  Based on the density matrix renormalization group (DMRG), strongly correlated
quantum many-body systems at finite temperatures can be simulated by sampling
over a certain class of pure matrix product states (MPS) called minimally
entangled typical thermal states (METTS). When a system features symmetries,
these can be utilized to substantially reduce MPS computation costs. It is
conceptually straightforward to simulate canonical ensembles using symmetric
METTS. In practice, it is important to alternate between different symmetric
collapse bases to decrease autocorrelations in the Markov chain of METTS. To
this purpose, we introduce symmetric Fourier and Haar-random block bases that
are efficiently mixing. We also show how grand-canonical ensembles can be
simulated efficiently with symmetric METTS. We demonstrate these approaches for
spin-1/2 XXZ chains and discuss how the choice of the collapse bases influences
autocorrelations as well as the distribution of measurement values and, hence,
convergence speeds.
