Here is a list of asymmetric quantum codes, i.e., codes that perform much better against one type of noise than another type.

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Code Relation
2D hyperbolic surface code Asymmetric 2D hyperbolic surface codes have been constructed [1].
3D lattice stabilizer code Applying Clifford deformations to various 3D stabilizer codes, including the 3D surface code, 3D color code, X-cube model code, and Sierpinski prism model code, yields a \(50\%\) code capacity threshold under infinitely biased Pauli noise [2].
Abelian LP code The Abelian LP construction has been adapted to accommodate noise bias, yielding bias-tailored LP codes [3].
Asymmetric quantum code (AQC)
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 [4]. A fault-tolerant teleportation-based computation scheme for asymmetric Bacon-Shor codes is effective against highly biased noise [5].
Binomial code Binomial code parameters against loss/gain errors and dephasing can be tuned.
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 \([[n,k,d_X/d_Z,w]]\).
Clifford-deformed surface code (CDSC) Random Clifford deformation can improve performance of surface codes against biased noise [2,6].
Compass code Families of random compass codes perform well against biased noise and spatially dependent (i.e., asymmetric) noise [7]. Clifford deformation can enhance the performance of compass codes against biased noise [8].
Concatenated Steane code Concatenating while taking into account noise bias can reduce resource overhead [9].
Cyclic quantum code Cyclic quantum codes can be adapted for asymmetric noise [10].
Distance-balanced code Distance balancing is a procedure that can convert an asymmetric CSS code into a less asymmetric one.
EA qubit stabilizer code Entanglement can help decode asymmetric quantum codes [11].
Finite-geometry LDPC (FG-LDPC) code FG-LDPC codes can be used to construct asymmetric CSS codes [13][12; Lemma 4.1].
Galois-qudit BCH code Asymmetric quantum BCH codes have been constructed [14,15,17][12; Lemma 4.4][16; Sec. 17.3], including subsystem BCH codes [18][16; Sec. 9.3].
Galois-qudit CSS code Most known Galois-qudit AQC families are derived from the asymmetric Galois-qudit CSS construction [19; Thm. 27.5.2], and assuming the MDS conjecture, all possible parameters for pure Galois-qudit CSS asymmetric MDS codes have been determined [19; Thm. 27.5.3].
Galois-qudit RS code Asymmetric Galois-qudit RS codes have been constructed [20,21][16; Sec. 17.3].
Hastings-Haah Floquet code Floquet codes can be adapted for asymmetric noise [22].
Higher-dimensional homological product code The \((a,b)\)-complex construction yields asymmetric CSS codes with \(d_X=\delta^a\) and \(d_Z=\delta^b\), allowing independent tuning of the two distances for channels with unequal bit-flip and phase-flip rates [23].
Kitaev surface code The surface code on the honeycomb tiling is an asymmetric CSS code [1].
Quantum Hermitian AG code One-point and two-point Hermitian codes can be used to construct asymmetric Galois-qudit CSS codes, and the two-point construction can improve on the corresponding one-point codes [24].
Quantum Reed-Muller (RM) code Asymmetric quantum RM codes have been constructed [12; Lemma 4.1].
Quantum maximum-distance-separable (MDS) code An asymmetric Singleton bound and linear programming bounds for asymmetric CSS codes have been formulated [12]. Asymmetric MDS codes have been characterized [25].
Quantum parity code (QPC) QPC parameters against bit- and phase-noise can be tuned.
Quantum spherical code (QSC) QSC code parameters against loss/gain errors and Gaussian rotations can be tuned.
Qubit QLDPC code There are recipes to determine transversal gates for asymmetric qubit QLDPC codes [26].
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).
Subsystem surface code Subsystem surface codes perform well against biased circuit-level noise [27].
Tensor-network code Quantum Lego and more general tensor-network code optimization for biased noise can be done using reinforcement learning [28,29].
Twisted XZZX toric code Twisted XZZX codes perform well against biased noise [30–32]; see also Ref. [33].
Two-component cat code Cat qubits provide an asymmetric-noise platform admitting bias-preserving \(X\), CNOT, and Toffoli gates [34,35]. A bias-preserving SWAP gate has also been proposed [36].
XY surface code XY surface codes perform well against biased noise [37].
XYZ color code XYZ color codes perform well against biased noise [38].
XYZ product code XYZ product codes can be used to protect against biased noise [39].
XYZ\(^2\) hexagonal stabilizer code The XYZ\(^2\) hexagonal stabilizer code has high thresholds under biased noise [40].
XZZX surface code The XZZX surface code can be foliated for a noise-bias preserving MBQC [41] or FBQC [42] protocol; see also [43].

References

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