YQI Talk - Grace Sommers - Princeton

 

Families of good (quantum) error-correcting codes exhibit a finite noise threshold separating a “coding phase,” in which logical information can be faithfully recovered, to a “noncoding phase.” Statistical mechanics is therefore a valuable tool for analyzing these phases and the transitions between them, often facilitated by mapping the decoding problem onto a classical spin model. In this talk, I offer a statistical mechanics perspective on three models. The first two models - dynamically generated concatenated codes and classical low-density parity-check (cLDPC) codes on expander graphs - both possess a spin glass coding phase, enabled by the expanding geometry. The bulk of the talk will focus on the third model, Haar-random codes, in which the logical information is encoded in a random subspace of the physical Hilbert space. Despite the lack of a corresponding local spin model, decohered Haar codes can be understood through analysis of band structure resulting from errors of different weights. A simple counting argument, together with explicit calculation, establishes that the threshold for Haar-random quantum codes saturates the hashing bound, coinciding with that of random stabilizer codes. Beyond the hashing bound, typical errors are uncorrectable, but postselected error correction remains possible until a much higher detection threshold, with a sequence of intervening phase transitions corresponding to different levels of postselection.