Valiant’s Holant Theorem and Matchgate Tensors

  • Jin-Yi Cai
  • Vinay Choudhary
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3959)


We propose matchgate tensors as a natural and proper language to develop Valiant’s new theory of Holographic Algorithms. We give a treatment of the central theorem in this theory—the Holant Theorem—in terms of matchgate tensors. Some generalizations are presented.


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  1. Dodson, C.T.J., Poston, T.: Tensor Geometry, Graduate Texts in Mathematics, 2nd edn., vol. 130. Springer, New York (1991)Google Scholar
  2. Jerrum, M., Snir, M.: Some Exact Complexity Results for Straight-Line Computations over Semirings. J. ACM 29(3), 874–897 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  3. Kasteleyn, P.W.: The statistics of dimers on a lattice. Physica 27, 1209–1225 (1961)CrossRefGoogle Scholar
  4. Kasteleyn, P.W.: Graph Theory and Crystal Physics. In: Harary, F. (ed.) Graph Theory and Theoretical Physics, pp. 43–110. Academic Press, London (1967)Google Scholar
  5. Strassen, V.: Gaussian Elimination is Not Optimal. Numerische Mathematik 13, 354–356 (1969)zbMATHCrossRefMathSciNetGoogle Scholar
  6. Tardos, É.: The gap between monotone and non-monotone circuit complexity is exponential. Combinatorica 8(1), 141–142 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  7. Temperley, H.N.V., Fisher, M.E.: Dimer problem in statistical mechanics –an exact result. Philosophical Magazine 6, 1061–1063 (1961)zbMATHCrossRefMathSciNetGoogle Scholar
  8. Valiant, L.G.: Negation can be Exponentially Powerful. Theor. Comput. Sci. 12, 303–314 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  9. Valiant, L.G.: Holographic Algorithms (Extended Abstract). In: Proc. 45th IEEE Symposium on Foundations of Computer Science, pp. 306–315 (2004) A more detailed version appeared in Electronic Colloquium on Computational Complexity Report TR05-099Google Scholar
  10. Valiant, L.G.: Holographic circuits. In: Proc. 32nd International Colloquium on Automata, Languages and Programming, pp. 1–15 (2005)Google Scholar
  11. Valiant, L.G.: Completeness for parity problems. In: Proc. 11th International Computing and Combinatorics Conference (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Jin-Yi Cai
    • 1
  • Vinay Choudhary
    • 1
  1. 1.Computer Sciences DepartmentUniversity of WisconsinMadisonUSA

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