Entanglement spectrum statistics in matchgate circuits with supplemental resources

We study the entanglement spectrum generated by matchgate circuits with various additional resources. We show that, for matchgate circuits acting on product states, Wigner-Dyson entanglement statistics emerges in the thermodynamic limit by virtue of a single SWAP gate. This is analogous to the Clifford case, in which only a single T-gate is needed to achieve universal entanglement spectrum statistics under similar system size scaling [1]. Additionally, we examine the entanglement complexity [2] of matchgate circuits with varied initial states, showing a jump in complexity when input states are composed of entangled products of three or more qubits. Our work clarifies the relationship between simulatability, entanglement entropy, and the complexity of entanglement in several different simulable gate sets.

[1] Zhou et. al., SciPost Phys. 9, 087 (2020)

[2] Shaffer et. al., J. Stat. Mech. (2014) P12007