  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 ab initio systems. We report a systemic
numerical study 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 $\mathcal{O}[(\exp(N))^2]$. We
find that, while DMQMC suffers a sign problem that scales with the size of the
outer product space, the interaction picture modification of DMQMC (IP-DMQMC)
has a sign problem which only scales with the size of original determinant
space. We achieve this by establishing that the critical walker population to
converge a simulation is the same between IP-DMQMC and full configuration
interaction quantum Monte Carlo (FCIQMC). This 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
ameliorates the cost of moving between a Slater determinant space and an outer
product basis.
