Categorical computation

TL;DR

This paper proposes a novel quantum computation scheme replacing linear operators with functors between categories, inspired by topological orders, aiming for a second quantization of computation.

Contribution

It introduces a theoretical framework for implementing higher functors physically in quantum computing, extending the concept beyond traditional linear operators.

Findings

Theoretical model connecting higher functors with topological orders.

Proposes a new paradigm for quantum computation based on category theory.

Lays groundwork for physical realization of higher 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.