  Fault-tolerant complexes describe surface-code fault-tolerant protocols from
a single geometric object. We first introduce fusion complexes that define a
general family of fusion-based quantum computing (FBQC) fault-tolerant quantum
protocols based on surface codes. We show that any 3-dimensional cell complex
where each edge has four incident faces gives a valid fusion complex. This
construction enables an automated search for fault tolerance schemes, allowing
us to identify 627 examples within a moderate search time. We implement this
using the open-source software tool Gavrog and present threshold results for a
variety of schemes, finding fusion networks with higher erasure and Pauli
thresholds than those existing in the literature. We then define more general
structures we call fault-tolerant complexes that provide a homological
description of fault tolerance from a large family of low-level error models,
which include circuit-based computation, floquet-based computation, and FBQC
with multi-qubit measurements. This extends the applicability of homological
descriptions of fault tolerance, and enables the generation of many new schemes
which have not been previously identified. We also define families of
fault-tolerant complexes for color codes and 3d single-shot subsystem codes,
which enables similar constructive methods, and we present several new examples
of each.
