Optimizing random local Hamiltonians by dissipation

TL;DR

This paper demonstrates that a simplified quantum Gibbs sampling algorithm can efficiently approximate low-energy states of complex many-body Hamiltonians, highlighting potential quantum advantages in optimization tasks.

Contribution

It proves a quantum Gibbs sampling method achieves a significant approximation ratio for complex Hamiltonians, surpassing previous classical and quantum algorithms.

Findings

Quantum Gibbs sampling achieves a /k approximation of the optimal state.

The results suggest quantum algorithms can efficiently find low-energy states in complex models.

Quantum Gibbs sampling may serve as an effective metaheuristic for optimization problems.

Abstract

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 $\Omega(\frac{1}{k})$-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.