  Advances in techniques for thermal sampling in classical and quantum systems
would deepen understanding of the underlying physics. Unfortunately, one often
has to rely solely on inexact numerical simulation, due to the intractability
of computing the partition function in many systems of interest. Emerging
hardware, such as quantum annealers, provide novel tools for such
investigations, but it is well known that studying general, non-native systems
on such devices requires graph minor embedding, at the expense of introducing
additional variables. The effect of embedding for sampling is more pronounced
than for optimization; for optimization one is just concerned with the ground
state physics, whereas for sampling one needs to consider states at all
energies. We argue that as the system size or the embedding size grows, the
chance of a sample being in the subspace of interest - the logical subspace -
can be exponentially suppressed. Though the severity of this scaling can be
lessened through favorable parameter choices, certain physical constraints
(such as a fixed temperature and range of couplings) provide hard limits on
what is currently feasible. Furthermore, we show that up to some practical and
reasonable assumptions, any type of post-processing to project samples back
into the logical subspace will bias the resulting statistics. We introduce a
new such technique, based on resampling, that substantially outperforms
majority vote, which is shown to fail quite dramatically at preserving
distribution properties.
