More details in the paper! Thanks to my amazing collaborators: Paolo Braccia, Élie Gouzien, @mvscerezo.bsky.social and @dgarciamartin.bsky.social 🙌

Meme credits go to Paolo

The previous result allows us to formulate exact matchgate synthesis as a Boolean Satisfiability (SAT) instance, enabling depth-optimal sequences.

When the target matchgates have additional algebraic structure, we can always find a sequence of gates from the generating set that implements the target exactly, without ancillas.

We also prove that an error in the SO(2n) compressed representation amplifies at most linearly in n in the corresponding unitary, enabling efficient approximate compilation schemes.

We provide a strategy to compile Rz and Rxx rotations using only matchgates by identifying an isomorphism between a two-qubit matchgate subgroup and SU(2).

We identify a finite gate set that densely generates the matchgate group, which corresponds to the generators of the matchgate-Clifford group (the intersection of the matchgate and Clifford groups) plus the T gate (up to a phase).

Matchgate circuits are connected to free-fermionic simulation and are useful primitives for benchmarking and verification. They admit a succinct representation in SO(2n) as poly(n)-sized matrices.