Complexity and order in approximate quantum error-correcting codes

• URL: http://arxiv.org/abs/2310.04710v2
• Date: Wed, 29 May 2024 02:57:17 GMT
• Title: Complexity and order in approximate quantum error-correcting codes
• Authors: Jinmin Yi, Weicheng Ye, Daniel Gottesman, Zi-Wen Liu,
• Abstract summary: We establish rigorous connections between quantum circuit complexity and approximate quantum error correction (AQEC) properties. Our key finding is 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. This theory of AQEC provides a versatile framework for understanding the quantum complexity and order of many-body quantum systems.
• Score: 1.1999555634662633
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We establish rigorous connections between quantum circuit complexity and approximate quantum error correction (AQEC) properties, covering both all-to-all and geometric scenarios including 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. Our key finding is 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 nontrivial ``phases'' of codes. Based on our results, we propose $O(k/n)$ as a boundary between subspaces that should and should not count as AQEC codes. This theory of AQEC provides a versatile framework for understanding the quantum complexity and order of many-body quantum systems, offering new insights for wide-ranging physical scenarios, in particular topological order and critical quantum systems which are of outstanding importance in many-body and high energy physics. We observe from various different perspectives that roughly $O(1/n)$ represents a common, physically significant ``scaling threshold'' of subsystem variance for features associated with nontrivial quantum order.

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