



01689nas a2200145   4500008004100000245007100041210006900112260001400181520124100195100001501436700001701451700002201468700001601490856003701506       2023                        eng d00aComplexity and order in approximate quantum error-correcting codes0 aComplexity and order in approximate quantum errorcorrecting code  c10/7/20233 a<p>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&#39;&#39; 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&#39;&#39; of subsystem variance for features associated with nontrivial quantum order.</p>1 aYi, Jinmin1 aYe, Weicheng1 aGottesman, Daniel1 aLiu, Zi-Wen  uhttps://arxiv.org/abs/2310.04710