On the Theory of Matchgate Computations

Abstract

Valiant has proposed a new theory of algorithmic computation based on perfect matchings and the Pfaffian. We study the properties of matchgates-the basic building blocks in this new theory. We give a set of algebraic identities which completely characterize these objects in terms of the Grassmann-Plücker identities. In the important case of 4 by 4 matchgate matrices, which was used in Valiant's classical simulation of a fragment of quantum computations, we further realize a group action on the character matrix of a matchgate, and relate this information to its compound matrix. Then we use Jacobi's theorem to prove that in this case the invertible matchgate matrices form a multiplicative group. These results are useful in establishing limitations on the ultimate capabilities of Valiant's theory of matchgate computations and his closely related theory of Holographic Algorithms.

Holographic algorithms

“…If a matchgate has arity n, the signature has size 2 n . However for symmetric signatures we have a compact form, and the running time of the decision algorithm is measured in n. Followed from this structural understanding we can give (i) a complete account of all the previous successes of holographic algorithms using symmetric signatures [29,5,32]; (ii) generalizations such as # 2 k −1 Pl-Rtw-Mon-kCNF and a similar problem for Vertex Cover, when this is possible; and (iii) a proof when this is not possible. This should be considered an important step in our understanding of holographic algorithms, from art to science.…”

“…These are part of the requirements for the bases to satisfy in order to be realizable. To be d-realizable is to have a d-dimensional solution subvariety in M for all realizability requirements, which include the parity requirements as well as the useful Grassmann-Plücker identities [5,28], called the matchgate identities. To have 0-realizability is a necessary condition.…”

Untitled

“…In the case of 2-input 2-output matchgates, these character values constitute a 4 by 4 matrix, called a character matrix. Subsequently in [1] and [2], this theory is generalized to matchgates of an arbitrary number of external nodes. The ultimate result is that there is an equivalence of matchgate characters (of not necessarily planar matchgates) and matchgate signatures (of planar matchgates).…”

“…These papers achieved the following general results: Firstly, there is essentially an equivalence between the character theory and the signature theory of matchgates, and secondly, a set of useful Grassmann-Plücker identities together with the Parity Condition are a necessary and sufficient condition for a sequence of values to be the signature of a planar matchgate. (The notion of "useful" was defined in [2].) This set of useful Grassmann-Plücker identities will be called Matchgate Identities (MGI) in the general sense.…”

“…Valiant used this to show that certain quantum gates cannot be simulated by the characters of general matchgates. In a sequence of two papers [1,2] a general study of the character theory and the signature theory of matchgates was undertaken. These papers achieved the following general results: Firstly, there is essentially an equivalence between the character theory and the signature theory of matchgates, and secondly, a set of useful Grassmann-Plücker identities together with the Parity Condition are a necessary and sufficient condition for a sequence of values to be the signature of a planar matchgate.…”