  The algorithm ``automated compression of environments'' (ACE) [Nat. Phys. 18,
662 (2022)] provides a versatile way of simulating an extremely broad class of
open quantum systems. This is achieved by encapsulating the influence of the
environment, which is determined by the interaction Hamiltonian(s) and initial
states, into compact process tensor matrix product operator (PT-MPO)
representations. The generality of the ACE method comes at high numerical cost.
Here, we demonstrate that orders-of-magnitude improvement of ACE is possible by
changing the order of PT-MPO contraction from a sequential to a tree-like
scheme. The problem of combining two partial PT-MPOs with large inner bonds is
solved by a preselection approach. The drawbacks of the preselection approach
are that the MPO compression is suboptimal and that it is more prone to error
accumulation than sequential combination and compression. We therefore also
identify strategies to mitigate these disadvantages by fine-tuning compression
parameters. This results in a scheme that is similar in compression efficiency
and accuracy to the original ACE algorithm, yet is significantly faster. Our
numerical experiments reach similar conclusions for bosonic and fermionic test
cases, suggesting that our findings are characteristic of the combination of
PT-MPOs more generally.
