms, we explicitly verify the impurity interpretation of MPS compression, as put forward by [V. Zauner et al., New J. Phys. 17, 053002 (2015)] for the transverse-field Ising model. Additionally, we motivate the conceptual usefulness of endowing MPS with an internal layered structure by studying restricted variational subspaces to describe elementary excitations on top of the ground state, which serves to elucidate a transparent renormalization group structure ingrained in MPS descriptions of ground states.