Haah's Cubic Code

Fractal logical operators and restricted excitations in a three-dimensional stabilizer model. The main concepts are polynomial stabilizers, fractal operators, finite size. Each must be defined in the context of a concrete physical model, mathematical representation, or experimental task. Terminology alone does not establish the behavior of the system.

Haah's cubic code is a three-dimensional local commuting stabilizer model with fractal operator support and no ordinary string logical operators. Its logical sectors and barrier behavior depend on lattice size and boundary conditions, so a claim of size-growing memory protection must carefully distinguish static energy barriers from finite-temperature dynamics. Restricted mobility does not automatically establish a fully self-correcting quantum memory at arbitrary temperature.

Represent its polynomial stabilizers over finite fields, verify the commutator matrix is zero, calculate finite-size ranks and enumerate small logical operators. Compare energy barriers for controlled partial logical operations, then study thermal dynamics under a fully specified local bath. The original paper provides a concrete algebraic starting point rather than an experimental performance demonstration.