TQFTs and quantum computing

Quantum computing is captured in the formalism of the monoidal subcategory of ${\mathbf{\text{Vect}}}_{\mathbb{C}}$ generated by ${\mathbb{C}}^2$ – in particular, quantum circuits are diagrams in ${\mathbf{\text{Vect}}}_{\mathbb{C}}$ – while topological quantum field theories, in the sense of Atiyah, are diagrams in ${\mathbf{\text{Vect}}}_{\mathbb{C}}$ 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 $\mathbb{F}{\mathbf{\text{Vect}}}_{\mathbb{C}}$. We realize quantum circuits as images of cobordisms under monoidal double functors from these modified cobordisms to $\mathbb{F}{\mathbf{\text{Vect}}}_{\mathbb{C}}$, which are computed by taking parallel transports of vectors and then combining the results in a pattern encoded in the domain double category.