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We study parallel fault-tolerant quantum computing for families ofhomological quantum low-density parity-check (LDPC) codes defined on3-manifolds with constant or almost-constant encoding rate. We derive genericformula for a transversal $T$ gate of color codes on general 3-manifolds, whichacts as collective non-Clifford logical CCZ gates on any triplet of logicalqubits with their logical-$X$ membranes having a $\mathbb{Z}_2$ tripleintersection at a single point. The triple intersection number is a topologicalinvariant, which also arises in the path integral of the emergent highersymmetry operator in a topological quantum field theory: the $\mathbb{Z}_2^3$gauge theory. Moreover, the transversal $S$ gate of the color code correspondsto a higher-form symmetry supported on a codimension-1 submanifold, giving riseto exponentially many addressable and parallelizable logical CZ gates. We havedeveloped a generic formalism to compute the triple intersection invariants for3-manifolds and also study the scaling of the Betti number and systoles withvolume for various 3-manifolds, which translates to the encoding rate anddistance. We further develop three types of LDPC codes supporting such logicalgates: (1) A quasi-hyperbolic code from the product of 2D hyperbolic surfaceand 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 3Dhyperbolic codes: the Torelli mapping torus code, constructed from mapping toriof a pseudo-Anosov element in the Torelli subgroup, which has constant ratewhile the distance scaling is currently unknown. We then show a genericconstant-overhead scheme for applying a parallelizable universal gate set withthe aid of logical-$X$ measurements.

Non-Clifford and parallelizable fault-tolerant logical gates on constant  and almost-constant rate homological quantum LDPC codes via higher symmetries