Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
134 tokens/sec
GPT-4o
10 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Non-Abelian Self-Correcting Quantum Memory (2405.11719v3)

Published 20 May 2024 in quant-ph, cond-mat.str-el, hep-th, and math.QA

Abstract: We construct a family of infinitely many new candidate non-Abelian self-correcting topological quantum memories in $D\geq 5+1$ spacetime dimensions without particle excitations using local commuting non-Pauli stabilizer lattice models and field theories of $\mathbb{Z}_23$ higher-form gauge fields with nontrivial topological action. We call such non-Pauli stabilizer models magic stabilizer codes. The family of topological orders have Abelian electric excitations and non-Abelian magnetic excitations that obey Ising-like fusion rules and non-Abelian braiding, including Borromean ring type braiding which is a signature of non-Abelian topological order, generalizing the dihedral group $\mathbb{D}_8$ gauge theory in (2+1)D. The simplest example includes a new non-Abelian self-correcting memory in (5+1)D with Abelian loop excitations and non-Abelian membrane excitations. We prove the self-correction property and the thermal stability, and devise a probabilistic local cellular-automaton decoder.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (68)
  1. M. Iqbal et al., “Non-Abelian topological order and anyons on a trapped-ion processor,” Nature 626 no. 7999, (2024) 505–511, arXiv:2305.03766 [quant-ph].
  2. Z. Nussinov and G. Ortiz, “Autocorrelations and thermal fragility of anyonic loops in topologically quantum ordered systems,” Physical Review B 77 no. 6, (Feb., 2008) . http://dx.doi.org/10.1103/PhysRevB.77.064302.
  3. S. Bravyi and B. Terhal, “A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes,” New Journal of Physics 11 no. 4, (Apr., 2009) 043029. http://dx.doi.org/10.1088/1367-2630/11/4/043029.
  4. M. B. Hastings, “Topological order at nonzero temperature,” Phys. Rev. Lett. 107 (Nov, 2011) 210501. https://link.aps.org/doi/10.1103/PhysRevLett.107.210501.
  5. B. J. Brown, D. Loss, J. K. Pachos, C. N. Self, and J. R. Wootton, “Quantum memories at finite temperature,” Reviews of Modern Physics 88 no. 4, (Nov., 2016) . http://dx.doi.org/10.1103/RevModPhys.88.045005.
  6. L. Guth and A. Lubotzky, “Quantum error correcting codes and 4-dimensional arithmetic hyperbolic manifolds,” Journal of Mathematical Physics 55 no. 8, (2014) 082202.
  7. N. P. Breuckmann and V. Londe, “Single-Shot Decoding of Linear Rate LDPC Quantum Codes with High Performance,” arXiv quant-ph (2020) , 2001.03568.
  8. M. Freedman and M. B. Hastings, “Building manifolds from quantum codes,” arXiv:2012.02249 (2020) .
  9. S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, “High-threshold and low-overhead fault-tolerant quantum memory,” Nature 627 no. 8005, (2024) 778–782. https://doi.org/10.1038/s41586-024-07107-7.
  10. N. P. Breuckmann, C. Vuillot, E. Campbell, A. Krishna, and B. M. Terhal, “Hyperbolic and semi-hyperbolic surface codes for quantum storage,” Quantum Science and Technology 2 no. 3, (Aug., 2017) 035007–21.
  11. M. Hastings, J. Haah, and R. O’Donnell, “Fiber bundle codes: breaking the n1/2⁢polylog⁢(n)superscript𝑛12polylog𝑛n^{1/2}\textrm{polylog}(n)italic_n start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT polylog ( italic_n ) barrier for quantum ldpc codes,” in Proc. ACM STOC, pp. 1276–1288. Association for Computing Machinery, New York, NY, USA, 2021.
  12. P. Panteleev and G. Kalachev, “Quantum ldpc codes with almost linear minimum distance,” IEEE Trans. Inf. Theo. 68 no. 1, (2022) 213–229.
  13. N. P. Breuckmann and J. N. Eberhardt, “Balanced product quantum codes,” IEEE Trans. Inf. Theo. 67 no. 10, (2021) 6653–6674.
  14. M. Hastings, “On quantum weight reduction,” arXiv:2102.10030 (2021) .
  15. P. Panteleev and G. Kalachev, “Asymptotically good quantum and locally testable classical ldpc codes,” in Proc. ACM STOC, pp. 375—388. Association for Computing Machinery, New York, NY, USA, 2022.
  16. A. Leverrier and G. Zemor, “Quantum tanner codes,” in Proc. IEEE FOCS, pp. 872–883. IEEE Computer Society, Los Alamitos, CA, USA, 2022. https://doi.ieeecomputersociety.org/10.1109/FOCS54457.2022.00117.
  17. T. Lin and M. Hsieh, “Good quantum ldpc codes with linear time decoder from lossless expanders,” arXiv:2203.03581 (2022) .
  18. S. Gu, C. Pattison, and E. Tang, “An efficient decoder for a linear distance quantum ldpc code,” arXiv:2206.06557 (2022) .
  19. I. Dinur, M. Hsieh, T. Lin, and T. Vidick, “Good quantum ldpc codes with linear time decoders,” in Proc. ACM STOC, pp. 905–918. Association for Computing Machinery, New York, NY, USA, 2023.
  20. 2023. https://epubs.siam.org/doi/abs/10.1137/1.9781611977554.ch45.
  21. S. Gu, E. Tang, L. Caha, S. Choe, Z. He, and A. Kubica, “Single-shot decoding of good quantum ldpc codes,” arXiv:2306.12470 (2023) .
  22. X. Chen, A. Dua, P.-S. Hsin, C.-M. Jian, W. Shirley, and C. Xu, “Loops in 4+1d Topological Phases,” arXiv:2112.02137 [cond-mat.str-el].
  23. B. Yoshida, “Feasibility of self-correcting quantum memory and thermal stability of topological order,” Annals of Physics 326 no. 10, (Oct., 2011) 2566–2633. http://dx.doi.org/10.1016/j.aop.2011.06.001.
  24. E. T. Campbell, B. M. Terhal, and C. Vuillot, “Roads towards fault-tolerant universal quantum computation,” Nature 549 no. 7671, (09, 2017) 172–179.
  25. E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory,” J. Math. Phys. 43 (2002) 4452–4505, arXiv:quant-ph/0110143.
  26. H. Bombin, “Single-shot fault-tolerant quantum error correction,” Phys. Rev. X 5 no. 3, (2015) 031043.
  27. G. Zhu, A. Lavasani, and M. Barkeshli, “Universal logical gates on topologically encoded qubits via constant-depth unitary circuits,” Phys. Rev. Lett. 125 (Jul, 2020) 050502. https://link.aps.org/doi/10.1103/PhysRevLett.125.050502.
  28. G. Zhu, A. Lavasani, and M. Barkeshli, “Instantaneous braids and dehn twists in topologically ordered states,” Phys. Rev. B 102 (Aug, 2020) 075105. https://link.aps.org/doi/10.1103/PhysRevB.102.075105.
  29. A. Lavasani, G. Zhu, and M. Barkeshli, “Universal logical gates with constant overhead: instantaneous dehn twists for hyperbolic quantum codes,” Quantum 3, 180 (2019) .
  30. H. Bombin, R. W. Chhajlany, M. Horodecki, and M. A. Martin-Delgado, “Self-correcting quantum computers,” New Journal of Physics 15 no. 5, (May, 2013) 055023–44.
  31. T. Johnson-Freyd and M. Yu, “Topological Orders in (4+1)-Dimensions,” SciPost Phys. 13 no. 3, (2022) 068, arXiv:2104.04534 [hep-th].
  32. C. Cordova, P.-S. Hsin, and C. Zhang, “Anomalies of Non-Invertible Symmetries in (3+1)d,” arXiv:2308.11706 [hep-th].
  33. L. Kong and X.-G. Wen, “Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions,” arXiv:1405.5858 [cond-mat.str-el].
  34. L. Kong, X.-G. Wen, and H. Zheng, “Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers,” 2015.
  35. L. Kong, X.-G. Wen, and H. Zheng, “Boundary-bulk relation in topological orders,” Nuclear Physics B 922 (Sept., 2017) 62–76. http://dx.doi.org/10.1016/j.nuclphysb.2017.06.023.
  36. T. Johnson-Freyd, “On the classification of topological orders,” Communications in Mathematical Physics 393 no. 2, (Apr., 2022) 989–1033. http://dx.doi.org/10.1007/s00220-022-04380-3.
  37. N. Madras and G. Slade, The Self-Avoiding Walk. Modern Birkhäuser Classics. Springer New York, 2012. https://books.google.com/books?id=xo32YMglDOcC.
  38. M. de Wild Propitius, “Confinement in partially broken abelian chern-simons theories,” Physics Letters B 410 no. 2, (1997) 188–194. https://www.sciencedirect.com/science/article/pii/S0370269397009854.
  39. A. Coste, T. Gannon, and P. Ruelle, “Finite group modular data,” Nucl. Phys. B 581 (2000) 679–717, arXiv:hep-th/0001158.
  40. P.-S. Hsin and A. Turzillo, “Symmetry-enriched quantum spin liquids in (3 + 1)d𝑑ditalic_d,” JHEP 09 (2020) 022, arXiv:1904.11550 [cond-mat.str-el].
  41. M. Barkeshli, Y.-A. Chen, P.-S. Hsin, and R. Kobayashi, “Higher-group symmetry in finite gauge theory and stabilizer codes,” SciPost Phys. 16 (2024) 089, arXiv:2211.11764 [cond-mat.str-el].
  42. E. Witten, “Five-brane effective action in M theory,” J. Geom. Phys. 22 (1997) 103–133, arXiv:hep-th/9610234.
  43. E. Witten, “AdS / CFT correspondence and topological field theory,” JHEP 12 (1998) 012, arXiv:hep-th/9812012.
  44. E. Witten, “Geometric Langlands From Six Dimensions,” arXiv:0905.2720 [hep-th].
  45. D. S. Freed and C. Teleman, “Relative quantum field theory,” Commun. Math. Phys. 326 (2014) 459–476, arXiv:1212.1692 [hep-th].
  46. To appear.
  47. B. Yoshida, “Topological color code and symmetry-protected topological phases,” Phys. Rev. B 91 (Jun, 2015) 245131. https://link.aps.org/doi/10.1103/PhysRevB.91.245131.
  48. B. Yoshida, “Topological phases with generalized global symmetries,” Phys. Rev. B 93 (Apr, 2016) 155131. https://link.aps.org/doi/10.1103/PhysRevB.93.155131.
  49. B. Yoshida, “Gapped boundaries, group cohomology and fault-tolerant logical gates,” Annals of Physics 377 (2017) 387–413. https://www.sciencedirect.com/science/article/pii/S0003491616302858.
  50. M. Barkeshli, Y.-A. Chen, S.-J. Huang, R. Kobayashi, N. Tantivasadakarn, and G. Zhu, “Codimension-2 defects and higher symmetries in (3+ 1) d topological phases,” SciPost Physics 14 no. 4, (2023) 065.
  51. F. Benini, C. Córdova, and P.-S. Hsin, “On 2-Group Global Symmetries and their Anomalies,” JHEP 03 (2019) 118, arXiv:1803.09336 [hep-th].
  52. L. Tsui and X.-G. Wen, “Lattice models that realize 𝕫nsubscript𝕫𝑛{\mathbb{z}}_{n}blackboard_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT-1 symmetry-protected topological states for even n𝑛nitalic_n,” Phys. Rev. B 101 (Jan, 2020) 035101.
  53. Y.-A. Chen and S. Tata, “Higher cup products on hypercubic lattices: application to lattice models of topological phases,” arXiv:2106.05274 [cond-mat.str-el].
  54. X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, “Symmetry protected topological orders and the group cohomology of their symmetry group,” Phys. Rev. B 87 no. 15, (2013) 155114, arXiv:1106.4772 [cond-mat.str-el].
  55. L. Tsui and X.-G. Wen, “Lattice models that realize ℤnsubscriptℤ𝑛\mathbb{Z}_{n}blackboard_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT-1 symmetry-protected topological states for even n𝑛nitalic_n,” Phys. Rev. B 101 no. 3, (2020) 035101, arXiv:1908.02613 [cond-mat.str-el].
  56. M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases,” Phys. Rev. B 86 (Sep, 2012) 115109. https://link.aps.org/doi/10.1103/PhysRevB.86.115109.
  57. L. Bhardwaj, D. Gaiotto, and A. Kapustin, “State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter,” JHEP 04 (2017) 096, arXiv:1605.01640 [cond-mat.str-el].
  58. W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries,” SciPost Phys. 6 no. 4, (2019) 041, arXiv:1806.08679 [cond-mat.str-el].
  59. A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code,” New J. Phys. 17 no. 8, (2015) 083026, arXiv:1503.02065 [quant-ph].
  60. H. Bombin and M. A. Martin-Delgado, “Topological quantum distillation,” Phys. Rev. Lett. 97 (2006) 180501, arXiv:quant-ph/0605138.
  61. G. Zhu, S. Sikander, E. Portnoy, A. W. Cross, and B. J. Brown, “Non-clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum ldpc codes via higher symmetries,” arXiv preprint arXiv:2310.16982 (2023) .
  62. M. Barkeshli, P.-S. Hsin, and R. Kobayashi, “Higher-group symmetry of (3+1)D fermionic ℤ2subscriptℤ2\mathbb{Z}_{2}blackboard_Z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT gauge theory: logical CCZ, CS, and T gates from higher symmetry,” SciPost Phys. 16 (2024) 122, arXiv:2311.05674 [cond-mat.str-el].
  63. P. Putrov, J. Wang, and S.-T. Yau, “Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions,” Annals Phys. 384 (2017) 254–287, arXiv:1612.09298 [cond-mat.str-el].
  64. B. Durhuus, J. Fröhlich, and T. Jonsson, “Self-avoiding and planar random surfaces on the lattice,” Nuclear Physics B 225 no. 2, (1983) 185–203. https://www.sciencedirect.com/science/article/pii/0550321383900482.
  65. R. Alicki, M. Horodecki, P. Horodecki, and R. Horodecki, “On thermal stability of topological qubit in kitaev’s 4d model,” Open Systems & Information Dynamics 17 no. 01, (2010) 1–20.
  66. A. Dua, T. Jochym-O’Connor, and G. Zhu, “Quantum error correction with fractal topological codes,” Quantum 7 (2023) 1122.
  67. Annals of mathematics studies. Princeton University Press, 1974. https://books.google.com/books?id=5zQ9AFk1i4EC.
  68. D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, “Generalized Global Symmetries,” JHEP 02 (2015) 172, arXiv:1412.5148 [hep-th].
Citations (2)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now