The Perils of Embedding for Quantum Sampling*

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 in the computational basis a Hamiltonian with non-zero off diagonal terms. In the quantum case, loosely speaking, it is even harder to preserve the correct distribution properties, due to the fact it is typically not possible to diagonalize the quantum Hamiltonian.
Using a quantum Monte-Carlo algorithm which is specifically adapted to take thermal samples from an embedded Hamiltonian, we i) provide theory and numerics showing there is an exponential reduction in the probability to sample the logical subspace directly as a function of the transverse field strength, and ii) show how certain observables can be biased by the embedding process. We study implications of this regarding criticality at a non-zero temperature phase transition in a 2D model.