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"A concise guide to complex Hadamard matrices", Wojciech Tadej, Karol Zyczkowski, 2006

Reviewed September 5, 2023

Citation: Tadej, Wojciech, and Karol ┼╗yczkowski. "A concise guide to complex Hadamard matrices." Open Systems & Information Dynamics 13.2 (2006): 133-177.

Web: https://arxiv.org/abs/quant-ph/0512154

Tags: Abelian-anyons, Hadamard-matrices

This paper gives a nice review of the theory of Hadamard matrices. What's beautiful here is that there are lots of open problems, many of which outlined in the paper

> J. H. Beder, Conjectures about Hadamard matrices, J. Stat. Plan. and Inference 72, 7-14 (1998).

Most of the problems come down to whether or not one can always construct Hadamard matrices. The beautiful fact is that Hadamard matrices are intimately connected with error correction, as originally noted in

> Heng, I., and C. H. Cooke. "Error correcting codes associated with complex Hadamard matrices." Applied mathematics letters 11.4 (1998): 77-80.
and even more than that, to to quantum error correction:
> Werner, Reinhard F. "All teleportation and dense coding schemes." Journal of Physics A: Mathematical and General 34.35 (2001): 7081.

You can see Hadamard matrices also appearing as the F-matrices of topologically ordered systems:

> Scruby, T. R., and Dan E. Browne. "A hierarchy of anyon models realised by twists in stacked surface codes." Quantum 4 (2020): 251.

It is natural to ask whether this can be brought together with the extensive literature on classification of topological phases presented in works such as

> Lan, Tian. "A classification of (2+ 1) D topological phases with symmetries." arXiv preprint arXiv:1801.01210 (2018).

To resolve the Hadamard conjecture. Namely, Hadamard matrices must exist since they appear implicitly in the constructions that have already been performed for making topological phases.