The computational difficulty of finding MPS ground states

TL;DR

This paper explores the computational complexity of finding ground states in 1D Hamiltonians represented as MPS, revealing that the problem can be as hard as factoring and NP-complete, challenging existing variational methods.

Contribution

It constructs classes of 1D Hamiltonians with MPS ground states that are computationally hard to solve, highlighting limitations of current algorithms like DMRG.

Findings

Finding MPS ground states can be as hard as factoring.

Determining ground states can be NP-complete.

Certifying ground states requires more than just MPS and a spectral gap.

Abstract

We determine the computational difficulty of finding ground states of one-dimensional (1D) Hamiltonians which are known to be Matrix Product States (MPS). To this end, we construct a class of 1D frustration free Hamiltonians with unique MPS ground states and a polynomial gap above, for which finding the ground state is at least as hard as factoring. By lifting the requirement of a unique ground state, we obtain a class for which finding the ground state solves an NP-complete problem. Therefore, for these Hamiltonians it is not even possible to certify that the ground state has been found. Our results thus imply that in order to prove convergence of variational methods over MPS, as the Density Matrix Renormalization Group, one has to put more requirements than just MPS ground states and a polynomial spectral gap.