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

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.

19 pages, 1 figure. A more detailed exposition can be found in arXiv:1904.03507. Erratum: Example 3.3 does not satisfy Assumption 3.1 (4)

References in corpus (3)

• Lieb-Robinson Bounds for Harmonic and Anharmonic Lattice Systems
• Stochastic Error Cancellation in Analog Quantum Simulation
• Low-Rank Approximability and Entropy Area Laws for Ground States of Unbounded Hamiltonians