近日，清华大学的刘子文及其研究团队取得一项新进展。经过不懈努力，他们揭示近似量子纠错码中的复杂性和有序性。相关研究成果已于2024年9月3日在国际知名学术期刊《自然—物理学》上发表。

本研究在量子电路复杂性与近似量子纠错（AQEC）能力之间建立了严格的联系。该研究的分析涵盖了具有全连接性以及诸如格点系统等几何情景的系统。为此，研究人员引入了一种称为子系统方差的代码参数，它与最优AQEC精度密切相关。对于一个在n个物理量子比特中编码k个逻辑量子比特的代码，研究发现，如果子系统方差低于O(k/n)的阈值，那么代码子空间中的任何状态都必须遵守特定的电路复杂性下界，这有助于识别代码的非平凡相。

这一AQEC理论为理解多体量子系统中的量子复杂性和有序性提供了一个多功能框架，为诸如拓扑序和临界量子系统等广泛重要的物理情景提供了新的见解。这项研究结果表明，O(1/n)代表了一个普遍且物理意义深远的子系统方差缩放阈值，该阈值与与非平凡量子序相关的特征相关联。在量子计算和多体物理学中，近似（而非精确）量子纠错是一个有用但相对未充分探索的概念。现在，基于与量子电路复杂性的联系，已经建立了一个理论框架。

据悉，为了制造出大规模容错量子计算机，必须采用某种形式的量子纠错，这一技术在物理学领域具有广泛的适用性。大多数研究通常假设能够实现精确纠错。然而，在许多实际和物理环境中，仅能实现近似量子纠错（AQEC）的编码可能同样有用，且本身具有重要意义。

附：英文原文

Title: Complexity and order in approximate quantum error-correcting codes

Author: Yi, Jinmin, Ye, Weicheng, Gottesman, Daniel, Liu, Zi-Wen

Issue&Volume: 2024-09-03

Abstract: Some form of quantum error correction is necessary to produce large-scale fault-tolerant quantum computers and finds broad relevance in physics. Most studies customarily assume exact correction. However, codes that may only enable approximate quantum error correction (AQEC) could be useful and intrinsically important in many practical and physical contexts. Here we establish rigorous connections between quantum circuit complexity and AQEC capability. Our analysis covers systems with both all-to-all connectivity and geometric scenarios like lattice systems. To this end, we introduce a type of code parameter that we call subsystem variance, which is closely related to the optimal AQEC precision. For a code encoding k logical qubits in n physical qubits, we find that if the subsystem variance is below an O(k/n) threshold, then any state in the code subspace must obey certain circuit complexity lower bounds, which identify non-trivial phases of codes. This theory of AQEC provides a versatile framework for understanding quantum complexity and order in many-body quantum systems, generating new insights for wide-ranging important physical scenarios such as topological order and critical quantum systems. Our results suggest that O(1/n) represents a common, physically profound scaling threshold of subsystem variance for features associated with non-trivial quantum order. Approximate—rather than exact—quantum error correction is a useful but relatively unexplored idea in quantum computing and many-body physics. A theoretical framework has now been established based on connections with quantum circuit complexity.

DOI: 10.1038/s41567-024-02621-x

Source: https://www.nature.com/articles/s41567-024-02621-x