Revised: December 17, 2013

Published: August 12, 2014

**Keywords:**complexity theory, matchgates, Pfaffian orientation

**ACM Classification:**F.1.3, F.2.2, G.2.1, G.2.2

**AMS Classification:**03D15, 05C70, 68R10

**Abstract:**
[Plain Text Version]

We study a collection of concepts and theorems that laid the foundation of matchgate computation. This includes the signature theory of planar matchgates, and the parallel theory of characters of not necessarily planar matchgates. Our aim is to present a unified and, whenever possible, simplified account of this challenging theory. Our results include: (1) A direct proof that the Matchgate Identities (MGI) are necessary and sufficient conditions for matchgate signatures. This proof is self-contained and does not go through the character theory. (2) A proof that the MGI already imply the Parity Condition. (3) A simplified construction of a crossover gadget. This is used in the proof of sufficiency of the MGI for matchgate signatures. This is also used to give a proof of equivalence between the signature theory and the character theory which permits omittable nodes. (4) A direct construction of matchgates realizing all matchgate-realizable symmetric signatures.