A Theory for Valiant's Matchcircuits (Extended Abstract)

TL;DR

This paper proves that nonsingular character matrices of matchgates form a group under inversion and establishes the universality of 1- and 2-bit matchgates for matchcircuits, extending prior results.

Contribution

It extends the group property of nonsingular matchgate matrices to all sizes and confirms the universality of small matchgates, answering longstanding open questions.

Findings

Nonsingular character matrices form a group under inversion.

1- and 2-bit matchgates are universal for matchcircuits.

Extended the result of Cai and Choudhary to all k-bit matchgates.

Abstract

The computational function of a matchgate is represented by its character matrix. In this article, we show that all nonsingular character matrices are closed under matrix inverse operation, so that for every $k$, the nonsingular character matrices of $k$-bit matchgates form a group, extending the recent work of Cai and Choudhary (2006) of the same result for the case of $k=2$, and that the single and the two-bit matchgates are universal for matchcircuits, answering a question of Valiant (2002).