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Some Results on Matchgates and Holographic Algorithms

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Automata, Languages and Programming (ICALP 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4051))

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Abstract

We establish a 1-1 correspondence between Valiant’s character theory of matchgate/matchcircuit [14] and his signature theory of planar-matchgate/matchgrid [16], thus unifying the two theories in expressibility. In [3], we had established a complete characterization of general matchgates, in terms of a set of useful Grassmann-Plücker identities. With this correspondence, we give a corresponding set of identities which completely characterizes planar-matchgates and their signatures. Applying this characterization we prove some negative results for holographic algorithms. On the positive side, we also give a polynomial time algorithm for a simultaneous node-edge deletion problem, using holographic algorithms. Finally we give characterizations of symmetric signatures realizable in the Hadamard basis.

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© 2006 Springer-Verlag Berlin Heidelberg

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Cai, JY., Choudhary, V. (2006). Some Results on Matchgates and Holographic Algorithms. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds) Automata, Languages and Programming. ICALP 2006. Lecture Notes in Computer Science, vol 4051. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11786986_61

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  • DOI: https://doi.org/10.1007/11786986_61

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-35904-3

  • Online ISBN: 978-3-540-35905-0

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