  We investigate topological order on fractal geometries embedded in $n$
dimensions. In particular, we diagnose the existence of the topological order
through the lens of quantum information and geometry, i.e., via its equivalence
to a quantum error-correcting code with a macroscopic code distance or the
presence of macroscopic systoles in systolic geometry. We first prove a no-go
theorem that $\mathbb{Z}_N$ topological order cannot survive on any fractal
embedded in 2D. For fractal lattice models embedded in 3D or higher spatial
dimensions, $\mathbb{Z}_N$ topological order survives if the boundaries of the
interior holes condense only loop or membrane excitations. Moreover, for a
class of models containing only loop or membrane excitations, and are hence
self-correcting on an $n$-dimensional manifold, we prove that topological order
survives on a large class of fractal geometries independent of the type of hole
boundaries. We further construct fault-tolerant logical gates using their
connection to global and higher-form topological symmetries. In particular, we
have discovered a logical CCZ gate corresponding to a global symmetry in a
class of fractal codes embedded in 3D with Hausdorff dimension asymptotically
approaching $D_H=2+\epsilon$ for arbitrarily small $\epsilon$, which hence only
requires a space-overhead $\Omega(d^{2+\epsilon})$ with $d$ being the code
distance. This in turn leads to the surprising discovery of certain exotic
gapped boundaries that only condense the combination of loop excitations and
gapped domain walls. We further obtain logical $\text{C}^{p}\text{Z}$ gates
with $p\le n-1$ on fractal codes embedded in $n$D. In particular, for the
logical $\text{C}^{n-1}\text{Z}$ in the $n^\text{th}$ level of Clifford
hierarchy, we can reduce the space overhead to $\Omega(d^{n-1+\epsilon})$.
Mathematically, our findings correspond to macroscopic relative systoles in
fractals.
