  We consider the computational task of sampling a bit string $x$ from a
distribution $\pi(x)=|\langle x|\psi\rangle|^2$, where $\psi$ is the unique
ground state of a local Hamiltonian $H$. Our main result describes a direct
link between the inverse spectral gap of $H$ and the mixing time of an
associated continuous-time Markov Chain with steady state $\pi$. The Markov
Chain can be implemented efficiently whenever ratios of ground state amplitudes
$\langle y|\psi\rangle/\langle x|\psi\rangle$ are efficiently computable, the
spectral gap of $H$ is at least inverse polynomial in the system size, and the
starting state of the chain satisfies a mild technical condition that can be
efficiently checked. This extends a previously known relationship between
sign-problem free Hamiltonians and Markov chains. The tool which enables this
generalization is the so-called fixed-node Hamiltonian construction, previously
used in Quantum Monte Carlo simulations to address the fermionic sign problem.
We implement the proposed sampling algorithm numerically and use it to sample
from the ground state of Haldane-Shastry Hamiltonian with up to 56 qubits. We
observe empirically that our Markov chain based on the fixed-node Hamiltonian
mixes more rapidly than the standard Metropolis-Hastings Markov chain.
