Quantum information processing, at its core, is effected through unitary transformations applied to states on the Bloch sphere, the standard geometric realization of a two-level, single-qubit system. That said, in viewing the application of a logic gate as a continuous process, it is natural to replace the original Hilbert space of the problem with a finite-rank Hermitian vector bundle on a smooth manifold, through which unitary transformations may be achieved in a natural geometric way via parallel transport along a unitary connection. It then follows that the construction of quantum circuits is captured in a topological quantum field theory (TQFT), decorated with flat bundles. Considering the curvature of the underlying manifold, one may describe the resultant quantum information theory as parabolic, elliptic, or hyperbolic, with the latter being generic but also fundamentally "new" in the context of quantum computing paradigms to date. An extension to the classification of 2-dimensional topological quantum matter — relying upon hyperbolic geometries — that has emerged in my collaborative work with theoretical physicists and experimentalists suggests that the hyperbolic regime of this setup may be realizable as a physical platform for computation.