Stabilizer Codes with Exotic Local-dimensions

Lane G. Gunderman

No affiliation for this work

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


Traditional stabilizer codes operate over prime power local-dimensions. In this work we extend the stabilizer formalism using the local-dimension-invariant setting to import stabilizer codes from these standard local-dimensions to other cases. In particular, we show that any traditional stabilizer code can be used for analog continuous-variable codes, and consider restrictions in phase space and discretized phase space. This puts this framework on an equivalent footing as traditional stabilizer codes. Following this, using extensions of prior ideas, we show that a stabilizer code originally designed with a finite field local-dimension can be transformed into a code with the same $n$, $k$, and $d$ parameters for any integral domain. This is of theoretical interest and can be of use for systems whose local-dimension is better described by mathematical rings, which permits the use of traditional stabilizer codes for protecting their information as well.

This work provides an extension of the traditional stabilizer formalism for encoding quantum information to other settings such as a continuous-variable systems (conjugate quadratures), discretized phase-space, phase encodings, as well as more mathematical settings that may correspond to physical settings.

► BibTeX data

► References

[1] Daniel Gottesman ``Class of quantum error-correcting codes saturating the quantum Hamming bound'' Physical Review A 54, 1862 (1996).

[2] Daniel Gottesman ``Stabilizer codes and quantum error correction'' California Institute of Technology (1997).

[3] A Robert Calderbankand Peter W Shor ``Good quantum error-correcting codes exist'' Physical Review A 54, 1098 (1996).

[4] Andrew M Steane ``Error correcting codes in quantum theory'' Physical Review Letters 77, 793 (1996).

[5] Lane G Gunderman ``Local-dimension-invariant qudit stabilizer codes'' Physical Review A 101, 052343 (2020).

[6] Lane G Gunderman ``Degenerate local-dimension-invariant stabilizer codes and an alternative bound for the distance preservation condition'' Physical Review A 105, 042424 (2022).

[7] Arun J Moorthyand Lane G Gunderman ``Local-dimension-invariant Calderbank–Shor–Steane codes with an improved distance promise'' Quantum Information Processing 22, 59 (2023).

[8] Seth Lloydand Jean-Jacques E Slotine ``Analog quantum error correction'' Physical Review Letters 80, 4088 (1998).

[9] Samuel L Braunstein ``Error correction for continuous quantum variables'' Springer (1998).

[10] Alexei Ashikhminand Emanuel Knill ``Nonbinary quantum stabilizer codes'' IEEE Transactions on Information Theory 47, 3065–3072 (2001).

[11] Vlad Gheorghiu ``Standard form of qudit stabilizer groups'' Physics Letters A 378, 505–509 (2014).

[12] Stephen S Bullockand Gavin K Brennen ``Qudit surface codes and gauge theory with finite cyclic groups'' Journal of Physics A: Mathematical and Theoretical 40, 3481 (2007).

[13] Tyler D Ellison, Yu-An Chen, Arpit Dua, Wilbur Shirley, Nathanan Tantivasadakarn, and Dominic J Williamson, ``Pauli stabilizer models of twisted quantum doubles'' PRX Quantum 3, 010353 (2022).

[14] Victor V Albert, Jacob P Covey, and John Preskill, ``Robust encoding of a qubit in a molecule'' Physical Review X 10, 031050 (2020).

[15] John Watrous ``The theory of quantum information'' Cambridge university press (2018).

[16] Daniel A Lidarand Todd A Brun ``Quantum error correction'' Cambridge university press (2013).

[17] Avanti Ketkar, Andreas Klappenecker, Santosh Kumar, and Pradeep Kiran Sarvepalli, ``Nonbinary stabilizer codes over finite fields'' IEEE transactions on information theory 52, 4892–4914 (2006).

[18] HF Chau ``Five quantum register error correction code for higher spin systems'' Physical Review A 56, R1 (1997).

[19] HF Chau ``Correcting quantum errors in higher spin systems'' Physical Review A 55, R839 (1997).

[20] Andrew Steane ``Multiple-particle interference and quantum error correction'' Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 452, 2551–2577 (1996).

[21] Daniel Gottesman ``Stabilizer codes with prime power qudits'' invited talk at Caltech IQIM seminar (Pasadena, California) 1, 12–13 (2014).

[22] Priya J Nadkarniand Shayan Srinivasa Garani ``$\mathbb{F}_p$-Linear and $\mathbb{F}_{p^m}$-Linear Qudit Codes From Dual-Containing Classical Codes'' IEEE Transactions on Quantum Engineering 2, 1–19 (2021).

[23] Shayan Srinivasa Garani, Priya J Nadkarni, and Ankur Raina, ``Theory Behind Quantum Error Correcting Codes: An Overview'' Journal of the Indian Institute of Science 1–47 (2023).

[24] Daniel Gottesman ``Fault-tolerant quantum computation with higher-dimensional systems'' Quantum Computing and Quantum Communications: First NASA International Conference, QCQC’98 Palm Springs, California, USA February 17–20, 1998 Selected Papers 302–313 (1999).

[25] Rahul Sarkarand Theodore J Yoder ``The qudit Pauli group: non-commuting pairs, non-commuting sets, and structure theorems'' arXiv preprint arXiv:2302.07966 (2023).

[26] Richard L Barnes ``Stabilizer codes for continuous-variable quantum error correction'' arXiv preprint quant-ph/​0405064 (2004).

[27] Victor V Albert ``Bosonic coding: introduction and use cases'' arXiv preprint arXiv:2211.05714 (2022).

[28] Pavel Panteleevand Gleb Kalachev ``Asymptotically good quantum and locally testable classical LDPC codes'' Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing 375–388 (2022).

[29] Anthony Leverrierand Gilles Zémor ``Quantum tanner codes'' 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS) 872–883 (2022).

[30] Irit Dinur, Min-Hsiu Hsieh, Ting-Chun Lin, and Thomas Vidick, ``Good quantum LDPC codes with linear time decoders'' Proceedings of the 55th Annual ACM Symposium on Theory of Computing 905–918 (2023).

[31] Markus Stroppel ``Locally compact groups'' European Mathematical Society (2006).

[32] Daniel Gottesman, Alexei Kitaev, and John Preskill, ``Encoding a qubit in an oscillator'' Physical Review A 64, 012310 (2001).

[33] Kyungjoo Noh, SM Girvin, and Liang Jiang, ``Encoding an oscillator into many oscillators'' Physical Review Letters 125, 080503 (2020).

[34] Jonathan Conrad, Jens Eisert, and Francesco Arzani, ``Gottesman-Kitaev-Preskill codes: A lattice perspective'' Quantum 6, 648 (2022).

[35] Jim Harringtonand John Preskill ``Achievable rates for the Gaussian quantum channel'' Physical Review A 64, 062301 (2001).

[36] Jonathan Conrad, Jens Eisert, and Jean-Pierre Seifert, ``Good Gottesman-Kitaev-Preskill codes from the NTRU cryptosystem'' arXiv preprint arXiv:2303.02432 (2023).

[37] Matthew B Hastings ``On quantum weight reduction'' arXiv preprint arXiv:2102.10030 (2021).

[38] Annika Niehage ``Quantum Goppa Codes over Hyperelliptic Curves'' arXiv preprint quant-ph/​0501074 (2005).

[39] Arne L Grimsmo, Joshua Combes, and Ben Q Baragiola, ``Quantum computing with rotation-symmetric bosonic codes'' Physical Review X 10, 011058 (2020).

[40] Philippe Faist, Sepehr Nezami, Victor V Albert, Grant Salton, Fernando Pastawski, Patrick Hayden, and John Preskill, ``Continuous symmetries and approximate quantum error correction'' Physical Review X 10, 041018 (2020).

[41] A Yu Kitaev ``Fault-tolerant quantum computation by anyons'' Annals of physics 303, 2–30 (2003).

[42] Lane Gunderman ``Collective Spin-Cavity Ensembles and the Protection of Higher-dimensional Quantum Information'' (2022).

[43] Haruki Watanabe, Meng Cheng, and Yohei Fuji, ``Ground state degeneracy on torus in a family of ZN toric code'' Journal of Mathematical Physics 64 (2023).

[44] Manu Mathurand Atul Rathor ``SU (N) toric code and non-Abelian anyons'' Physical Review A 105, 052423 (2022).

[45] Christophe Vuillot, Alessandro Ciani, and Barbara M Terhal, ``Homological Quantum Rotor Codes: Logical Qubits from Torsion'' arXiv preprint arXiv:2303.13723 (2023).

Cited by

On Crossref's cited-by service no data on citing works was found (last attempt 2024-02-26 15:30:15). On SAO/NASA ADS no data on citing works was found (last attempt 2024-02-26 15:30:16).