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

TL;DR

This paper investigates a parameterized quantum problem related to local Hamiltonians with constraints on quantum states, establishing its complexity class and linking it to a quantum exponential time hypothesis.

Contribution

It introduces the weighted local Hamiltonian problem, proves its placement in the quantum weft hierarchy, and connects its complexity to a quantum exponential time hypothesis.

Findings

The weighted local Hamiltonian problem is in QW[1].

It is hard for QM[1], indicating high complexity.

The problem's tractability depends on a quantum ETH.

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.