## Description

## Protection

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] and Hamming and Singleton bounds for general asymmetric subsystem codes [6]. Asymmetric MDS codes have been characterized [6,7].

## Gates

## Fault Tolerance

## Threshold

## Notes

## Parent

## Cousins

- Distance-balanced code — Distance balancing is a procedure that can convert an asymmetric CSS code into a less asymmetric one.
- Subsystem surface code — Subsystem surface codes perform well against biased circuit-level noise [15].
- Clifford-deformed surface code (CDSC) — Random Clifford deformation can improve performance of surface codes agaisnt biased noise [16,17].
- XY surface code — XY surface codes perform well against biased noise [18].
- XYZ color code — XYZ color codes perform well against biased noise [19].
- Twisted XZZX toric code — Twisted XZZX codes perform well against biased noise [20–22]; see also Ref. [23].
- XZZX surface code — The XZZX surface code can be foliated for a noise-bias preserving MBQC [24] or FBQC [25] protocol; see also [26].
- Concatenated Steane code — Concatenating while taking into account noise bias can reduce resource overhead [12].
- Quantum maximum-distance-separable (MDS) code — An asymmetric Singleton bound and linear programming bounds for asymmetric CSS codes have been formulated [4]. Asymmetric MDS codes have been characterized [7].
- 2D hyperbolic surface code — Asymmetric 2D hyperbolic surface codes have been constructed [27].
- Kitaev surface code — The surface code on a hexagonal lattice is an asymmetric CSS code [27].
- Compass code — Families of random compass codes perform well against biased noise and spatially dependent (i.e., asymmetric) noise [28].
- Calderbank-Shor-Steane (CSS) stabilizer code — 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 \([[n,k,(d_X,d_Z),w]]\) or \([[n,k,d_X/d_Z,w]]\).
- Quantum Reed-Muller code — Asymmetric quantum RM codes have been constructed [4; Lemma 4.1].
- Galois-qudit BCH code — Asymmetric quantum BCH codes have been constructed [2,29–31][4; Lemma 4.4], including subsystem BCH codes [6].
- Finite-geometry LDPC (FG-LDPC) code — FG-LDPC codes can be used to construct asymmetric CSS codes [31][4; Lemma 4.1].
- Hermitian code — Hermitian codes can be used to construct asymmetric Galois-qudit CSS codes [32].
- Galois-qudit RS code — Asymmetric Galois-qudit RS codes have been constructed [33,34].
- Bacon-Shor code — Bacon-Shor code parameters against bit- and phase-noise can be optimized by changing the block geometry, yielding good performance against biased noise [35]. A fault-tolerant teleportation-based computation scheme for asymmetric Bacon-Shor codes is effective against highly biased noise [36].
- Two-component cat code — Two-component cat codes admit a bias-preserving CNOT gate that is continuously connected to the identity [10,37].
- Square-lattice GKP code — GKP code parameters against position and momentum displacements can be tuned by the choice of lattice (e.g., square vs rectangular).
- Binomial code — Binomial code parameters against loss/gain errors and dephasing can be tuned.
- Quantum spherical code (QSC) — QSC code parameters against loss/gain errors and Gaussian rotations can be tuned.
- Quantum parity code (QPC) — QPC parameters against bit- and phase-noise can be tuned.
- EA qubit stabilizer code — Entanglement can help decode asymmetric quantum codes [38].
- GKP-surface code — Using rectangular lattices accounts for asymmetric noise and improves the GKP-surface threshold to \(0.58\) [39].
- Tensor-network code — Quantum Lego and more general tensor-network code optimization for biased noise can be done using reinforcement learning [40,41].
- Abelian LP code — The Abelian LP construction has been adapted to accommodate noise bias, yielding bias-tailored LP codes [42].

## References

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## Page edit log

- Victor V. Albert (2024-04-04) — most recent

## Cite as:

“Asymmetric quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/asymmetric_qecc