Asymmetric quantum code[1,2] 

Also known as Noise-biased quantum code.


Quantum systems can be roughly characterized by two types of noise, a bit-flip noise that maps canonical basis states into each other, and a phase-flip noise that induces relative phases between superpositions of such basis states. A code cannot protect against both types of noise arbitrarily well, and there is a tradeoff between the two types of protection. An asymmetric quantum code is one that performs much better against one type of noise than the other type. Such codes typically have tunable distances against each noise type and include CSS codes, GKP codes, and QSCs.


Noise channels for which one type of noise is more prominent than the other are called asymmetric-noise channels or biased-noise channels. An example of a noise-biased channel is a Pauli channel of independent \(X\) and \(Z\)-type noise with \(p_X \gg p_Z\) or visa versa.

In the context of comparing weight as well as of determining distances for noise models biased toward \(X\)- or \(Z\)-type errors, an extended notation for asymmetric CSS block quantum codes is \([[n,k,(d_X,d_Z)]]\) or \([[n,k,d_X/d_Z]]\), where \(d_{X,Z}\) are the \(X\)- and \(Z\)-distances, respectively [3]. An asymmetric Singleton bound and linear programming bounds for asymmetric CSS codes have been formulated [4], as well as asymmetric quantum GV bounds [5]. Such codes have been characterized [6].


Taking into account noise bias can reduce resource overhead in magic-state distillation schemes [7].A CNOT gate continuously connected to the identity cannot be noise-bias-preserving in finite dimensions [8][9; Appx. A].

Fault Tolerance

Fault-tolerant noise-bias-preserving computation scheme [8].Fault-tolerant circuits converting between asymmetric and symmetric subsystem codes [10].


A lower bound on concatenated thresholds with CSS codes under biased noise [11].


See Ref. [12] for a brief review of asymmetric quantum codes.




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“Asymmetric quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.
@incollection{eczoo_asymmetric_qecc, title={Asymmetric quantum code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Asymmetric quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.