TQFTs and quantum computing
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866913374266195968 |
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| author | Azam, Mahmud Rayan, Steven |
| author_facet | Azam, Mahmud Rayan, Steven |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_03556 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | TQFTs and quantum computing Azam, Mahmud Rayan, Steven Quantum Physics High Energy Physics - Theory Category Theory Differential Geometry 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. |
| title | TQFTs and quantum computing |
| topic | Quantum Physics High Energy Physics - Theory Category Theory Differential Geometry |
| url | https://arxiv.org/abs/2210.03556 |