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  • Open Access

Quantum Subspace Verification for Error Correction Codes

Junjie Chen1, Pei Zeng2, Qi Zhao3, Xiongfeng Ma1, and You Zhou4,5,*

  • *Contact author: you_zhou@fudan.edu.cn
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PRX Quantum 6, 040353 – Published 3 December, 2025

DOI: https://doi.org/10.1103/fyw6-lq1l

Abstract

Quantum error correction is pivotal in advancing toward large-scale quantum computation, and efficient verification is crucial for ensuring the high fidelity of code states. Traditional methodologies, such as state tomography, direct fidelity estimation, and state verification, either fall short in measurement efficiency, especially for large-scale systems, or are restricted to some specific states. In this work, we introduce a general framework for quantum subspace verification, enabling efficient and measurement-noise-robust fidelity estimation between a given state and the target subspace with a specified confidence level. By integrating the proposed subspace verification with direct fidelity estimation, we develop a composite protocol that significantly improves the efficiency of verifying the fidelity of general magic logical states, as demonstrated by intuitive numerical results. This improvement stems from the use of subspace verification, which leverages the knowledge of code subspaces to significantly reduce measurement costs. Additionally, we detail the construction of verification operators for typical error correction codes, including general stabilizer codes and quantum low-density parity-check codes, enabling their efficient implementation using practical local measurements. Notably, for certain codes, such as the Calderbank-Shor-Steane codes and quantum low-density parity-check stabilizer codes, we reduce the number of required measurement settings and sample complexity to a constant level using graphical methods. Our approach facilitates efficient and feasible verification of error correction codes and generic magic logical states, advancing their practical implementations on quantum platforms.

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Popular Summary

Quantum information processing and quantum computing offer potential advantages in various areas, but their realization is hindered by noise. Quantum error correction (QEC) is essential for building large-scale quantum computers, though it introduces substantial overhead by requiring many physical qubits. Verifying the fidelity of encoded states becomes increasingly difficult as system size grows. Traditional approaches, such as direct fidelity estimation and state verification, are either resource intensive or applicable only to specific target states. We develop a general and efficient framework for subspace verification and integrate it with direct fidelity estimation to fill the gaps.

The framework of subspace verification we introduce can efficiently and robustly estimate how close a prepared state is to its target code subspace. Our framework extends the conventional hypothesis testing setup by tolerating measurement failures, making it robust against measurement noise. We design verification strategies for stabilizer and low-density parity-check code subspace using tools from graph theory. When integrated with direct fidelity estimation, our method greatly reduces measurement cost and supports the verification of generic “magic” logical states.

Our approach facilitates efficient and feasible verification of error correction codes and generic magic logical states, advancing their practical implementations on quantum platforms.

Article Text

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Outline

Information

PRX Quantum 6, 040353– Published 3 December, 2025
  • Received 12 March 2025
  • Revised 28 May 2025
  • Accepted 10 November 2025
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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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