Quantum codes achieving approximate quantum error correction (AQEC) are useful, often fundamentally important, from both practical and physical perspectives but remain little understood. In this work, 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. Our theory of AQEC provides a versatile framework for understanding the quantum complexity and order of manybody 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.
Zi-Wen Liu is an assistant professor at Yau Mathematical Sciences Center, Tsinghua University. Prior to this, he was a 5-Year Senior Postdoctoral Fellow at Perimeter Institute for Theoretical Physics. He obtained his PhD in Physics from MIT. His main recent interests are the interactions between quantum information/computation and physics and mathematics.