Fermionic anyons: Entanglement and quantum computation from a resource-theoretic perspective

Quantum computational models can be approached via the lens of resources needed to perform computational tasks, where a computational advantage is achieved by consuming specific forms of quantum resources, or, conversely, resource-free computations are classically simulable. Can we similarly identify quantum computational resources in the setting of more general quasi-particle statistics? In this work, we develop a framework to characterize the separability of a specific type of one-dimensional quasiparticle known as a fermionic anyon. As we evince, the usual notion of partial trace fails in this scenario, so we build the notion of separability through a fractional Jordan-Wigner transformation, leading to an entanglement description of fermionic-anyon states. We apply this notion of fermionic-anyon separability and the unitary operations that preserve it, mapping it to the free resources of matchgate circuits. We also identify how entanglement between two qubits encoded in a dual-rail manner, as standard for matchgate circuits, corresponds to the notion of entanglement between fermionic anyons.