Ranks of Tensor Networks for Eigenspace Projections and the Curse of Dimensionality

TL;DR

This paper demonstrates that the ground state projection of certain unbounded Hamiltonians can be efficiently approximated using low-rank tensor networks, overcoming the typical curse of dimensionality in high-dimensional tensor representations.

Contribution

It shows that for a class of unbounded Hamiltonians, the ground state projection can be approximated with low effective dimensionality, independent of the tensor's high dimension.

Findings

Low effective dimensionality of GSP for certain Hamiltonians

Approximation of GSP avoids exponential complexity with dimension

Tensor network ranks can be controlled independently of dimension

Abstract

The hierarchical (multi-linear) rank of an order-$d$ tensor is key in determining the cost of representing a tensor as a (tree) Tensor Network (TN). In general, it is known that, for a fixed accuracy, a tensor with random entries cannot be expected to be efficiently approximable without the curse of dimensionality, i.e., a complexity growing exponentially with $d$. In this work, we show that the ground state projection (GSP) of a class of unbounded Hamiltonians can be approximately represented as an operator of low effective dimensionality that is independent of the (high) dimension $d$ of the GSP. This allows to approximate the GSP without the curse of dimensionality.