  The entanglement level statistics of a quantum state have recently been
proposed to be a signature of universality in the underlying quantum circuit.
This is a consequence of level repulsion in the entanglement spectra being tied
to the integrability of entanglement generated. However, such studies of the
level-spacing statistics in the entanglement spectrum have thus far been
limited to the output states of Clifford and Haar random circuits on product
state inputs. In this work, we provide the first example of a circuit which is
composed of a simulable gate set, yet has a Wigner-Dyson distributed
entanglement level spectrum without any perturbing universal element. We first
show that, for matchgate circuits acting on random product states, Wigner-Dyson
statistics emerge by virtue of a single SWAP gate, in direct analog to previous
studies on Clifford circuits. We then examine the entanglement spectrum of
matchgate circuits with varied input states, and find a sharp jump in the
complexity of entanglement as we go from two- to three-qubit entangled inputs.
Studying Clifford and matchgate hybrid circuits, we find examples of
classically simulable circuits whose output states exhibit Wigner-Dyson
entanglement level statistics in the absence of universal quantum gate
elements. Our study thus provides strong evidence that entanglement spectrum is
not strongly connected to notions of simulability in any given quantum circuit.
