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On the Theory of Matchgate Computations

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Abstract

Valiant has proposed a new theory of algorithmic computation based on perfect matchings and Pfaffians. We study the properties of matchgates—the basic building blocks in this new theory. We give a set of algebraic identities which completely characterizes these objects for arbitrary numbers of inputs and outputs. These identities are derived from Grassmann-Plücker identities. The 4 by 4 matchgate character matrices are of particular interest. These were used in Valiant’s classical simulation of a fragment of quantum computations. For these 4 by 4 matchgates, we use Jacobi’s theorem on compound matrices to prove that the invertible matchgate matrices form a multiplicative group. Our results can also be expressed in the theory of Holographic Algorithms in terms of realizable standard signatures. These results are useful in establishing limitations on the ultimate capabilities of Valiant’s theory of matchgate computations and Holographic Algorithms.

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Authors and Affiliations

  1. Computer Sciences Dept., University of Wisconsin, Madison, WI, 53706, USA

    Jin-Yi Cai & Vinay Choudhary

  2. Dept. of Computer Sc. & Tech., Tsinghua University, Beijing, 100084, People’s Republic of China

    Pinyan Lu

Authors
  1. Jin-Yi Cai
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  2. Vinay Choudhary
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  3. Pinyan Lu
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Correspondence to Pinyan Lu.

Additional information

J.-Y. Cai supported by NSF CCR-0208013 and CCR-0511679. A preliminary version of this paper will appear in Proceedings of IEEE Conference on Computational Complexity 2007, pp. 305–318.

V. Choudhary supported by NSF CCR-0208013.

P. Lu supported by NSF CCR-0511679 and by the National Natural Science Foundation of China Grant 60553001 and the National Basic Research Program of China Grant 2007CB807900, 2007CB807901.

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Cai, JY., Choudhary, V. & Lu, P. On the Theory of Matchgate Computations. Theory Comput Syst 45, 108–132 (2009). https://doi.org/10.1007/s00224-007-9092-8

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  • DOI: https://doi.org/10.1007/s00224-007-9092-8

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