Complexity and order in approximate quantum error-correcting codes

We establish rigorous connections between quantum circuit complexity and approximate quantum error correction (AQEC) capability, two properties of fundamental importance to the physics and practical use of quantum many-body systems, covering systems with both all-to-all connectivity and geometric scenarios like lattice systems in finite spatial dimensions. 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, for a code encoding $k$ logical qubits in $n$ physical qubits, 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 quantum complexity and order in many-body quantum systems, generating 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.