  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 particle number in a system. We prove that
this problem is in QW[1], the first level of the quantum weft hierarchy and
that it is hard for QM[1], the quantum analogue of M[1]. 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.
