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  • Open Access

Mixed-State Topological Order under Coherent Noise

Seunghun Lee and Eun-Gook Moon*

  • *Contact author: egmoon@kaist.ac.kr
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PRX Quantum 6, 030355 – Published 18 September, 2025

DOI: https://doi.org/10.1103/hchr-rqq9

Abstract

Mixed-state phases of matter under local decoherence have recently garnered significant attention due to the ubiquitous presence of noise in current quantum processors. One of the key issues is understanding how topological quantum memory is affected by realistic coherent noise, such as random rotation noise and amplitude-damping noise. In this work, we investigate the intrinsic error threshold of the two-dimensional toric code (TC), a paradigmatic topological quantum memory, under these types of coherent noise by employing both analytical and numerical methods based on the doubled-Hilbert-space formalism. A connection between the mixed-state phase of the decohered TC and a non-Hermitian Ashkin-Teller-type statistical-mechanics model is established, and the mixed-state phase diagrams under the coherent noise are obtained. We find remarkable stability of mixed-state topological order under random rotation noise with axes near the Y axis of qubits. We also identify intriguing extended critical regions at the phase boundaries, highlighting a connection with non-Hermitian physics. We argue that these phase boundaries provide upper bounds for the intrinsic error threshold, beyond which quantum error correction becomes impossible. We complement these findings by estimating the error thresholds for random rotation noise under standard quantum error correction, thereby providing lower bounds on the intrinsic error threshold.

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Corrections

25 February, 2026

Correction: Support statements in the Acknowledgments section have been clarified.

Popular Summary

Topologically ordered phases have recently been realized in quantum processors and offer a promising platform for topological quantum memory, which can protect quantum information against local decoherence. Previously, studies of their stability have mostly focused on simple incoherent noise, such as random bit or phase flips. However, real quantum devices also suffer coherent noise, including systematic unitary rotations from imperfect gate operations and energy loss via spontaneous photon emission. Understanding how these coherent errors impact topological quantum memories is therefore essential.

In this work, we address this question for the prototypical topological code—a toric code (TC)—through the lens of mixed-state phases.

We examine two coherent-noise models: random rotation noise and amplitude-damping noise. Employing the doubled-Hilbert-space formalism, we map the mixed-state phase of the decohered TC onto a non-Hermitian Ashkin-Teller model and argue that its phase boundary provides an upper bound on the intrinsic error threshold of the code, beyond which quantum error correction fails. We complement this with numerical estimates of a lower bound on the threshold. For random rotation noise, we also uncover remarkably wide stability regions and identify intriguing novel mixed-state critical phases. We expect that our mixed-state approach will provide fresh insights into how topological quantum memories behave under realistic coherent noise.

Article Text

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PRX Quantum 6, 030355– Published 18 September, 2025
  • Received 17 November 2024
  • Revised 4 March 2025
  • Accepted 7 July 2025
  • Corrected 25 February 2026
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