On the logical operators of quantum codes

Abstract

I show how applying a symplectic Gram-Schmidt orthogonalization to the normalizer of a quantum code gives a different way of determining the code's logical operators. This approach may be more natural in the setting where we produce a quantum code from classical codes because the generator matrices of the classical codes form the normalizer of the resulting quantum code. This technique is particularly useful in determining the logical operators of an entanglement-assisted code produced from two classical binary codes or from one classical quaternary code. Finally, this approach gives additional formulas for computing the amount of entanglement that an entanglement-assisted code requires.

5 pages, sequel to the findings in arXiv:0804.1404

References in corpus (8)

• Good Quantum Error-Correcting Codes Exist
• Improved Simulation of Stabilizer Circuits
• Correcting Quantum Errors with Entanglement
• Stabilizer Formalism for Operator Quantum Error Correction
• Optimal Entanglement Formulas for Entanglement-Assisted Quantum Coding
• Catalytic quantum error correction
• Convolutional and tail-biting quantum error-correcting codes
• Entanglement in the stabilizer formalism

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