Quantum Codes from Toric Surfaces

TL;DR

This paper develops a method to construct quantum error correcting codes from toric surfaces, specifically Hirzebruch surfaces, by extending classical code constructions using cohomology and intersection theory.

Contribution

It generalizes the construction of linear codes from toric varieties to quantum codes via the Calderbank-Shor-Steane method, utilizing dualizing differential forms on toric surfaces.

Findings

Constructed quantum codes from toric Hirzebruch surfaces.

Established a dualizing differential form for toric surfaces.

Merged classical code constructions with quantum code theory.

Abstract

A theory for constructing quantum error correcting codes from Toric surfaces by the Calderbank-Shor-Steane method is presented. In particular we study the method on toric Hirzebruch surfaces. The results are obtained by constructing a dualizing differential form for the toric surface and by using the cohomology and the intersection theory of toric varieties. In earlier work the author developed methods to construct linear error correcting codes from toric varieties and derive the code parameters using the cohomology and the intersection theory on toric varieties. This method is generalized in section to construct linear codes suitable for constructing quantum codes by the Calderbank-Shor-Steane method. Essential for the theory is the existence and the application of a dualizing differential form on the toric surface. A.R. Calderbank, P.W. Shor and A.M. Steane produced stabilizer codes from linear codes containing their dual codes. These two constructions are merged to obtain results for toric surfaces. Similar merging has been done for algebraic curves with different methods by A. Ashikhmin, S. Litsyn and M.A. Tsfasman.