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).
@InProceedings{li_et_al:LIPIcs.STACS.2008.1368, author = {Li, Angsheng and Xia, Mingji}, title = {{A Theory for Valiant's Matchcircuits}}, booktitle = {25th International Symposium on Theoretical Aspects of Computer Science}, pages = {491--502}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-06-4}, ISSN = {1868-8969}, year = {2008}, volume = {1}, editor = {Albers, Susanne and Weil, Pascal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1368}, URN = {urn:nbn:de:0030-drops-13686}, doi = {10.4230/LIPIcs.STACS.2008.1368}, annote = {Keywords: Pfaffian, Matchgate, Matchcircuit} }
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