We study a parameterized version of the local Hamiltonian problem, called the weighted local Hamiltonian problem, where the relevant quantum states are superpositions of computational basis states of Hamming weight k. The Hamming weight constraint can have a physical interpretation as a constraint on the number of excitations allowed or the particle number in a system. We prove that this problem is in \ (QW1 \), the first level of the quantum weft hierarchy, and that it is hard for \ (QM1 \), the quantum analogue of \ (M1 \). Our results show that this problem cannot be fixed parameter quantum tractable (FPQT) unless certain natural quantum analogue of the exponential time hypothesis (ETH) is false.
Bremner et al. (Fri,) studied this question.