On the efficiency of nondegenerate quantum error correction codes for Pauli channels

TL;DR

This paper analyzes the efficiency of nondegenerate quantum error correction codes for Pauli channels, showing that correcting multiple errors does not improve efficiency below certain error probabilities and that existing codes are near optimal.

Contribution

It demonstrates that multiple-error correction codes do not outperform single-error codes in efficiency for fixed code size and error probabilities, and evaluates the optimality of existing codes.

Findings

Multiple-error correction does not increase efficiency below a certain error probability.

Existing codes with up to 256 qubits are less efficient than optimal single-error codes.

Proposed single-error correcting codes are close to theoretical efficiency limits.

Abstract

We examine the efficiency of pure, nondegenerate quantum-error correction-codes for Pauli channels. Specifically, we investigate if correction of multiple errors in a block is more efficient than using a code that only corrects one error per block. Block coding with multiple-error correction cannot increase the efficiency when the qubit error-probability is below a certain value and the code size fixed. More surprisingly, existing multiple-error correction codes with a code length equal or less than 256 qubits have lower efficiency than the optimal single-error correcting codes for any value of the qubit error-probability. We also investigate how efficient various proposed nondegenerate single-error correcting codes are compared to the limit set by the code redundancy and by the necessary conditions for hypothetically existing nondegenerate codes. We find that existing codes are close to optimal.