TQFTs and quantum computing

TL;DR

This paper explores a formal connection between topological quantum field theories and quantum computing by developing a categorical framework that represents quantum circuits as cobordisms with parallel transport, unifying these concepts.

Contribution

It introduces a double categorical structure for cobordisms and demonstrates how quantum circuits can be realized as images of cobordisms under monoidal double functors.

Findings

Established a double categorical framework for cobordisms.

Formalized quantum circuits as images of cobordisms via functors.

Connected TQFTs with quantum computing through parallel transport mechanisms.

Abstract

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.