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Free-fermion dynamics with measurements: Topological classification and adaptive preparation of topological states

Asadullah Bhuiyan1,*, Haining Pan2, and Chao-Ming Jian1

  • 1Department of Physics, Cornell University, Ithaca, New York 14853, USA
  • 2Department of Physics and Astronomy, Center for Materials Theory, Rutgers University, Piscataway, New Jersey 08854, USA

  • *Contact author: ab2398@cornell.edu
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Phys. Rev. Research 8, 023147 – Published 11 May, 2026

DOI: https://doi.org/10.1103/nk38-ygyq

Abstract

We develop a general framework for classifying fermionic dynamical systems with measurements using symmetry and topology. We introduce two complementary classification schemes based on the Altland-Zirnbauer tenfold way: (1) the many-body evolution operator (mEO) symmetry class, which classifies fermionic dynamics at the many-body level and naturally extends to interacting dynamics, and (2) the single-particle transfer matrix symmetry class, which classifies free-fermion dynamics at the single-particle level and connects to Anderson localization physics. In the free-fermion limit, we show that these two frameworks are equivalent via a dynamical bulk-boundary correspondence: The topology of the dynamical system's spacetime bulk determines the topology of the area-law-entangled steady-state ensemble living on its temporal boundary. Next, we prove that symmetry-invariant, postselection-free Gaussian measurements are realizable in only four of the ten mEO classes (A, AI, BDI, D); the remaining six require either postselection or interacting (non-Gaussian) measurements. Building on these results, we construct general postselection-free topological adaptive circuits that realize topological dynamical phases in any spatial dimension for the four admissible mEO classes. These circuits simultaneously provide a protocol for preparing and stabilizing free-fermion topological states in all ten symmetry classes. As a concrete demonstration, we construct and simulate 2+1d adaptive circuits that realize mEO-class-A topological dynamics, steering toward a steady-state ensemble of Chern insulators in O(1) circuit depth. Finally, we numerically characterize topological phase transitions, dynamical domain-wall modes, and robustness to coherent noise, identifying finite error thresholds at which trajectory-resolved and trajectory-averaged quantities undergo distinct phase transitions.

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Phys. Rev. Research 8, 023147– Published 11 May, 2026
  • Received 9 August 2025
  • Accepted 6 April 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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