Local Hamiltonian problem (${\lambda }_{\min }({\sum }_j{h}_j)\le a\text{or}{\lambda }_{\min }({\sum }_j{h}_j)\ge b$)

Given a polynomial list of fixed-locality Hermitian terms and thresholds $a<b$ separated by at least an inverse polynomial, distinguish whether the minimum eigenvalue of their sum is at most $a$ or at least $b$. Terms may be normalized to $0\le h_j\le I$ with polynomial rescaling. This promise problem is in QMA by local-energy measurement verification.

Uniformly sample one positive local term and measure the effect $h_j$. The averaged energy-flag effect on a witness is $H/M$, where $M$ is the number of terms. Repetition with a threshold halfway between the promised flag probabilities uses $O(M^2/(b-a{)}^2)$ witness registers for constant error. A spectral tensor-product analysis, as in QMA parallel repetition with entangled witnesses, proves soundness against entangled witnesses. Constant-locality positive operator-valued measures are efficiently implementable using one ancilla qubit.