Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 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

Diagonal Coset Approach to Topological Quantum Computation with Fibonacci Anyons (2404.01779v2)

Published 2 Apr 2024 in quant-ph

Abstract: We investigate a promising conformal field theory realization scheme for topological quantum computation based on the Fibonacci anyons, which are believed to be realized as quasiparticle excitations in the $\mathbb{Z}_3$ parafermion fractional quantum Hall state in the second Landau level with filling factor $\nu=12/5$. These anyons are non-Abelian and are known to be capable of universal topological quantum computation. The quantum information is encoded in the fusion channels of pairs of such non-Abelian anyons and is protected from noise and decoherence by the topological properties of these systems.The quantum gates are realized by braiding of these anyons. We propose here an implementation of the $n$-qubit topological quantum register in terms of $2n+2$ Fibonacci anyons. The matrices emerging from the anyon exchanges, i.e. the generators of the braid group for one qubit are derived from the coordinate wave functions of a large number of electron holes and 4 Fibonacci anyons which can furthermore be represented as correlation functions in $\mathbb{Z}_3$ parafermionic two-dimensional conformal field theory. The representations of the braid groups for more than 4 anyons are obtained by fusing pairs of anyons before braiding, thus reducing eventually the system to 4 anyons.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (45)
  1. J. P. Dowling and G. J. Milburn, “Quantum technology: the second quantum revolution,” Phil. Trans. R. Soc. A 361 (2003) 1655–1674.
  2. L. Jaeger, The Second Quantum Revolution. Copernicus Cham, 2018.
  3. M. Nielsen and I. Chuang, Quantum Computation and Quantum Information. Cambridge University Press, 2000.
  4. A. F., A. K., and R. e. a. Babbush, “Quantum supremacy using a programmable superconducting processor,” Nature 574 (2019) 505–510.
  5. Y. W. et al., “Strong quantum computational advantage using a superconducting quantum processor,” Phys. Rev. Lett. 127 (2021) 180501.
  6. A. Kitaev, “Fault-tolerant quantum computation by anyons,” Ann. of Phys. (N.Y.) 303 (2003) 2.
  7. J. Preskill, “Topological quantum computation,” Lecture Notes for Physics 219 (2004) http://www.theory.caltech.edu/∼similar-to\sim∼preskill/ph219.
  8. S. D. Sarma, M. Freedman, C. Nayak, S. H. Simon, and A. Stern, “Non-Abelian Anyons and Topological Quantum Computation,” Rev. Mod. Phys. 80 (2008) 1083, arXiv:0707.1889.
  9. S. H. Simon, Topological Quantum. Oxford University Press, Oxford UK, 2023.
  10. F. Wilczek, Fractional statisitcs and anyon superconductivity. World Scientific, Singapore, 1990.
  11. G. Moore and N. Read, “Nonabelions in the fractional quantum Hall effect,” Nucl. Phys. B360 (1991) 362.
  12. Elsevier, 2nd ed., 2024.
  13. A. Ahlbrecht, L. S. Georgiev, and R. F. Werner, “Implementation of Clifford gates in the Ising-anyon topological quantum computer,” Phys. Rev. A 79 (2009) 032311, arXiv:0812.2338.
  14. H. C. Choi, W. Kang, S. D. Sarma, L. N. Pfeiffer, and K. W. West, “Activation gaps of fractional quantum Hall effect in the second Landau level,” Phys. Rev. B 77 (2008) 081301.
  15. N. Read and E. Rezayi, “Beyond paired quantum Hall states: parafermions and incompressible states in the first excited Landau level,” Phys. Rev. B59 (1998) 8084.
  16. A. Cappelli, L. S. Georgiev, and I. T. Todorov, “Parafermion Hall states from coset projections of Abelian conformal theories,” Nucl. Phys. B 599 [FS] (2001) 499–530, hep-th/0009229.
  17. M. Freedman, M. Larsen, and Z. Wang, “A modular functor which is universal for quantum computation,” Commun. Math. Phys. 227 (2002) 605, quant-ph/0001108.
  18. N. Bonesteel, L. Hormozi, G. Zikos, and S. Simon, “Braid topologies for quantum computation,” Phys. Rev. Lett. 95 (2005) 140503.
  19. L. Hormozi, G. Zikos, N. E. Bonesteel, and S. H. Simon, “Topological quantum compiling,” Phys. Rev. B 75 (2007) 165310.
  20. L. Hormozi, N. Bonesteel, and S. Simon, “Topological quantum computing with Read–Rezayi states,” Phys. Rev. Lett. 103 (2009) 160501.
  21. S. D. Sarma, M. Freedman, and C. Nayak, “Topologically-protected qubits from a possible non-Abelian fractional quantum Hall state,” Phys. Rev. Lett. 94 (2005) 166802, cond-mat/0412343.
  22. L. S. Georgiev, “Topologically protected gates for quantum computation with non-Abelian anyons in the Pfaffian quantum Hall state,” Phys. Rev. B 74 (2006) 235112, cond-mat/0607125.
  23. L. S. Georgiev, “Towards a universal set of topologically protected gates for quantum computation with Pfaffian qubits,” Nucl. Phys. B 789 (2008) 552–590, hep-th/0611340.
  24. L. Hadjiivanov and L. S. Georgiev, “Braiding Fibonacci anyons,” arXiv:2403*****.
  25. E. Ardonne and K. Schoutens, “Wavefunctions for topological quantum registers,” Ann. Phys. 322 (2007) 201–235.
  26. Springer–Verlag, New York, 1997.
  27. L. S. Georgiev, “Exact Modular S𝑆Sitalic_S Matrix for ℤksubscriptℤ𝑘{\mathbb{Z}}_{k}blackboard_Z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT Parafermion Quantum Hall Islands and Measurement of Non-Abelian Anyons,” J. of Geom. and Symm. in Phys. 62 (2021) 1–28.
  28. W. Pan, J.-S. Xia, V. Shvarts, D. E. Adams, H. L. Störmer, D. C. Tsui, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, “Exact quantization of the even-denominator fractional quantum Hall state at ν=5/2𝜈52\nu=5/2italic_ν = 5 / 2 Landau level filling factor,” Phys. Rev. Lett. 83 (1999) 3530, cond-mat/9907356.
  29. J. Xia, W. Pan, C. Vicente, E. Adams, N. Sullivan, H. Stormer, D. Tsui, L. Pfeiffer, K. Baldwin, and K. West, “Electron correlation in the second Landau level: a competition between many nearly degenerated quantum phases,” Phys. Rev. Lett. 93 (2004) 176809.
  30. W. Pan, J. S. Xia, H. L. Stormer, D. C. Tsui, C. Vicente, E. D. Adams, N. S. Sullivan, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, “Experimental studies of the fractional quantum Hall effect in the first excited Landau level,” Phys. Rev. B 77 (Feb, 2008) 075307.
  31. J. Fröhlich, U. M. Studer, and E. Thiran, “A classification of quantum Hall fluids,” J. Stat. Phys. 86 (1997) 821, cond-mat/9503113.
  32. L. S. Georgiev, “Hilbert space decomposition for Coulomb blockade in Fabry–Pérot interferometers,” in Lie Theory and Its Applications in Physics: IX International Workshop, V. Dobrev, ed., Springer Proceedings in Mathematics & Statistics 36, pp. 439–450. 2011. arXiv:1112.5946. Proceedings of the 9-th International Workshop ”Lie Theory and Its Applications in Physics”, 20-26 June 2011, Varna, Bulgaria.
  33. L. S. Georgiev, “Thermopower and thermoelectric power factor of ℤksubscriptℤ𝑘{{\mathbb{Z}}}_{k}blackboard_Z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT parafermion quantum dots,” Nucl. Phys. B 899 (2015) 289–311, arXiv:1505.02538.
  34. A. Cappelli and G. R. Zemba, “Modular invariant partition functions in the quantum Hall effect,” Nucl. Phys. B490 (1997) 595, hep-th/9605127.
  35. L. S. Georgiev, “A universal conformal field theory approach to the chiral persistent currents in the mesoscopic fractional quantum Hall states,” Nucl. Phys. B 707 (2005) 347–380, hep-th/0408052.
  36. L. S. Georgiev, “Thermoelectric properties of Coulomb-blockaded fractional quantum Hall islands,” Nucl. Phys. B 894 (2015) 284–306, arXiv:1406.6177.
  37. S. Schweber, An Introduction to Relativistic Quantum Field Theory. Row, Peterson and Company, 1961.
  38. McGraw-Hill, New York, 1953.
  39. R. B. Laughlin, “Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations,” Phys. Rev. Lett. 50 (May, 1983) 1395–1398.
  40. P. Bonderson, A. Kitaev, and K. Shtengel, “Detecting non-abelian statistics in the ν=5/2𝜈52\nu=5/2italic_ν = 5 / 2 fractional quantum Hall state,” Phys. Rev. Lett. 96 (2006) 016803, cond-mat/0508616.
  41. A. Stern and B. I. Halperin, “Proposed experiments to probe the non-Abelian ν=5/2𝜈52\nu=5/2italic_ν = 5 / 2 quantum Hall state,” Phys. Rev. Lett. 96 (2006) 016802.
  42. P. Bonderson, K. Shtengel, and J. K. Slingerland, “Probing non-abelian statistics with two-particle interferometry,” Phys. Rev. Lett. 97 (2006) 016401, cond-mat/0601242.
  43. V. G. Knizhnik and A. B. Zamolodchikov, “Current Algebra and Wess–Zumino Model in Two Dimensions,” Nucl. Phys. B 247 (1984) 83–103.
  44. M. T. Rouabah, N. E. Belaloui, and A. Tounsi, “Compiling single-qubit braiding gate for Fibonacci anyons topological quantum computation,” J. of Phys: Conference Series 1766 (2021) 012029.
  45. Y.-H. Zhang, P.-L. Zheng, Y. Zhang, and D.-L. Deng, “Topological quantum compiling with reinforcement learning,” Phys. Rev. Lett. 125 (2020) 170501.
Citations (1)

Summary

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

X Twitter Logo Streamline Icon: https://streamlinehq.com