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  • Open Access

Fundamental Thresholds for Computational and Erasure Errors via the Coherent Information

Luis Colmenarez1,2,*, Seyong Kim3, and Markus Müller1,2

  • *Contact author: colmenarez@physik.rwth-aachen.de
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PRX Quantum 6, 040327 – Published 7 November, 2025

DOI: https://doi.org/10.1103/d8rx-srpn

Abstract

Quantum error correcting (QEC) codes protect quantum information against environmental noise. Computational errors caused by the environment change the quantum state within the qubit subspace, whereas quantum erasures correspond to the loss of qubits at known positions. Correcting either type of error involves different correction mechanisms, which makes studying the interplay between erasure and computational errors particularly challenging. In this work, we propose a framework based on the coherent information (CI) of the mixed-state density operator associated to noisy QEC codes, for treating both types of errors together. We show how to rigorously derive different families of statistical mechanics mappings for generic stabilizer QEC codes in the presence of both types of errors. We observe that the erasure errors enter as a classical average over fully depolarizing channels. Further, we show that computing the CI for erasure errors only can be done efficiently upon sampling over erasure configurations. We then test our approach on the two-dimensional toric and color codes and compute optimal thresholds for erasure errors only, finding a 50% threshold for both codes. This strengthens the notion that both codes share the same optimal thresholds. When considering both computational and erasure errors, the CI of small-size codes yields thresholds in very accurate agreement with established results that have been obtained in the thermodynamic limit. Next, we perform a similar analysis for a low-density parity-check (LDPC) code—the lift-connected surface code. We find a 50% threshold under erasure errors alone and, for the first time, derive the exact statistical mechanics mappings in the presence of both computational and erasure errors. We thereby further establish the CI as a practical tool for studying optimal thresholds for code classes beyond topological codes under realistic noise, and as a means for uncovering new relations between QEC codes and statistical physics models.

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Popular Summary

Quantum computers are powerful but extremely delicate—they store information in qubits, which are easily disrupted. Two main disturbances can occur: computational errors, where the qubit’s value changes by mistake, and erasure errors, where a qubit disappears completely. Most quantum error correction methods handle just one type of error at a time. This research introduces a new approach that can analyse settings where both errors appear at once.

We use a concept called coherent information, which helps one to measure how much useful information can still be recovered after errors happen. We also borrow ideas from classical statistical physics by turning the problem into a model of interacting spins, which makes the mathematical analysis more manageable. This method can accurately estimate how many errors a system can handle at best, when using optimal ways to recover the encoded quantum information. Our work not only confirms that two common types of quantum error correcting codes—called toric codes and color codes—and a new code-called lift-connected surface code can still function properly if up to 50% of the qubits are erased. Even when both types of errors appear together, the results found for small systems match very closely what is expected for very large systems, which shows the reliability of the method.

This work offers a new and practical way to test how well quantum error correcting codes can perform in real-world conditions, and it also builds a stronger link between quantum computing and the concepts and tools used in classical statistical physics.

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PRX Quantum 6, 040327– Published 7 November, 2025
  • Received 21 February 2025
  • Revised 30 June 2025
  • Accepted 18 September 2025
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