  Given quantum hardware that enables sampling from a family of natively
implemented Hamiltonians, how well can one use that hardware to sample from a
Hamiltonian outside that family? A common approach is to minor embed the
desired Hamiltonian in a native Hamiltonian. In Phys. Rev. Research 2, 023020
(2020) it was shown that minor embedding can be detrimental for classical
thermal sampling. Here, we generalize these results by considering quantum
thermal sampling in the transverse-field Ising model, i.e. sampling a
Hamiltonian with non-zero off diagonal terms. To study these systems
numerically we introduce a modification to standard cluster update quantum
Monte-Carlo (QMC) techniques, which allows us to much more efficiently obtain
thermal samples of an embedded Hamiltonian, enabling us to simulate systems of
much larger sizes and larger transverse-field strengths than would otherwise be
possible. Our numerics focus on models that can be implemented on current
quantum devices using planar two-dimensional lattices, which exhibit
finite-temperature quantum phase transitions. Our results include: i) An
estimate on the probability to sample the logical subspace directly as a
function of transverse-field, temperature, and total system size, which agrees
with QMC simulations. ii) We show that typically measured observables (diagonal
energy and magnetization) are biased by the embedding process, in the regime of
intermediate transverse field strength, meaning that the extracted values are
not the same as in the native model. iii) We provide numerical evidence
suggesting that, as the embedding size is increased, the critical point shifts
to increasingly large values of the transverse-field, which qualitatively
agrees with a simple perturbation theory argument we put forward.
