Perils of embedding for quantum sampling
Abstract
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), 10.1103/PhysRevResearch.2.023020 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 transversefield Ising model, i.e., sampling a Hamiltonian with nonzero off diagonal terms. In the quantum case, loosely speaking, it is even harder to preserve the correct distribution properties, since the local transverse fields affect the physical qubits in the embedding in a manner that cannot be lifted by setting an appropriate energy scale, as in the classical case. 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 transversefield strengths than would otherwise be possible. Our numerics focus on models that can be implemented on current quantum devices using planar twodimensional lattices, which exhibit a phase transition driven by the transversefield strength. Our results include the following: (1) We give an estimate on the probability to sample the logical subspace directly as a function of transversefield, temperature, and total system size, which agrees with QMC simulations. (2) We show that typically measured observables (diagonal energy and magnetization) are biased by the embedding process, in the regime of intermediate transversefield strength, meaning that the extracted values are not the same as in the native model. (3) By considering individual embedding realizations akin to "realizations of disorder," we provide numerical evidence suggesting that as the embedding size is increased, the critical point shifts to increasingly large values of the transverse field.
 Publication:

Physical Review A
 Pub Date:
 February 2022
 DOI:
 10.1103/PhysRevA.105.022615
 arXiv:
 arXiv:2103.07036
 Bibcode:
 2022PhRvA.105b2615M
 Keywords:

 Quantum Physics
 EPrint:
 11+5 pages, 16 Figures. V2: updated to published version, including new results by averaging over embedding realizations