A Theory for Valiant's Matchcircuits (Extended 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 [1] 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 [4].