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

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 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.