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

Michael J. Bremner; Zhengfeng Ji; Xingjian Li; Luke Mathieson; Mauro E. S. Morales

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arXiv · 2211.05325

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

Michael J. Bremner·Zhengfeng Ji·Xingjian Li·Luke Mathieson·Mauro E. S. Morales

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Abstract

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.

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Michael J. Bremner, Zhengfeng Ji, Xingjian Li, Luke Mathieson, Mauro E. S. Morales. 2022-11-10. Parameterized Complexity of Weighted Local Hamiltonian Problems and the Quantum Exponential Time Hypothesis. https://arxiv.org/abs/2211.05325

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