Using arbitrary parity-check matrices for quantum error correction assisted by less noisy qubits

TL;DR

This paper generalizes a quantum error correction framework to utilize any parity-check matrix and supports asymmetric error correction, enhancing flexibility and performance in quantum coding.

Contribution

It introduces a generalized scheme that allows the use of arbitrary parity-check matrices and pairs of codes for asymmetric error correction in quantum systems.

Findings

Enables use of any parity-check matrix in quantum error correction

Supports importing pairs of codes for asymmetric error correction

Improves error correction capabilities for asymmetric noise models

Abstract

Recently a framework for assisted quantum error correction was proposed in which a specific type of error is allowed to occur on auxiliary qubits, which is in contrast to standard entanglement assistance that requires noiseless auxiliary qubits. However, while the framework maintains the ability to import any binary or quaternary linear code without sacrificing active error correction power, it requires the code designer to turn a parity-check matrix of the underlying classical code into an equivalent one in standard form. This means that classical coding theoretic techniques that require parity-check matrices to be in specific form may not fully be exploitable. Another issue of the recently proposed scheme is that the error correction capabilities for bit errors and phase errors are generally equal, which is not ideal for asymmetric error models. This paper addresses these two problems. We generalize the framework in such a way that any parity-check matrix of any binary or quaternary linear code can be exploited. Our generalization also allows for importing a pair of distinct linear codes so that error correction capabilities become suitably asymmetric.