Algebraic quantum codes: linking quantum mechanics and discrete mathematics

Abstract

We discuss the connection between quantum error-correcting codes (QECCS) and algebraic coding theory. We start with an introduction to the relevant concepts of quantum mechanics, including the general error model. A quantum error-correcting code is a subspace of a complex Hilbert space, and its error-correcting properties are characterized by the Knill-Laflamme conditions. Using the stabilizer formalism, we illustrate how QECCs for can be constructed using techniques from algebraic coding theory. We also sketch how the information obtained via a quantum measurement can be interpreted as syndrome of the related classical code. Additionally, we present secondary constructions for QECCs, leading to propagation rules for the parameters of QECCs. This includes the puncture code by Rains and construction X for quantum codes.

Keywords:

• Quantum error-correction
• stabilizer codes
• algebraic coding theory
• puncture code
• quantum construction X

Acknowledgments

The author acknowledges discussions with Frederic Ezerman, Petr Lisoněk, Buket Özkaya, and Martin Rötteler, as well as discussions during the Oberwolfach Workshop 1912 on Contemporary Coding Theory, March 2019. The ‘International Centre for Theory of Quantum Technologies’ project (contract no. 2018/MAB/5) is carried out within the International Research Agendas Programme of the Foundation for Polish Science co-financed by the European Union from the funds of the Smart Growth Operational Programme, axis IV: Increasing the research potential (Measure 4.3).

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 For simplicity, we exclude the case K = 1 here.

2 For simplicity, we assume that the phase factor ${\omega }_p^{\gamma }=1$.

Additional information

Funding

This work was supported by Foundation for Polish Science [2018/MAB/5].