论文标题
TQFT和量子计算
TQFTs and quantum computing
论文作者
论文摘要
量子计算是在$ \ textbf {vect} _ {\ mathbb c} $的单体子类别的形式上捕获的$ \ textbf {vect} _ {\ mathbb c} $ in Cobordism索引的图。我们启动一个正式的该连接的程序。在此过程中，我们配备了与机械配置，用于在连接下沿曲线并行传输产生线性图，然后将这些结构组装成双重类别。有限维的复杂矢量空间和它们之间的线性图具有合适的双分类结构，我们称为$ \ Mathbb f \ textbf {vect} _ {\ Mathbb c} $。我们将量子回路视为从这些修改的Coobordism到$ \ Mathbb f \ textbf {vect} _ {\ Mathbb c} $的单型双函子下的cobordism的图像，通过将矢量的并行运输量进行计算，然后将结果合并为模式，并将结果组合成域模式。
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.
