Lower bounds to variational problems with guarantees

TL;DR

This paper introduces efficiently computable lower bounds for ground state energies in quantum many-body problems, enabling better validation of variational methods in translationally invariant lattice systems.

Contribution

It provides new scalable lower bounds and hierarchies that complement existing variational upper bounds for quantum lattice Hamiltonians.

Findings

Anderson bound and semi-definite relaxations scale with constant performance in energy density.

Hierarchies improve the Anderson bound systematically.

Lower bounds can be efficiently computed for translationally invariant systems.

Abstract

Variational methods play an important role in the study of quantum many-body problems, both in the flavor of classical variational principles based on tensor networks as well as of quantum variational principles in near-term quantum computing. This work stresses that for translationally invariant lattice Hamiltonians with periodic boundary conditions, one can easily derive efficiently computable lower bounds to ground state energies that can and should be compared with variational principles providing upper bounds. As small technical results, it is shown that (i) the Anderson bound and a (ii) common hierarchy of semi-definite relaxations both provide approximations with performance guarantees that scale like a constant in the energy density for cubic lattices. (iii) Also, the Anderson bound is systematically improved as a hierarchy of semi-definite relaxations inspired by the quantum marginal problem.