A Subpolynomial-Time Algorithm for the Free Energy of One-Dimensional Quantum Systems in the Thermodynamic Limit

Authors Hamza Fawzi, Omar Fawzi, Samuel O. Scalet

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Author Details

Hamza Fawzi
  • Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK
Omar Fawzi
  • Univ Lyon, Inria, ENS Lyon, UCBL, LIP, Lyon, France
Samuel O. Scalet
  • Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK

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Hamza Fawzi, Omar Fawzi, and Samuel O. Scalet. A Subpolynomial-Time Algorithm for the Free Energy of One-Dimensional Quantum Systems in the Thermodynamic Limit. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 49:1-49:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We introduce a classical algorithm to approximate the free energy of local, translation-invariant, one-dimensional quantum systems in the thermodynamic limit of infinite chain size. While the ground state problem (i.e., the free energy at temperature T = 0) for these systems is expected to be computationally hard even for quantum computers, our algorithm runs for any fixed temperature T > 0 in subpolynomial time, i.e., in time O((1/ε)^c) for any constant c > 0 where ε is the additive approximation error. Previously, the best known algorithm had a runtime that is polynomial in 1/ε where the degree of the polynomial is exponential in the inverse temperature 1/T. Our algorithm is also particularly simple as it reduces to the computation of the spectral radius of a linear map. This linear map has an interpretation as a noncommutative transfer matrix and has been studied previously to prove results on the analyticity of the free energy and the decay of correlations. We also show that the corresponding eigenvector of this map gives an approximation of the marginal of the Gibbs state and thereby allows for the computation of various thermodynamic properties of the quantum system.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum information theory
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Quantum complexity theory
  • One-dimensional quantum systems
  • Free energy


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