Citation: Kong, Liang, and Hao Zheng. "Categorical computation." Frontiers of Physics 18.2 (2023): 21302.
Web: https://arxiv.org/abs/2102.04814
Tags: Philosophical, Defects/boundaries, Universal-scheme
In this paper, the authors present a proposal for making a quantum computer based on higher categorical structures in n-categories. The point is that sets give classical computing, and vector spaces give quantum computing. Vector spaces are a categorification of sets. Maybe if you categorify more you get something interesting?
I have a hard time seeing the merit of the idea. One of the big points of discussion in the article is that topological order is described by higher categorical structures, so by manipulating topological order one can implement higher categorical computing. However, any computation performed this way cannot be better than BQP. Namely, topological quantum systems are at the base level quantum systems so they cannot efficiently simulate anything beyond BQP! In general, any higher categorical structures which can be implemented using quantum materials will necessarily be in BQP since simulating quantum materials is in BQP. The only advantage can be good constants, and doing exotic forms of computation for better constants does not seem like a prudent idea.
The other approach could be finding n-categories which natively have a model of computation more powerful than quantum computation. This could be interesting from a complexity theory point of view, but has no practical bearing for building a computer unless we discover some physics beyond quantum mechanics which happens to be described by these exotic n-categories.