Quantum error correction for a subgroup of the Pauli group using higher rank numerical range

Abstract

Suppose noise is characterized by a quantum channel whose error operators are members of an arbitrary (not necessarily abelian) Pauli subgroup of -qubit Pauli group, where . In this paper, using a maximal abelian subgroup of , we obtained the largest , for which the joint rank- numerical range associated with members of is non-empty. It is proved that qubits codeword encodes data qubits. Moreover, we obtain recovery operators such that the decoded state automatically appears in the output with no syndrome measurements for such channels.

Keywords:

• quantum error correction
• higher rank numerical range
• Pauli group
• recovery operator

AMS Subject Classifications:

• 15A60
• 81P45
• 15A69
• 15A90

Notes

The authors declare that there is no conflict of interests regarding the publication of this article.