  Quantum computing is captured in the formalism of the monoidal subcategory of
$\textbf{Vect}_{\mathbb C}$ generated by $\mathbb C^2$ -- in particular,
quantum circuits are diagrams in $\textbf{Vect}_{\mathbb C}$ -- while
topological quantum field theories, in the sense of Atiyah, are diagrams in
$\textbf{Vect}_{\mathbb C}$ indexed by cobordisms. We initiate a program that
formalizes this connection. In doing so, we equip cobordisms with machinery for
producing linear maps by parallel transport along curves under a connection and
then assemble these structures into a double category. Finite-dimensional
complex vector spaces and linear maps between them are given a suitable double
categorical structure which we call $\mathbb F\textbf{Vect}_{\mathbb C}$. We
realize quantum circuits as images of cobordisms under monoidal double functors
from these modified cobordisms to $\mathbb F\textbf{Vect}_{\mathbb C}$, which
are computed by taking parallel transports of vectors and then combining the
results in a pattern encoded in the domain double category.
