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    Coherent error induced phase transition

    Hanchen Liu* and Xiao Chen†

    • Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA

    • *Contact author: hanchen.liu@bc.edu
    • †Contact author: chenaad@bc.edu

    Phys. Rev. B 114, 214106 – Published 19 October, 2026

    DOI: https://doi.org/10.1103/jly5-q9mx

    Abstract

    We investigate the stability of logical information in quantum stabilizer codes subject to coherent unitary errors. Beginning with a logical state, we sample a quenched coherent-unitary error realization and subsequently measure stabilizer checks, resulting in a syndrome-dependent post-measurement state. By examining both this syndrome state and the associated syndrome distribution, we identify a phase transition in the behavior of the logical information. In the Clifford/stabilizer setting, the change of logical stabilizer structure in a syndrome branch determines the state-level MAP Pauli-frame return probability for a chosen logical-basis input. We separately use quantum coherent information to diagnose recoverability of arbitrary encoded inputs at the channel level. Below a critical error threshold pc, the syndrome branches remain compatible with the original logical sector, enabling a high state-return probability. Above pc, the data are consistent with effective scrambling of the encoded subspace relative to the stabilizer checks, where the syndrome-resolved logical maps approach a global-Clifford-induced stabilizer instrument. This common post-threshold picture organizes both the toric-code and finite-rate random-stabilizer-code results within the same scrambling mechanism. The syndrome distribution supplies complementary local and global structural diagnostics whose scaling depends on the code family. We refer to this phenomenon as a coherent error induced phase transition. To illustrate this transition, we study two classes of quantum error-correcting codes—the toric code and finite-rate random stabilizer codes—thereby shedding light on the design and performance limits of quantum error correction under coherent errors.

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