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"Non-binary unitary error bases and quantum codes", Emanuel Knill, 1996

Reviewed September 1, 2023

Citation: Knill, Emanuel. "Non-binary unitary error bases and quantum codes." arXiv preprint quant-ph/9608048 (1996).

Web: https://arxiv.org/abs/quant-ph/9608048

Tags: Computer-scientific, Abelian-anyons, Hadamard-matrices, Error-correcting-codes

This paper is a very interesting case. It is a very early work (1996) trying to abstract the properties which allowed the Pauli group to perform error correction. The goal is to construct other systems with the same properties as the Pauli group, which could be used for next-generation error correction schemes.

The fascinating part is that there is a deep connection between generalizations of the Pauli group and topological phases. For instance, there is a deep connection between the Pauli group and the toric code. The commutation relations and the number of quasiparticles in the toric code come exactly from the commutation relationships on the Pauli group. Every Pauli-like group can be used to make error correcting codes, and these exited states generate a topological phase.

You can see Knill writing down lots of formulas in this paper which are very reminiscent of fusion systems and tensor categories. Really, this is a paper about the classification of topological phases via quantum computing from before topological quantum computing was a thing!

This paper is part-1 of a series, with the follow up being

> Knill, Emanuel. "Group representations, error bases and quantum codes." arXiv preprint quant-ph/9608049 (1996).

There are some explicit examples of error bases given in this paper. It would be interesting to see if anybody has looked at these examples and put together what topological phase they correspond to.