Zero-Energy Problems for Supersymmetric Hamiltonians on a Chain Are $\mathrm{QMA}_1$-Complete

Abstract

We study exact zero modes of supersymmetric quantum systems whose interactions are arranged on a one-dimensional chain. For Hermitian supercharges, deciding whether a zero mode exists is $\mathrm{QMA}_1$-complete, and the hard instances are geometrically local on a chain. We also classify an explicitly encoded nilpotent $\mathcal{N}=2$ formulation: its exact-zero problem is $\mathrm{QMA}_1$-complete, including the restriction in which the associated Hamiltonian is local on a chain. Both classifications use a verification theorem for simultaneous sparse linear constraints and the same one-dimensional frustration-free Hamiltonian construction. The construction gives an explicit inverse-polynomial lower bound on the ground energy of every reduced NO instance, separating it from zero.