  In the era of noisy, intermediate-scale quantum (NISQ) devices, the efficient
preparation of many-body resource states is a task of paramount importance. In
this paper we focus on the deterministic preparation of matrix-product states
(MPS) in constant depth by utilizing measurements and classical communication
to fuse smaller states into larger ones. We place strong constraints on the MPS
that can be prepared using this method, which we refer to as MPS fusion.
Namely, we establish that it is necessary for the MPS to have a flat
entanglement spectrum. Using the recently introduced split-index MPS (SIMPS)
representation, we then introduce a family of states that belong to interesting
phases of matter protected by non-onsite symmetries and serve as resources for
long-range quantum teleportation, but which lie beyond the scope of ordinary
MPS fusion. It is shown constructively that these states can be prepared in
constant depth using a broader class of measurement-assisted protocols, which
we dub SIMPS fusion. Even in cases when MPS fusion is possible, using SIMPS
fusion can give rise to significantly reduced resource overhead. Our results
therefore simultaneously establish the boundaries of conventional MPS fusion
and push the envelope of which states can be prepared using
measurement-assisted protocols.
