Perils of embedding for sampling problems

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.