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

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