  Density matrix quantum Monte Carlo (DMQMC) is a recently-developed method for
stochastically sampling the $N$-particle thermal density matrix to obtain
exact-on-average energies for model and \emph{ab initio} systems. We report a
systematic numerical study of the sign problem in DMQMC based on simulations of
atomic and molecular systems. In DMQMC, the density matrix is written in an
outer product basis of Slater determinants and has a size of space which is the
square of the number of Slater determinants. In principle this means DMQMC
needs to sample a space which scales in the system size, $N$, as
$\mathcal{O}[(\exp(N))^2]$. In practice, there is a system-dependent critical
walker population ($N_c$) which must be exceeded in order to remove the sign
problem, and this imposes limitations by way of storage and computer time. We
establish that $N_c$ for DMQMC is the square of $N_c$ for FCIQMC. By contrast,
the minimum $N_c$ in the interaction picture modification of DMQMC (IP-DMQMC)
only is directly proportionate to the $N_c$ for FCIQMC. We find that this comes
from the asymmetric propagation of IP-DMQMC compared to the symmetric
propagation of canonical DMQMC. An asymmetric mode of propagation is
prohibitively expensive for DMQMC because it has a much greater stochastic
error. Finally, we find that the equivalence between IP-DMQMC and FCIQMC seems
to extend to the initiator approximation, which is often required to study
larger basis sets and other systems. This suggests IP-DMQMC offers a way to
ameliorate the cost of moving between a Slater determinant space and an outer
product basis.
