Approximation quality of finite-level local-consistency relaxations

Determine how accurately the t-th level of the locally consistent marginal hierarchy approximates the exact ground-state energy of a general quantum Hamiltonian.

Background

The paper defines a hierarchy of semidefinite relaxations based on locally consistent reduced density matrices. For each fixed level t, the relaxation is efficiently computable and provides a lower bound to the ground-state energy, while the exact value is recovered only at the full level. The authors establish exactness for several restricted families, including commuting Hamiltonians on trees, but do not characterize the approximation error for general Hamiltonians.

This unresolved question concerns the tightness of the hierarchy as a function of the level t and the structure of the Hamiltonian. Understanding it would determine when low-level local consistency constraints provide useful certified estimates of quantum ground-state energies.