Matrix Product State and mean field solutions for one-dimensional systems can be found efficiently

TL;DR

This paper demonstrates that optimal solutions for mean field and Matrix Product State approximations of ground states in one-dimensional quantum systems can be found efficiently, with polynomial scaling, including for commuting Hamiltonians.

Contribution

It proves that both mean field and Matrix Product State solutions of fixed bond dimension can be computed in polynomial time, improving understanding of their computational complexity.

Findings

Optimal solutions for mean field and MPS can be found efficiently.

Ground states of 1D commuting Hamiltonians are efficiently computable.

Variational methods for these states scale polynomially.

Abstract

We consider the problem of approximating ground states of one-dimensional quantum systems within the two most common variational ansatzes, namely the mean field ansatz and Matrix Product States. We show that both for mean field and for Matrix Product States of fixed bond dimension, the optimal solutions can be found in a way which is provably efficient (i.e., scales polynomially). This implies that the corresponding variational methods can be in principle recast in a way which scales provably polynomially. Moreover, our findings imply that ground states of one-dimensional commuting Hamiltonians can be found efficiently.