  We study parallel fault-tolerant quantum computing for families of
homological quantum low-density parity-check (LDPC) codes defined on
3-manifolds with constant or almost-constant encoding rate. We derive generic
formula for a transversal $T$ gate of color codes on general 3-manifolds, which
acts as collective non-Clifford logical CCZ gates on any triplet of logical
qubits with their logical-$X$ membranes having a $\mathbb{Z}_2$ triple
intersection at a single point. The triple intersection number is a topological
invariant, which also arises in the path integral of the emergent higher
symmetry operator in a topological quantum field theory: the $\mathbb{Z}_2^3$
gauge theory. Moreover, the transversal $S$ gate of the color code corresponds
to a higher-form symmetry supported on a codimension-1 submanifold, giving rise
to exponentially many addressable and parallelizable logical CZ gates. We have
developed a generic formalism to compute the triple intersection invariants for
3-manifolds and also study the scaling of the Betti number and systoles with
volume for various 3-manifolds, which translates to the encoding rate and
distance. We further develop three types of LDPC codes supporting such logical
gates: (1) A quasi-hyperbolic code from the product of 2D hyperbolic surface
and a circle, with almost-constant rate $k/n=O(1/\log(n))$ and $O(\log(n))$
distance; (2) A homological fibre bundle code with $O(1/\log^{\frac{1}{2}}(n))$
rate and $O(\log^{\frac{1}{2}}(n))$ distance; (3) A specific family of 3D
hyperbolic codes: the Torelli mapping torus code, constructed from mapping tori
of a pseudo-Anosov element in the Torelli subgroup, which has constant rate
while the distance scaling is currently unknown. We then show a generic
constant-overhead scheme for applying a parallelizable universal gate set with
the aid of logical-$X$ measurements.
