Quantum codes from Hadamard matrices

Abstract

From each q × q unitary matrix Φ, we construct a family of quantum codes , t ≥ 1, for q-state systems which encode (2t + 1)² q-states into one q-state. We show that such codes are capable of correcting the errors of weight up to t if and only if Φ is a complex Hadamard matrix.

Keywords:

• complex Hadamard matrix
• quantum code

Acknowledgements

W.F. Ke was partially supported by The National Science Council, Taiwan, under grants NSC-94-2115-M-006-008 and NSC-95-2115-M-006-004.