The strongest evidence for superiority of quantum annealing on spin glass problems has come from comparing simulated quantum annealing using quantum Monte Carlo (QMC) methods to simulated classical annealing [G. Santoro et al., Science 295, 2427(2002
)]. Motivated by experiments on programmable quantum annealing devices we revisit the question of when quantum speedup may be expected for Ising spin glass problems. We find that even though a better scaling compared to simulated classical annealing can be achieved for QMC simulations, this advantage is due to time discretization and measurements which are not possible on a physical quantum annealing device. QMC simulations in the physically relevant continuous time limit, on the other hand, do not show superiority. Our results imply that care has to be taken when using QMC simulations to assess quantum speedup potential and are consistent with recent arguments that no quantum speedup should be expected for two-dimensional spin glass problems.
The classical Heisenberg model on the trillium and distorted windmill lattices exhibits a degenerate ground state within large-$N$ theory, where the degenerate wavevectors form a surface and line, in 3-dimensional space, respectively. We name such st
ates partially ordered to represent the existence of long-range order along the direction normal to these degenerate manifolds. We investigate the effects of thermal fluctuations using Monte Carlo (MC) methods, and find a first order transition to a magnetically ordered state for both cases. We further show that the ordering on the distorted windmill lattice is due to order by disorder, while the ground state of the trillium lattice is unique. Despite these different routes to the realization of low temperature ordered phases, the static structure factors obtained by large-$N$ theory and MC simulations for each lattice show quantitative agreement in the cooperative paramagnetic regime at finite temperatures. This suggests that a remnant of the characteristic angle-dependent spin correlations of partial order remains above the transition temperatures for both lattices. The possible relevance of these results to $beta$-Mn, CeIrSi, and MnSi is discussed.
