Complexity in approximate quantum error correction

Quantum codes achieving approximate quantum error correction (AQEC) are useful, and often fundamentally important, from both practical and physical perspectives, yet still lack a systematic understanding. In this talk, we connect AQEC capability to quantum circuit complexity in both all-to-all and geometric settings. We introduce subsystem variance, a parameter closely tied to optimal AQEC precision, and show: (i) if subsystem variance is below an O(k/n) threshold, then every code state obeys circuit-complexity lower bounds; (ii) if two code states admit circuit-complexity upper bounds, then subsystem variance must be Ω(1). Together, these results yield a complete “phase diagram” of AQEC codes. Our theory of AQEC provides a versatile framework for understanding the quantum complexity and order of many-body quantum systems. Applications include complexity constraints for (approximate) QEC codes with transversal logical gates and W-state preparation, as well as new perspectives on topological order, criticality, and Lieb–Schultz–Mattis–type constraints. Ref: arXiv:2310.04710, 2510.04453.