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Cited by 1Quantum Computing Algorithms and ArchitectureQuantum Information and CryptographyCoding theory and cryptography

Gottesman-Kitaev-Preskill codes: A lattice perspective

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Conrad, Jonathan, Eisert, Jens, Arzani, Francesco·2022·Quantum
DOI 10.22331/q-2022-02-10-648
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Abstract

We examine general Gottesman-Kitaev-Preskill (GKP) codes for continuous-variable quantum error correction, including concatenated GKP codes, through the lens of lattice theory, in order to better understand the structure of this class of stabilizer codes. We derive formal bounds on code parameters, show how different decoding strategies are precisely related, propose new ways to obtain GKP codes by means of glued lattices and the tensor product of lattices and point to natural resource savings that have remained hidden in recent approaches. We present general results that we illustrate through examples taken from different classes of codes, including scaled self-dual GKP codes and the concatenated surface-GKP code.

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  • GKP-code stabilizers are isomorphic to symplectic integral lattices.

    “In general, GKP-code stabilizers are isomorphic to symplectic integral lattices [149].”

    From Quantum error correction and fault tolerance: A comprehensive tutorial (2026) · cites this paper as [149]

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Research works printed but not linked: 47

  1. B. M. Terhal, J. Conrad, and C. Vuillot. "Towards scalable bosonic quantum error correction". Quantum Science and Technology 5, 043001 (2020). · DOI · Paper page
  2. A. L. Grimsmo and S. Puri. "Quantum error correction with the Gottesman-Kitaev-Preskill code". PRX Quantum 2, 020101 (2021). · DOI
  3. J. E. Bourassa, R. N. Alexander, M. Vasmer, A. Patil, I. Tzitrin, T. Matsuura, D. Su, B. Q. Baragiola, S. Guha, G. Dauphinais, and et al. "Blueprint for a scalable photonic fault-tolerant quantum computer". Quantum 5, 392 (2021). · DOI
  4. S. Bartolucci, P. Birchall, H. Bombin, H. Cable, C. Dawson, M. Gimeno-Segovia, E. Johnston, K. Kieling, N. Nickerson, M. Pant, F. Pastawski, T. Rudolph, and C. Sparrow. "Fusion-based quantum computation" (2021). arXiv:2101.09310. · arXiv
  5. K. Noh, C. Chamberland, and F. G. S. L. Brandão. "Low-overhead fault-tolerant quantum error correction with the surface-GKP code". PRX Quantum 3, 010315 (2022). · DOI
  6. C. Flühmann, T. L. Nguyen, M. Marinelli, V. Negnevitsky, K. Mehta, and J. P. Home. "En- coding a qubit in a trapped-ion mechanical oscillator". Nature 566, 513-517 (2019). · DOI · Paper page
  7. P. Campagne-Ibarcq, A. Eickbusch, S. Touzard, E. Zalys-Geller, N. E. Frattini, V. V. Sivak, P. Reinhold, S. Puri, S. Shankar, R. J. Schoelkopf, and et al. "Quantum error correction of a qubit encoded in grid states of an oscillator". Nature 584, 368-372 (2020). · DOI · Paper page
  8. K. Fukui, A. Tomita, and A. Okamoto. "Analog quantum error correction with encoding a qubit into an oscillator". Phys. Rev. Lett. 119, 180507 (2017). · DOI · Paper page
  9. C. Vuillot, H. Asasi, Y. Wang, L. P. Pryadko, and B. M. Terhal. "Quantum error correction with the toric Gottesman-Kitaev-Preskill code". Phys. Rev. A 99, 032344 (2019). · DOI
  10. K. Noh and C. Chamberland. "Fault-tolerant bosonic quantum error correction with the surface-Gottesman-Kitaev-Preskill code". Phys. Rev. A 101, 012316 (2020). · DOI · Paper page
  11. L. Hänggli, M. Heinze, and R. König. "Enhanced noise resilience of the surface-gottesman- kitaev-preskill code via designed bias". Phys. Rev. A 102 (2020). · DOI
  12. J. W. Harrington. "Analysis of quantum error-correcting codes: Symplectic lattice codes and toric codes". PhD thesis. California Institute of Technology. (2004). · DOI
  13. J. Harrington and J. Preskill. "Achievable rates for the Gaussian quantum channel". Phys. Rev. A 64, 062301 (2001). · DOI
  14. L. Hänggli and R. König. "Oscillator-to-oscillator codes do not have a threshold". IEEE Transactions on Information Theory 68, 1068-1084 (2022). · DOI
  15. J. Conway and N. Sloane. "Sphere packings, lattices and groups". Volume 290. Springer, New York, NY (1988). · DOI
  16. P. Shor and R. Laflamme. "Quantum analog of the macwilliams identities for classical coding theory". Phys. Rev. Lett. 78, 1600-1602 (1997). · DOI
  17. E. M. Rains. "Quantum weight enumerators". IEEE Trans. Inf. Th. 44, 1388-1394 (1998). · DOI
  18. D. E. Gottesman. "Stabilizer codes and quantum error correction". PhD thesis. California Institute of Technology. (1997). · DOI · Paper page
  19. Arvind, B. Dutta, N. Mukunda, and R. Simon. "The real symplectic groups in quantum mechanics and optics". Pramana 45, 471-497 (1995). · DOI
  20. K. Duivenvoorden, B.M. Terhal, and D. Weigand. "Single-mode displacement sensor". Phys. Rev. A 95, 012305 (2017). · DOI
  21. K. Noh, S. M. Girvin, and L. Jiang. "Encoding an oscillator into many oscillators". Phys. Rev. Lett. 125, 080503 (2020). · DOI
  22. N. P. Breuckmann. "Homological quantum codes beyond the toric code". PhD thesis. RWTH Aachen University. (2017). · DOI
  23. S. B. Bravyi and A. Yu. Kitaev. "Quantum codes on a lattice with boundary" (1998). arXiv:quant-ph/9811052. · arXiv
  24. Y. Tomita and K. M. Svore. "Low-distance surface codes under realistic quantum noise". Phys. Rev. A 90 (2014). · DOI
  25. D. Micciancio and B. Warinschi. "A linear space algorithm for computing the Hermite normal form". In Proceedings of the 2001 International Symposium on Symbolic and Algebraic Com- putation. Page 231-236. ISSAC '01New York, NY, USA (2001). Association for Computing Machinery. · DOI
  26. D. Micciancio. "Improving lattice based cryptosystems using the Hermite normal form". In Lecture Notes in Computer Science. Pages 126-145. Springer Berlin Heidelberg (2001). · DOI
  27. C. Hermite. "Sur l'introduction des variables continues dans la théorie des nombres.". J. Reine Angew. Math 1851, 191-216 (1851). · DOI
  28. A. Lenstra, H. Lenstra, and L. Lovász. "Factoring polynomials with rational coefficients". Mathematische Annalen 261, 515-534 (1982). · DOI
  29. K. Chandrasekaran, V. Gandikota, and E. Grigorescu. "Deciding orthogonality in Construction-A Lattices". In P. Harsha and G. Ramalingam, editors, 35th IARCS Annual Con- ference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Volume 45 of Leibniz International Proceedings in Informatics (LIPIcs), pages 151-162. · DOI
  30. A. Y. Kitaev. "Fault-tolerant quantum computation by anyons". Ann. Phys. 303, 2-30 (2003). · DOI · Paper page
  31. K. Noh, V. V. Albert, and L. Jiang. "Quantum capacity bounds of Gaussian thermal loss channels and achievable rates with Gottesman-Kitaev-Preskill codes". IEEE Trans. Inf. Th. 65, 2563-2582 (2019). · DOI
  32. S. Glancy and E. Knill. "Error analysis for encoding a qubit in an oscillator". Phys. Rev. A 73 (2006). · DOI
  33. I. Tzitrin, J. E. Bourassa, N. C. Menicucci, and K. K. Sabapathy. "Progress towards practical qubit computation using approximate Gottesman-Kitaev-Preskill codes". Phys. Rev. A 101, 032315 (2020). · DOI
  34. J. Conrad. "Twirling and Hamiltonian engineering via dynamical decoupling for Gottesman- Kitaev-Preskill quantum computing". Phys. Rev. A 103 (2021). · DOI
  35. S. L. Braunstein. "Squeezing as an irreducible resource". Phys. Rev. A 71, 055801 (2005). · DOI
  36. L. Babai. "On lovász' lattice reduction and the nearest lattice point problem". Combinatorica 6, 1-13 (1986). · DOI
  37. E. Knill, R. Laflamme, and L. Viola. "Theory of quantum error correction for general noise". Phys. Rev. Lett. 84, 2525-2528 (2000). · DOI · Paper page
  38. Y. Tian, X. Zhu, and R. Sun. "Modular form approach to solving lattice problems". In T. V. Gopal, M Agrawal, A Li, and S. B. Cooper, editors, Theory and Applications of Models of Computation. Pages 401-421. Cham (2014). Springer International Publishing. · DOI
  39. S. Bravyi, M. Suchara, and A. Vargo. "Efficient algorithms for maximum likelihood decoding in the surface code". Phys. Rev. A 90 (2014). · DOI · Paper page
  40. T. Gannon. "Lattices and theta functions". PhD thesis. McGill University (Canada). (1991). url: escholarship.mcgill.ca/concern/theses/rr171z009.
  41. J.-P. Tillich and G. Zemor. "Quantum LDPC codes with positive rate and minimum dis- tance proportional to the square root of the block length". IEEE Trans. Inf. Th. 60, 1193-1202 (2014). · DOI
  42. S. Bravyi and M. B. Hastings. "Homological product codes". In Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing. Page 273-282. STOC '14New York, NY, USA (2014). Association for Computing Machinery. · DOI
  43. E. Dennis, A. Kitaev, A. Landahl, and J. Preskill. "Topological quantum memory". J. Math. Phys. 43, 4452-4505 (2002). · DOI · Paper page
  44. C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T.C. Ralph, J.H. Shapiro, and S. Lloyd. "Gaussian quantum information". Rev. Mod. Phys. 84, 621-669 (2012). · DOI
  45. J. Eisert and M. B. Plenio. "Introduction to the basics of entanglement theory in continuous- variable systems". Int. J. Quant. Inf. 1, 479 (2003). · DOI
  46. L. J. Mensen, B. Q. Baragiola, and N. C. Menicucci. "Phase-space methods for represent- ing, manipulating, and correcting Gottesman-Kitaev-Preskill qubits". Phys. Rev. A 104, 022408 (2021). · DOI
  47. K. Duivenvoorden, B. M. Terhal, and D. Weigand. "Single-mode displacement sensor". Phys. Rev. A 95, 012305 (2017). · DOI

Cited by

Sorted by citations

Authors
Conrad, J., Eisert, J., & Arzani, F.
Affiliations
  • Helmholtz-Zentrum Berlin für Materialien und Energie (DE)
  • Freie Universität Berlin (DE)
ORCID
Venue
Quantum
Publisher
Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften
Published
2022-02-10
Volume
6
Pages
648
ISSN
2521-327X (ISSN-L)
DOI
10.22331/q-2022-02-10-648
OpenAlex
W3202555143
Cited by
0
References
1
FWCI
0.00
Type
journal article
Language
en
Open access
Licence
Locations
  • Quantum (publishedVersion, cc-by)
  • arXiv (Cornell University) (submittedVersion)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
  • HZB Repository (Helmholtz-Zentrum Berlin für Materialien und Energie GmbH (HZB)) (submittedVersion, other-oa)
  • arXiv (Cornell University) (cc-by)
Funders
  • BMBF (RealistiQ, PhoQuant, QPIC-1, QSolid)
  • DFG (CRC 183)
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
  • Physical Sciences › Computer Science › Artificial Intelligence › Coding theory and cryptography
Keywords
Gottesman-Kitaev-Preskill code, continuous-variable quantum error correction, stabilizer codes, lattice theory, self-dual lattices, surface code, decoding strategies

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