Matchgate hierarchy: A Clifford-like hierarchy for deterministic gate teleportation in matchgate circuits

The Clifford hierarchy, introduced by Gottesman and Chuang in 1999, is an increasing sequence of sets of quantum gates crucial to the gate teleportation model for fault-tolerant quantum computation. Gates in the hierarchy can be deterministically implemented, with increasing complexity, via gate teleportation using (adaptive) Clifford circuits with access to magic states. We propose an analogous gate teleportation protocol and a related hierarchy in the context of matchgate circuits, another restricted class of quantum circuits that can be efficiently classically simulated but are promoted to quantum universality via access to `matchgate-magic' states. The protocol deterministically implements any $n$-qubit gate in the hierarchy using adaptive matchgate circuits with magic states, with the level in the hierarchy indicating the required depth of adaptivity and thus number of magic states consumed. It also provides a whole family of novel deterministic matchgate-magic states. We completely characterise the gates in the matchgate hierarchy for two qubits, with the consequence that, in this case, the required number of resource states grows linearly with the target gate's level in the hierarchy. For an arbitrary number of qubits, we propose a characterisation of the matchgate hierarchy by leveraging the fermionic Stone$\unicode{x2013}$von Neumann theorem. It places a polynomial upper bound on the space requirements for representing gates at each level.