A central challenge in quantum simulation is to prepare low-energy states of strongly interacting many-body systems. In this work, we study the problem of preparing a quantum state that optimizes a random all-to-all, sparse or dense, spin or fermionic k-local Hamiltonian. We prove that a simplified quantum Gibbs sampling algorithm achieves a Ω(k1)-fraction approximation of the optimum, giving an exponential improvement on the k-dependence over the prior best (both classical and quantum) algorithmic guarantees. Combined with the circuit lower bound for such states, our results suggest that finding low-energy states for sparsified (quasi)local spin and fermionic models is quantumly easy but classically nontrivial. This further indicates that quantum Gibbs sampling may be a suitable metaheuristic for optimization problems.
@article{arxiv.2411.02578,
title = {Optimizing random local Hamiltonians by dissipation},
author = {Joao Basso and Chi-Fang Chen and Alexander M. Dalzell},
journal= {arXiv preprint arXiv:2411.02578},
year = {2024}
}