Chapter 6
Asymmetric Quantum Codes
To make reliable the transmission or storage of quantum information against noise
caused by the environment, there exist many works available in the literature dealing with constructions of efficient quantum error-correcting codes (QECCs) over
unbiased quantum channels [11, 12, 25, 55, 80, 89, 92, 133, 148].
In the past 10 years, these constructions have been extended to asymmetric quantum channels in a natural way [3, 35–37, 70, 92, 94, 95, 97, 101, 139, 140, 149,
157].
Asymmetric quantum error-correcting codes (AQECCs) are quantum codes
defined over quantum channels where qudit-flip errors and phase-shift errors may
have different probabilities. Steane [147] was the first author who introduced the
notion of asymmetric quantum errors. As usual, the parameters of an AQECC is
given by [[n, k, d z /d x ]] q , where n is the length, k means that the code has dimension
q
k , d z is the minimum distance corresponding to phase-shift errors and d x is the
minimum distance corresponding to qudit-flip errors. As an example of a quantum
channel such that d z > d x (i.e., a channel presenting asymmetry) is the combined
amplitude damping and dephasing channel (specific to binary systems; see [139]).
To put the reader into context, we give here a brief summary of the papers available
in the literature dealing with investigations of AQECCs. In [70], the authors utilized
BCH codes to correct qubit-flip errors and LDPC codes to correct more frequently
phase-shift errors. In [149], an investigation of AQECCs via code conversion was
presented. In references [3, 92], several families of AQECCs derived from BCH codes
were constructed. Asymmetric stabilizer codes obtained from LDPC codes were
constructed in [139]; in [140], the same authors constructed several families of both
binary and nonbinary AQECCs as well as they derived bounds such as the (quantum)
Singleton and the linear programming bound to AQECCs. In [157], constructions
of nonadditive AQECCs as well as constructions of asymptotically good AQECCs
derived from algebraic geometry codes were presented. In [35], the CSS construction
[25, 80, 121] was extended to include codes endowed with the Hermitian and also
© Springer Nature Switzerland AG 2020
G. G. La Guardia, Quantum Error Correction, Quantum Science and Technology,
https://doi.org/10.1007/978-3-030-48551-1_6
125
Asymmetric Quantum Codes
To make reliable the transmission or storage of quantum information against noise
caused by the environment, there exist many works available in the literature dealing with constructions of efficient quantum error-correcting codes (QECCs) over
unbiased quantum channels [11, 12, 25, 55, 80, 89, 92, 133, 148].
In the past 10 years, these constructions have been extended to asymmetric quantum channels in a natural way [3, 35–37, 70, 92, 94, 95, 97, 101, 139, 140, 149,
157].
Asymmetric quantum error-correcting codes (AQECCs) are quantum codes
defined over quantum channels where qudit-flip errors and phase-shift errors may
have different probabilities. Steane [147] was the first author who introduced the
notion of asymmetric quantum errors. As usual, the parameters of an AQECC is
given by [[n, k, d z /d x ]] q , where n is the length, k means that the code has dimension
q
k , d z is the minimum distance corresponding to phase-shift errors and d x is the
minimum distance corresponding to qudit-flip errors. As an example of a quantum
channel such that d z > d x (i.e., a channel presenting asymmetry) is the combined
amplitude damping and dephasing channel (specific to binary systems; see [139]).
To put the reader into context, we give here a brief summary of the papers available
in the literature dealing with investigations of AQECCs. In [70], the authors utilized
BCH codes to correct qubit-flip errors and LDPC codes to correct more frequently
phase-shift errors. In [149], an investigation of AQECCs via code conversion was
presented. In references [3, 92], several families of AQECCs derived from BCH codes
were constructed. Asymmetric stabilizer codes obtained from LDPC codes were
constructed in [139]; in [140], the same authors constructed several families of both
binary and nonbinary AQECCs as well as they derived bounds such as the (quantum)
Singleton and the linear programming bound to AQECCs. In [157], constructions
of nonadditive AQECCs as well as constructions of asymptotically good AQECCs
derived from algebraic geometry codes were presented. In [35], the CSS construction
[25, 80, 121] was extended to include codes endowed with the Hermitian and also
© Springer Nature Switzerland AG 2020
G. G. La Guardia, Quantum Error Correction, Quantum Science and Technology,
https://doi.org/10.1007/978-3-030-48551-1_6
125