Polynomial-Time Approximation of Zero-Free Partition Functions

Authors Penghui Yao, Yitong Yin, Xinyuan Zhang

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Penghui Yao
  • State Key Laboratory for Novel Software Technology, Nanjing University, China
Yitong Yin
  • State Key Laboratory for Novel Software Technology, Nanjing University, China
Xinyuan Zhang
  • State Key Laboratory for Novel Software Technology, Nanjing University, China

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Penghui Yao, Yitong Yin, and Xinyuan Zhang. Polynomial-Time Approximation of Zero-Free Partition Functions. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 108:1-108:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Zero-free based algorithms are a major technique for deterministic approximate counting. In Barvinok’s original framework [Barvinok, 2017], by calculating truncated Taylor expansions, a quasi-polynomial time algorithm was given for estimating zero-free partition functions. Patel and Regts [Patel and Regts, 2017] later gave a refinement of Barvinok’s framework, which gave a polynomial-time algorithm for a class of zero-free graph polynomials that can be expressed as counting induced subgraphs in bounded-degree graphs. In this paper, we give a polynomial-time algorithm for estimating classical and quantum partition functions specified by local Hamiltonians with bounded maximum degree, assuming a zero-free property for the temperature. Consequently, when the inverse temperature is close enough to zero by a constant gap, we have a polynomial-time approximation algorithm for all such partition functions. Our result is based on a new abstract framework that extends and generalizes the approach of Patel and Regts.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • partition function
  • zero-freeness
  • local Hamiltonian


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