## 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]. Such codes have been characterized [6].

## 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 [13].
- Clifford-deformed surface code (CDSC) — Random Clifford deformation can improve performance of surface codes agaisnt biased noise [14,15].
- XY surface code — XY surface codes perform well against biased noise [16].
- XYZ color code — XYZ color codes perform well against biased noise [17].
- Twisted XZZX toric code — Twisted XZZX codes perform well against biased noise [18–20]; see also Ref. [21].
- XZZX surface code — The XZZX surface code can be foliated for a noise-bias preserving MBQC [22] or FBQC [23] protocol; see also [24].
- Concatenated Steane code — Concatenating while taking into account noise bias can reduce resource overhead [10].
- 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 [6].
- 2D hyperbolic surface code — Asymmetric 2D hyperbolic surface codes have been constructed [25].
- Kitaev surface code — The surface code on a hexagonal lattice is an asymmetric CSS code [25].
- Compass code — Families of random compass codes perform well against biased noise and spatially dependent (i.e., asymmetric) noise [26].
- 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].
- Qubit BCH code — Asymmetric quantum BCH codes have been constructed [2,27][4; Lemma 4.4].
- Finite-geometry LDPC (FG-LDPC) code — EG-LDPC codes can be used to construct asymmetric CSS codes [4; Lemma 4.1].
- Galois-qudit RS code — Asymmetric Galois-qudit RS codes have been constructed [28].
- 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 [29]. A fault-tolerant teleportation-based computation scheme for asymmetric Bacon-Shor codes is effective against highly biased noise [30].
- Two-component cat code — Two-component cat codes admit a bias-preserving CNOT gate that is continuously connected to the identity [9,31].
- 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.
- Abelian LP code — The Abelian LP construction has been adapted to accommodate noise bias, yielding bias-tailored LP codes [32].

## References

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- [2]
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- [3]
- M. F. Ezerman et al., “CSS-Like Constructions of Asymmetric Quantum Codes”, IEEE Transactions on Information Theory 59, 6732 (2013) arXiv:1207.6512 DOI
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- R. Matsumoto, “Two Gilbert–Varshamov-type existential bounds for asymmetric quantum error-correcting codes”, Quantum Information Processing 16, (2017) arXiv:1705.04087 DOI
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- M. F. EZERMAN et al., “PURE ASYMMETRIC QUANTUM MDS CODES FROM CSS CONSTRUCTION: A COMPLETE CHARACTERIZATION”, International Journal of Quantum Information 11, 1350027 (2013) arXiv:1006.1694 DOI
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- [13]
- O. Higgott and N. P. Breuckmann, “Subsystem Codes with High Thresholds by Gauge Fixing and Reduced Qubit Overhead”, Physical Review X 11, (2021) arXiv:2010.09626 DOI
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- A. Dua et al., “Clifford-Deformed Surface Codes”, PRX Quantum 5, (2024) arXiv:2201.07802 DOI
- [15]
- E. Huang et al., “Tailoring Three-Dimensional Topological Codes for Biased Noise”, PRX Quantum 4, (2023) arXiv:2211.02116 DOI
- [16]
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- J. F. S. Miguel, D. J. Williamson, and B. J. Brown, “A cellular automaton decoder for a noise-bias tailored color code”, Quantum 7, 940 (2023) arXiv:2203.16534 DOI
- [18]
- A. Robertson et al., “Tailored Codes for Small Quantum Memories”, Physical Review Applied 8, (2017) arXiv:1703.08179 DOI
- [19]
- J. P. Bonilla Ataides et al., “The XZZX surface code”, Nature Communications 12, (2021) arXiv:2009.07851 DOI
- [20]
- Q. Xu et al., “Tailored XZZX codes for biased noise”, (2022) arXiv:2203.16486
- [21]
- B. Röthlisberger et al., “Incoherent dynamics in the toric code subject to disorder”, Physical Review A 85, (2012) arXiv:1112.1613 DOI
- [22]
- J. Claes, J. E. Bourassa, and S. Puri, “Tailored cluster states with high threshold under biased noise”, npj Quantum Information 9, (2023) arXiv:2201.10566 DOI
- [23]
- H. Bombín et al., “Increasing error tolerance in quantum computers with dynamic bias arrangement”, (2023) arXiv:2303.16122
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- A. M. Stephens, W. J. Munro, and K. Nemoto, “High-threshold topological quantum error correction against biased noise”, Physical Review A 88, (2013) arXiv:1308.4776 DOI
- [25]
- C. D. de Albuquerque et al., “Euclidean and Hyperbolic Asymmetric Topological Quantum Codes”, (2021) arXiv:2105.01144
- [26]
- M. Li et al., “2D Compass Codes”, Physical Review X 9, (2019) arXiv:1809.01193 DOI
- [27]
- G. G. La Guardia, “New families of asymmetric quantum BCH codes”, Quantum Information and Computation 11, 239 (2011) DOI
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- G. G. La Guardia, “Asymmetric quantum Reed-Solomon and generalized Reed-Solomon codes”, Quantum Information Processing 11, 591 (2011) DOI
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- J. Napp and J. Preskill, “Optimal Bacon-Shor codes”, (2012) arXiv:1209.0794
- [30]
- P. Brooks and J. Preskill, “Fault-tolerant quantum computation with asymmetric Bacon-Shor codes”, Physical Review A 87, (2013) arXiv:1211.1400 DOI
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- S. Puri et al., “Bias-preserving gates with stabilized cat qubits”, Science Advances 6, (2020) arXiv:1905.00450 DOI
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- J. Roffe et al., “Bias-tailored quantum LDPC codes”, Quantum 7, 1005 (2023) arXiv:2202.01702 DOI

## 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