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We study a parameterized version of the local Hamiltonian problem, called theweighted local Hamiltonian problem, where the relevant quantum states aresuperpositions of computational basis states of Hamming weight $k$. The Hammingweight constraint can have a physical interpretation as a constraint on thenumber of excitations allowed or particle number in a system. We prove thatthis problem is in QW[1], the first level of the quantum weft hierarchy andthat it is hard for QM[1], the quantum analogue of M[1]. Our results show thatthis problem cannot be fixed-parameter quantum tractable (FPQT) unless certainnatural quantum analogue of the exponential time hypothesis (ETH) is false.

Parameterized Complexity of Weighted Local Hamiltonian Problems and the Quantum Exponential Time Hypothesis