Dequantization Barriers for Guided Stoquastic Hamiltonians

TL;DR

This paper demonstrates that certain stoquastic Hamiltonian ground-state problems cannot be efficiently solved or sampled by classical algorithms, highlighting a fundamental quantum advantage in these cases.

Contribution

It constructs a specific problem based on spectral expanders that proves classical algorithms cannot efficiently perform guided ground-state preparation for a broad class of stoquastic Hamiltonians.

Findings

Classical algorithms fail to sample the constructed distribution efficiently.

The problem extends previous work by ruling out any classical approach for guided ground-state preparation.

Quantum algorithms can efficiently solve the problem, indicating a quantum advantage.

Abstract

We construct a probability distribution, induced by the Perron--Frobenius eigenvector of an exponentially large graph, which cannot be efficiently sampled by any classical algorithm, even when provided with the best-possible warm-start distribution. In the quantum setting, this problem can be viewed as preparing the ground state of a stoquastic Hamiltonian given a guiding state as input, and is known to be efficiently solvable on a quantum computer. Our result suggests that no efficient classical algorithm can solve a broad class of stoquastic ground-state problems. Our graph is constructed from a class of high-degree, high-girth spectral expanders to which self-similar trees are attached. This builds on and extends prior work of Gily\'en, Hastings, and Vazirani [Quantum 2021, STOC 2021], which ruled out dequantization for a specific stoquastic adiabatic path algorithm. We strengthen their result by ruling out any classical algorithm for guided ground-state preparation.