Built on

M. Borello, J. De La Cruz, and W. Willems, On checkable codes in group algebras, Journal of Algebra and Its Applications 21
Original
1901
Earlier work this paper cites.
P. Panteleev and G. Kalachev, Degenerate quantum LDPC codes with good finite length performance, Quantum 5
Original
1904
Earlier work this paper cites.
S. Lin and E. J. Weldon, Long BCH codes are bad, Information and Control 11
1967
Earlier work this paper cites.
E. Berlekamp and J. Justesen, Some long cyclic linear binary codes are not so bad, IEEE Transactions on Information Theory 20
1974
Earlier work this paper cites.
Y. A. Drozd and V. V. Kirichenko, Finite Dimensional Algebras (Springer-Verlag, 1994)
1994
Earlier work this paper cites.
P. W. Shor, Fault-tolerant quantum computation, in Proc. 37th Ann. Symp. on Fundamentals of Comp. Sci. , IEEE (IEEE Comp. Soc. Press, Los Alamitos, 1996) pp. 56–65, quant-ph/9605011
1996
Earlier work this paper cites.
A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Phys. Rev. A 54
1996
Earlier work this paper cites.
A. M. Steane, Simple quantum error-correcting codes, Phys. Rev. A 54
1996
Earlier work this paper cites.
D. Gottesman, Stabilizer Codes and Quantum Error Correction , Ph.D. thesis , Caltech (1997)
1997
Earlier work this paper cites.
S. B. Bravyi and A. Y. Kitaev, Quantum codes on a lattice with boundary, quant-ph/9811052 (1998), unpublished
1998
Earlier work this paper cites.
E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, J. Math. Phys. 43
2002
Earlier work this paper cites.
A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303
2003
Earlier work this paper cites.
L. M. Bazzi and S. K. Mitter, Some randomized code constructions from group actions, IEEE Transactions on Information Theory 52
2006
Earlier work this paper cites.
W. Zeng and L. P. Pryadko, Minimal distances for certain quantum product codes and tensor products of chain complexes, Phys. Rev. A 102
Original
2007
Earlier work this paper cites.
J.-P. Tillich and G. Zémor, Quantum LDPC codes with positive rate and minimum distance proportional to n \sqrt{n} , in Proc. IEEE Int. Symp. Inf. Theory (ISIT) (2009) pp. 799–803
2009
Earlier work this paper cites.
P. Panteleev and G. Kalachev, Quantum LDPC codes with almost linear minimum distance, IEEE Transactions on Information Theory 68
Original
2012
Earlier work this paper cites.
H. Bombin, R. W. Chhajlany, M. Horodecki, and M. A. Martin-Delgado, Self-correcting quantum computers, New Journal of Physics 15
2013
Earlier work this paper cites.
Y. Fujiwara, Ability of stabilizer quantum error correction to protect itself from its own imperfection, Phys. Rev. A 90
2014
Earlier work this paper cites.
A. Ashikhmin, C. Y. Lai, and T. A. Brun, Robust quantum error syndrome extraction by classical coding, in 2014 IEEE International Symposium on Information Theory (2014) pp. 546–550
2014
Earlier work this paper cites.
Y. Tomita and K. M. Svore, Low-distance surface codes under realistic quantum noise, Phys. Rev. A 90
Original
2014
Earlier work this paper cites.

Similar

A. M. Stephens, Fault-tolerant thresholds for quantum error correction with the surface code, Phys. Rev. A 89
Original
2014
Cited alongside, same era.
M. B. Hastings, Decoding in hyperbolic spaces: LDPC codes with linear rate and efficient error correction, Quantum Information and Computation 14
Original
2014
Cited alongside, same era.
H. Bombín, Single-shot fault-tolerant quantum error correction, Phys. Rev. X 5
Original
2015
Cited alongside, same era.
I. Dumer, A. A. Kovalev, and L. P. Pryadko, Thresholds for correcting errors, erasures, and faulty syndrome measurements in degenerate quantum codes, Phys. Rev. Lett. 115
Original
2015
Cited alongside, same era.
B. J. Brown, N. H. Nickerson, and D. E. Browne, Fault-tolerant error correction with the gauge color code, Nature Communications 7

Then

J. Roffe, LDPC: Python tools for low density parity check codes , PyPi repository (2022)
2022
Later among the works it cites.
Google Quantum AI, Suppressing quantum errors by scaling a surface code logical qubit, Nature 614
Original
2023
Later among the works it cites.
L. Skoric, D. E. Browne, K. M. Barnes, N. I. Gillespie, and E. T. Campbell, Parallel window decoding enables scalable fault tolerant quantum computation, Nature Communications 14
2023
Later among the works it cites.
O. Higgott and N. P. Breuckmann, Improved single-shot decoding of higher-dimensional hypergraph-product codes, PRX Quantum 4
2023
Later among the works it cites.
R. Wang, H.-K. Lin, and L. P. Pryadko, Abelian and non-abelian quantum two-block codes, in 2023 12th International Symposium on Topics in Coding (ISTC) (2023) pp. 1–5, arXiv:2305.06890
Original

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Reading the bibliography…

2025

Abelian multi-cycle codes for single-shot error correction

Lin, Hsiang-Ku, Lim, Pak Kau, Kovalev, Alexey A. et al.

Understand

We construct a family of quantum low-density parity-check codes locally equivalent to higher-dimensional quantum hypergraph-product (QHP) codes.

Similarly to QHP codes, the proposed codes have highly redundant sets of low-weight stabilizer generators, which improves decoding accuracy in a fault-tolerant regime and gives them single-shot properties.
The advantage of the new construction is that it gives shorter codes.
We derive simple expressions for the dimension of the proposed codes in two important special cases, give bounds on the distances, and explicitly construct some relatively short codes.

Built on

M. Borello, J. De La Cruz, and W. Willems, On checkable codes in group algebras, Journal of Algebra and Its Applications 21
Original
1901
Earlier work this paper cites.
P. Panteleev and G. Kalachev, Degenerate quantum LDPC codes with good finite length performance, Quantum 5
Original
1904
Earlier work this paper cites.
S. Lin and E. J. Weldon, Long BCH codes are bad, Information and Control 11
1967
Earlier work this paper cites.
E. Berlekamp and J. Justesen, Some long cyclic linear binary codes are not so bad, IEEE Transactions on Information Theory 20
1974
Earlier work this paper cites.
Y. A. Drozd and V. V. Kirichenko, Finite Dimensional Algebras (Springer-Verlag, 1994)
1994
Earlier work this paper cites.
P. W. Shor, Fault-tolerant quantum computation, in Proc. 37th Ann. Symp. on Fundamentals of Comp. Sci. , IEEE (IEEE Comp. Soc. Press, Los Alamitos, 1996) pp. 56–65, quant-ph/9605011
1996
Earlier work this paper cites.
A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Phys. Rev. A 54
1996
Earlier work this paper cites.
A. M. Steane, Simple quantum error-correcting codes, Phys. Rev. A 54
1996
Earlier work this paper cites.
D. Gottesman, Stabilizer Codes and Quantum Error Correction , Ph.D. thesis , Caltech (1997)
1997
Earlier work this paper cites.
S. B. Bravyi and A. Y. Kitaev, Quantum codes on a lattice with boundary, quant-ph/9811052 (1998), unpublished
1998
Earlier work this paper cites.
E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, J. Math. Phys. 43
2002
Earlier work this paper cites.
A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303
2003
Earlier work this paper cites.
L. M. Bazzi and S. K. Mitter, Some randomized code constructions from group actions, IEEE Transactions on Information Theory 52
2006
Earlier work this paper cites.
W. Zeng and L. P. Pryadko, Minimal distances for certain quantum product codes and tensor products of chain complexes, Phys. Rev. A 102
Original
2007
Earlier work this paper cites.
J.-P. Tillich and G. Zémor, Quantum LDPC codes with positive rate and minimum distance proportional to n \sqrt{n} , in Proc. IEEE Int. Symp. Inf. Theory (ISIT) (2009) pp. 799–803
2009
Earlier work this paper cites.
P. Panteleev and G. Kalachev, Quantum LDPC codes with almost linear minimum distance, IEEE Transactions on Information Theory 68
Original
2012
Earlier work this paper cites.
H. Bombin, R. W. Chhajlany, M. Horodecki, and M. A. Martin-Delgado, Self-correcting quantum computers, New Journal of Physics 15
2013
Earlier work this paper cites.
Y. Fujiwara, Ability of stabilizer quantum error correction to protect itself from its own imperfection, Phys. Rev. A 90
2014
Earlier work this paper cites.
A. Ashikhmin, C. Y. Lai, and T. A. Brun, Robust quantum error syndrome extraction by classical coding, in 2014 IEEE International Symposium on Information Theory (2014) pp. 546–550
2014
Earlier work this paper cites.
Y. Tomita and K. M. Svore, Low-distance surface codes under realistic quantum noise, Phys. Rev. A 90
Original
2014
Earlier work this paper cites.

Similar

A. M. Stephens, Fault-tolerant thresholds for quantum error correction with the surface code, Phys. Rev. A 89
Original
2014
Cited alongside, same era.
M. B. Hastings, Decoding in hyperbolic spaces: LDPC codes with linear rate and efficient error correction, Quantum Information and Computation 14
Original
2014
Cited alongside, same era.
H. Bombín, Single-shot fault-tolerant quantum error correction, Phys. Rev. X 5
Original
2015
Cited alongside, same era.
I. Dumer, A. A. Kovalev, and L. P. Pryadko, Thresholds for correcting errors, erasures, and faulty syndrome measurements in degenerate quantum codes, Phys. Rev. Lett. 115
Original
2015
Cited alongside, same era.
B. J. Brown, N. H. Nickerson, and D. E. Browne, Fault-tolerant error correction with the gauge color code, Nature Communications 7

Then

J. Roffe, LDPC: Python tools for low density parity check codes , PyPi repository (2022)
2022
Later among the works it cites.
Google Quantum AI, Suppressing quantum errors by scaling a surface code logical qubit, Nature 614
Original
2023
Later among the works it cites.
L. Skoric, D. E. Browne, K. M. Barnes, N. I. Gillespie, and E. T. Campbell, Parallel window decoding enables scalable fault tolerant quantum computation, Nature Communications 14
2023
Later among the works it cites.
O. Higgott and N. P. Breuckmann, Improved single-shot decoding of higher-dimensional hypergraph-product codes, PRX Quantum 4
2023
Later among the works it cites.
R. Wang, H.-K. Lin, and L. P. Pryadko, Abelian and non-abelian quantum two-block codes, in 2023 12th International Symposium on Topics in Coding (ISTC) (2023) pp. 1–5, arXiv:2305.06890
Original

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

2016

Cited alongside, same era.

A. Ashikhmin, C. Y. Lai, and T. A. Brun, Correction of data and syndrome errors by stabilizer codes, in 2016 IEEE International Symposium on Information Theory (ISIT) (2016) pp. 2274–2278, arXiv:1602.01545

Original

2016

Cited alongside, same era.

N. P. Breuckmann, K. Duivenvoorden, D. Michels, and B. M. Terhal, Local decoders for the 2D and 4D toric code, Quantum Inf. Comput. 17

Original

2017

Cited alongside, same era.

I. Haviv, M. Langberg, M. Schwartz, and E. Yaakobi, Non-linear cyclic codes that attain the Gilbert-Varshamov bound, in 2017 IEEE International Symposium on Information Theory (ISIT) (2017) pp. 586–588

2017

Cited alongside, same era.

I. Dumer, A. A. Kovalev, and L. P. Pryadko, Distance verification for classical and quantum LDPC codes, IEEE Trans. Inf. Th. 63

2017

Cited alongside, same era.

E. T. Campbell, A theory of single-shot error correction for adversarial noise, Quantum Science and Technology 4

Original

2019

Cited alongside, same era.

W. Zeng and L. P. Pryadko, Higher-dimensional quantum hypergraph-product codes with finite rates, Phys. Rev. Lett. 122

Original

2019

Cited alongside, same era.

N. P. Breuckmann and V. Londe, Single-shot decoding of linear rate LDPC quantum codes with high performance, arXiv:2001.03568 (2020), unpublished

Original

2020

Cited alongside, same era.

2023