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Fracton models from product codes (2312.08462v2)

Published 13 Dec 2023 in quant-ph, cond-mat.stat-mech, and cond-mat.str-el

Abstract: We explore a deep connection between fracton order and product codes. In particular, we propose and analyze conditions on classical seed codes which lead to fracton order in the resulting quantum product codes. Depending on the properties of the input codes, product codes can realize either Type-I or Type-II fracton models, in both nonlocal and local constructions. For the nonlocal case, we show that a recently proposed model of lineons on an irregular graph can be obtained as a hypergraph product code. Interestingly, constrained mobility in this model arises only from glassiness associated with the graph. For the local case, we introduce a novel type of classical LDPC code defined on a planar aperiodic tiling. By considering the specific example of the pinwheel tiling, we demonstrate the systematic construction of local Type-I and Type-II fracton models as product codes. Our work establishes product codes as a natural setting for exploring fracton order.

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References (41)
  1. C. Chamon, Quantum glassiness in strongly correlated clean systems: An example of topological overprotection, Phys. Rev. Lett. 94, 040402 (2005).
  2. J. Haah, Local stabilizer codes in three dimensions without string logical operators, Phys. Rev. A 83, 042330 (2011).
  3. B. Yoshida, Exotic topological order in fractal spin liquids, Phys. Rev. B 88, 125122 (2013).
  4. S. Vijay, J. Haah, and L. Fu, A new kind of topological quantum order: A dimensional hierarchy of quasiparticles built from stationary excitations, Phys. Rev. B 92, 235136 (2015).
  5. K. Slagle and Y. B. Kim, Quantum field theory of X𝑋Xitalic_X-cube fracton topological order and robust degeneracy from geometry, Phys. Rev. B 96, 195139 (2017).
  6. M. Pretko, The fracton gauge principle, Phys. Rev. B 98, 115134 (2018).
  7. A. Gromov, Towards classification of fracton phases: The multipole algebra, Phys. Rev. X 9, 031035 (2019).
  8. W. Shirley, K. Slagle, and X. Chen, Foliated fracton order from gauging subsystem symmetries, SciPost Phys. 6, 041 (2019).
  9. R. M. Nandkishore and M. Hermele, Fractons, Annual Review of Condensed Matter Physics 10, 295 (2019), https://doi.org/10.1146/annurev-conmatphys-031218-013604 .
  10. M. Pretko, X. Chen, and Y. You, Fracton phases of matter, International Journal of Modern Physics A 35, 2030003 (2020), https://doi.org/10.1142/S0217751X20300033 .
  11. N. P. Breuckmann and J. N. Eberhardt, Balanced product quantum codes, IEEE Transactions on Information Theory 67, 6653 (2021a).
  12. N. P. Breuckmann and J. N. Eberhardt, Quantum low-density parity-check codes, PRX Quantum 2, 040101 (2021b).
  13. J. Haah, Commuting Pauli Hamiltonians as maps between free modules, Communications in Mathematical Physics 324, 351 (2013).
  14. J. Haah, Algebraic methods for quantum codes on lattices, Revista colombiana de matematicas 50, 299 (2016).
  15. Supplemental material.
  16. P. Gorantla, H. T. Lam, and S.-H. Shao, Fractons on graphs and complexity, Phys. Rev. B 106, 195139 (2022).
  17. M. B. Hastings, J. Haah, and R. O’Donnell, Fiber bundle codes: Breaking the n1/2⁢polylog⁡(n)superscript𝑛12polylog𝑛n^{1/2}\operatorname{polylog}(n)italic_n start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT roman_polylog ( italic_n ) barrier for quantum ldpc codes (2020), arXiv:2009.03921 [quant-ph] .
  18. P. Panteleev and G. Kalachev, Quantum LDPC codes with almost linear minimum distance, IEEE Transactions on Information Theory 68, 213 (2022b).
  19. J.-P. Tillich and G. Zémor, Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength, IEEE Transactions on Information Theory 60, 1193 (2014).
  20. H. Bombín, Single-shot fault-tolerant quantum error correction, Phys. Rev. X 5, 031043 (2015).
  21. A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Phys. Rev. A 54, 1098 (1996).
  22. A. M. Steane, Error correcting codes in quantum theory, Phys. Rev. Lett. 77, 793 (1996).
  23. R. Tanner, A recursive approach to low complexity codes, IEEE Transactions on Information Theory 27, 533 (1981).
  24. Note that this scaling is distinct from that of local fractons, which merely follows a polynomial envelope.
  25. Finite-dimensional local fracton models with immobile loop excitations are known [57]. However such a case is not expected for the nonlocal models we consider.
  26. I. H. Kim and J. Haah, Localization from superselection rules in translationally invariant systems, Phys. Rev. Lett. 116, 027202 (2016).
  27. A. Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303, 2 (2003).
  28. A. Kitaev, Anyons in an exactly solved model and beyond, Annals of Physics 321, 2 (2006), january Special Issue.
  29. F. R. Chung, Spectral graph theory, Vol. 92 (American Mathematical Soc., 1997).
  30. N. Delfosse, Tradeoffs for reliable quantum information storage in surface codes and color codes, in 2013 IEEE International Symposium on Information Theory (2013) pp. 917–921.
  31. C. Radin and M. Wolff, Space tilings and local isomorphism, Geometriae Dedicata 42, 355 (1992).
  32. C. Radin, The pinwheel tilings of the plane, Annals of Mathematics 139, 661 (1994).
  33. D. Frettlöh, Substitution tilings with statistical circular symmetry, Eur. J. Comb. 29, 1881–1893 (2008).
  34. We observe that for small values of p𝑝pitalic_p short logical operators attached to the boundary can appear, thus we take large enough values to suppress these.
  35. S. Bravyi, D. Poulin, and B. Terhal, Tradeoffs for reliable quantum information storage in 2d systems, Phys. Rev. Lett. 104, 050503 (2010).
  36. D. J. Williamson and N. Baspin, Layer codes (2023), arXiv:2309.16503 [quant-ph] .
  37. G. Vidal, Entanglement renormalization, Phys. Rev. Lett. 99, 220405 (2007).
  38. J. Haah, Bifurcation in entanglement renormalization group flow of a gapped spin model, Phys. Rev. B 89, 075119 (2014).
  39. J. F. San Miguel, A. Dua, and D. J. Williamson, Bifurcating subsystem symmetric entanglement renormalization in two dimensions, Phys. Rev. B 103, 035148 (2021).
  40. A. Strikis and L. Berent, Quantum low-density parity-check codes for modular architectures, PRX Quantum 4, 020321 (2023).
  41. M.-Y. Li and P. Ye, Fracton physics of spatially extended excitations, Phys. Rev. B 101, 245134 (2020).
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