Fault Complexes

Fault complexes are a framework for understanding quantum error correction protocols through the language of algebraic homology and chain complexes. While homology initially provided a language that explained already known properties of codes, it has since proven to be tremendously productive, yielding various quantum low-density parity-check (qLDPC) codes with good properties. Fault complexes aim to extend the power of this framework from (CSS) codes at an instantaneous moment in time, to codes in space-time, or more generally, fault-tolerant protocols. It intends to serve as a unifying umbrella that incorporates multiple previously distinct viewpoints, that of gate-based and measurement-based computing, while making the full suite of mathematical tools from homology theory available to analyze these systems.