Computational power of matchgates with supplementary resources

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 multiple-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 the
case of limited access to 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 noncomputational bases are allowed, even with just computational basis inputs, we get quantum
universal power again.