  Measurement-based quantum computing (MBQC) is a promising alternative to
traditional circuit-based quantum computing predicated on the construction and
measurement of cluster states. Recent work has demonstrated that MBQC provides
a more general framework for fault-tolerance that extends beyond foliated
quantum error-correcting codes. We systematically expand on that paradigm, and
use combinatorial tiling theory to study and construct new examples of
fault-tolerant cluster states derived from crystal structures. Included among
these is a robust self-dual cluster state requiring only degree-3 connectivity.
We benchmark several of these cluster states in the presence of circuit-level
noise, and find a variety of promising candidates whose performance depends on
the specifics of the noise model. By eschewing the distinction between data and
ancilla, this malleable framework lays a foundation for the development of
creative and competitive fault-tolerance schemes beyond conventional
error-correcting codes.
