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

Understand

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.