A category theoretic bridge from classical error correction to quantum error correction

Abstract

Both the category Set of sets and functions as well as the category EVect of Euclidean vector spaces and their linear maps form monoidal categories under the Cartesian product and the tensor product, respectively. We categorically model classical computing within a full monoidal subcategory of Set, and similarly quantum computing within a full monoidal subcategory of EVect. The free functor offers a map expressing how all classical computing algorithms can be functorially mapped to a corresponding quantum algorithm. Error correction codes are included in this model using natural transformations and shown by examples, i.e Hamming codes and Toric codes, to maintain the objects and morphisms of their respective categories. Keywords: mathematics, math, computer science, linear algebra, Deustch-Josza algorithm, Toric codes, Hamming codes, Three-Bit Repetition code, Three-Qubit Repetition code, category theory, quan-tum computing, classical computing, error correction, classical error correction, quantum error correction, free functor, monoidal category, category