  All known examples confirming the possibility of an exponential separation
between classical simulation algorithms and stoquastic adiabatic quantum
computing (AQC) exploit symmetries that constrain adiabatic dynamics to
effective, symmetric subspaces. The symmetries produce large effective
eigenvalue gaps, which in turn make adiabatic computation efficient. We present
a classical algorithm to efficiently sample from the effective subspace of a
$k$-local stoquastic Hamiltonian $H$, without a priori knowledge of its
symmetries (or near-symmetries). Our algorithm maps any $k$-local Hamiltonian
to a graph $G=(V,E)$ with $\lvert V \rvert = O\left(\mathrm{poly}(n)\right)$
where $n$ is the number of qubits. Given the well-known result of Babai, we
exploit graph isomorphism to study the automorphisms of $G$ and arrive at an
algorithm quasi-polynomial in $\lvert V\rvert$ for producing samples from the
effective subspace eigenstates of $H$. Our results rule out exponential
separations between stoquastic AQC and classical computation that arise from
hidden symmetries in $k$-local Hamiltonians. Furthermore, our graph
representation of $H$ is not limited to stoquastic Hamiltonians and may rule
out corresponding obstructions in non-stoquastic cases, or be useful in
studying additional properties of $k$-local Hamiltonians.
