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  • Open Access

Approximate Quantum Error Correction with 1D Log-Depth Circuits

Guoding Liu1,*, Zhenyu Du1,*, Zi-Wen Liu2,†, and Xiongfeng Ma1,‡

  • 1Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
  • 2Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China

  • *These authors contributed equally to this work.
  • †Contact author: zwliu0@tsinghua.edu.cn
  • ‡Contact author: xma@tsinghua.edu.cn
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PRX Quantum 7, 010331 – Published 13 February, 2026

DOI: https://doi.org/10.1103/7rzk-2jyh

Abstract

Efficient and high-performance quantum error correction is essential for achieving fault-tolerant quantum computing. Low-depth random circuits offer a promising approach to identifying effective and practical encoding strategies. In this work, we rigorously prove through information-theoretic analysis that one-dimensional logarithmic-depth random Clifford encoding circuits can achieve high quantum error correction performance. We demonstrate that these random codes typically exhibit good approximate quantum error correction capability by proving that their encoding rate achieves the hashing bound for Pauli noise and the channel capacity for erasure errors. We show that the error correction inaccuracy decays once a threshold of logarithmic depth is exceeded, resulting in negligible recovery errors. This threshold is shown to be lower than that of the simple separate block encoding, and the decay rate is higher. We further establish that these codes are optimal by proving that logarithmic depth is necessary to maintain a constant encoding rate and high error correction performance. To prove our results, we propose decoupling theorems tailored for one-dimensional low-depth circuits. These results also imply strong decoupling and rapid thermalization properties in low-depth random circuits and have potential applications in quantum information science and physics.

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Popular Summary

Quantum computing promises advantages over classical computing, with potential breakthroughs in material science, artificial intelligence, and cryptography. However, current quantum devices are susceptible to noise, making reliable long-term calculations difficult. Protecting quantum information through efficient error correction is essential to making quantum technologies practical.

In this work, we introduce an error correction scheme that uses low-dimensional, shallow quantum circuits to protect quantum information. Rather than aiming for perfect recovery, we focus on approximate correction, where small recovery errors are acceptable. This flexibility allows for greater efficiency and practicality. Our scheme is built from one-dimensional, logarithmic-depth random Clifford circuits and achieves high error-correcting performance using far fewer resources than conventional approaches.

We show that the error decreases rapidly as circuit depth increases while maintaining a high encoding rate. Importantly, we also prove that our scheme is mathematically optimal: no shallower circuit can match this performance. This result strengthens the case for low-depth architectures in quantum error correction and points to a path toward fault-tolerant quantum computing using simpler hardware.

Our findings also shed light on the power of low-depth random circuits, which are gaining interest in quantum many-body physics. This opens avenues for research at the intersection of quantum information, condensed matter, and quantum technologies.

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PRX Quantum 7, 010331– Published 13 February, 2026
  • Received 7 April 2025
  • Revised 14 September 2025
  • Accepted 6 January 2026
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