  Building on a recent investigation of the Shastry-Sutherland model [S. Wessel
et al., Phys. Rev. B 98, 174432 (2018)], we develop a general strategy to
eliminate the Monte Carlo sign problem near the zero temperature limit in
frustrated quantum spin models. If the Hamiltonian of interest and the
sign-problem-free Hamiltonian---obtained by making all off-diagonal elements
negative in a given basis---have the same ground state and this state is a
member of the computational basis, then the average sign returns to one as the
temperature goes to zero. We illustrate this technique by studying the
triangular and kagome lattice Heisenberg antiferrromagnet in a magnetic field
above saturation, as well as the Heisenberg antiferromagnet on a modified
Husimi cactus in the dimer basis. We also provide detailed appendices on using
linear programming techniques to automatically generate efficient directed loop
updates in quantum Monte Carlo simulations.
