Asymmetric quantum code[1,2] 

Also known as Noise-biased quantum code.

Description

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.

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

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].

Threshold

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

Notes

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

Parent

Cousins

References

[1]
A. M. Steane, “Simple quantum error-correcting codes”, Physical Review A 54, 4741 (1996) arXiv:quant-ph/9605021 DOI
[2]
L. Ioffe and M. Mézard, “Asymmetric quantum error-correcting codes”, Physical Review A 75, (2007) arXiv:quant-ph/0606107 DOI
[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
[4]
P. K. Sarvepalli, A. Klappenecker, and M. Rötteler, “Asymmetric quantum codes: constructions, bounds and performance”, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, 1645 (2009) DOI
[5]
R. Matsumoto, “Two Gilbert–Varshamov-type existential bounds for asymmetric quantum error-correcting codes”, Quantum Information Processing 16, (2017) arXiv:1705.04087 DOI
[6]
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
[7]
P. Webster, S. D. Bartlett, and D. Poulin, “Reducing the overhead for quantum computation when noise is biased”, Physical Review A 92, (2015) arXiv:1509.05032 DOI
[8]
P. Aliferis et al., “Fault-tolerant computing with biased-noise superconducting qubits: a case study”, New Journal of Physics 11, 013061 (2009) arXiv:0806.0383 DOI
[9]
J. Guillaud and M. Mirrahimi, “Repetition Cat Qubits for Fault-Tolerant Quantum Computation”, Physical Review X 9, (2019) arXiv:1904.09474 DOI
[10]
Z. W. E. Evans et al., “Error correction optimisation in the presence of X/Z asymmetry”, (2007) arXiv:0709.3875
[11]
P. Aliferis and J. Preskill, “Fault-tolerant quantum computation against biased noise”, Physical Review A 78, (2008) arXiv:0710.1301 DOI
[12]
L. Wang and S. Zhu, “On the Construction of Optimal Asymmetric Quantum Codes”, (2014) arXiv:1403.7755
[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
[14]
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]
D. K. Tuckett, S. D. Bartlett, and S. T. Flammia, “Ultrahigh Error Threshold for Surface Codes with Biased Noise”, Physical Review Letters 120, (2018) arXiv:1708.08474 DOI
[17]
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
[24]
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
[28]
G. G. La Guardia, “Asymmetric quantum Reed-Solomon and generalized Reed-Solomon codes”, Quantum Information Processing 11, 591 (2011) DOI
[29]
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
[31]
S. Puri et al., “Bias-preserving gates with stabilized cat qubits”, Science Advances 6, (2020) arXiv:1905.00450 DOI
[32]
J. Roffe et al., “Bias-tailored quantum LDPC codes”, Quantum 7, 1005 (2023) arXiv:2202.01702 DOI
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Zoo Code ID: asymmetric_qecc

Cite as:
“Asymmetric quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/asymmetric_qecc
BibTeX:
@incollection{eczoo_asymmetric_qecc, title={Asymmetric quantum code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/asymmetric_qecc} }
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“Asymmetric quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/asymmetric_qecc

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