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.

Graphical abstract

Keywords

quantum computation / categorical computation / topological order

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Acknowledgements

We thank Zheng-Wei Liu, Ce Shen, Xiao-Ming Sun, Zhong Wang and Bo Yang for comments. We are supported by Guangdong Provincial Key Laboratory (Grant No. 2019B121203002). L.K. is also supported by the National Natural Science Foundation of China under Grant No. 11971219 and Guangdong Basic and Applied Basic Research Foundation under Grant No. 2020B1515120100. H.Z. is also supported by the National Natural Science Foundation of China under Grant No. 11871078.

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