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Efficient and simple Gibbs state preparation of the two-dimensional toric code via duality to classical Ising chains

Pablo Páez-Velasco1,2,*, Niclas Schilling3,†, Samuel O. Scalet4,5,‡, Frank Verstraete4,6,§, and Ángela Capel4,7,∥

  • *Contact author: pablopaez@ucm.es
  • †Contact author: niclas.schilling@student.uni-tuebingen.de
  • ‡Contact author: sos25@cam.ac.uk
  • §Contact author: fv285@cam.ac.uk
  • ∥Contact author: ac2722@cam.ac.uk
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Phys. Rev. Research 8, 023277 – Published 12 June, 2026

DOI: https://doi.org/10.1103/sbxq-sqvp

Abstract

We introduce the notion of polynomial-depth duality transformations, which relates two sets of operator algebras through a conjugation by a poly-depth quantum circuit, and make use of this to construct efficient Gibbs samplers for a variety of interesting quantum Hamiltonians as they are poly-depth dual to classical Hamiltonians. This is, for example, the case for the two-dimensional toric code, which is demonstrated to be poly-depth dual to two decoupled classical Ising spin chains for any system size, and we give evidence that such dualities hold for a wide class of stabilizer Hamiltonians. Additionally, we extend the above notion of duality to Lindbladians in order to show that mixing times and other quantities such as the spectral gap or the modified logarithmic Sobolev inequality are preserved under duality.

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Phys. Rev. Research 8, 023277– Published 12 June, 2026
  • Received 17 September 2025
  • Accepted 15 May 2026
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