Categorical computation

Abstract

In quantum computing, the computation is achieved by linear operators in or between Hilbert spaces. In this work, we explore a new computation scheme, in which the linear operators in quantum computing are replaced by (higher) functors between two (higher) categories. If from Turing computing to quantum computing is the first quantization of computation, then this new scheme can be viewed as the second quantization of computation. The fundamental problem in realizing this idea is how to realize a (higher) functor physically. We provide a theoretical idea of realizing (higher) functors physically based on the physics of topological orders.

Figures (4)

• Figure 1: The physical realization of a 1-functor $\EuScript{F}: \EuScript{X}\to\EuScript{X}$.
• Figure 2: These figures illustrate the idea of fixing the unstable problem in Figure \ref{['fig:1-functor']}.
• Figure 3: the idea of physically realizing a monoidal 1-functor
• Figure 4: the idea of realizing a monoidal functor $\EuScript{F}: \EuScript{A} \to \EuScript{B}$ defined by $a \mapsto \mathbb{1}_{\EuScript{F}'} \boxtimes_\EuScript{A} a \boxtimes_\EuScript{A} \mathbb{1}_{\EuScript{F}"} \in \EuScript{B}$ in a 2d fractional quantum Hall system

Theorems & Definitions (17)

• Remark 2.0.1
• Definition 2.0.2
• Example 2.0.3
• Remark 2.0.4
• Definition 2.0.5
• Example 2.0.6
• Remark 2.0.7
• Remark 2.0.8
• Remark 2.0.9
• Remark 3.0.1