A New Holant Dichotomy Inspired by Quantum Computation

Author Miriam Backens

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Miriam Backens

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Miriam Backens. A New Holant Dichotomy Inspired by Quantum Computation. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


Holant problems are a framework for the analysis of counting complexity problems on graphs. This framework is simultaneously general enough to encompass many counting problems on graphs and specific enough to allow the derivation of dichotomy results, partitioning all problems into those which are in FP and those which are #P-hard. The Holant framework is based on the theory of holographic algorithms, which was originally inspired by concepts from quantum computation, but this connection appears not to have been explored before. Here, we employ quantum information theory to explain existing results in a concise way and to derive a dichotomy for a new family of problems, which we call Holant^+. This family sits in between the known families of Holant^*, for which a full dichotomy is known, and Holant^c, for which only a restricted dichotomy is known. Using knowledge from entanglement theory -- both previously existing work and new results of our own -- we prove a full dichotomy theorem for Holant^+, which is very similar to the restricted Holant^c dichotomy and may thus be a stepping stone to a full dichotomy for that family.
  • computational complexity
  • counting complexity
  • Holant
  • dichotomy
  • entanglement


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