  We study the classical simulation complexity in both the weak and strong
senses, of matchgate (MG) computations supplemented with all combinations of
settings involving inclusion of intermediate adaptive or nonadaptive
computational basis measurements, product state or magic and general entangled
state inputs, and single- or multi-line outputs. We find a striking parallel to
known results for Clifford circuits, after some rebranding of resources. We
also give bounds on the amount of classical simulation effort required in case
of limited access intermediate measurements and entangled inputs. In further
settings we show that adaptive MG circuits remain classically efficiently
simulable if arbitrary two-qubit entangled input states on consecutive lines
are allowed, but become quantum universal for three or more lines. And if
adaptive measurements in non-computational bases are allowed, even with just
computational basis inputs, we get quantum universal power again.
