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Non-Abelian statistics and topological quantum information processing in 1D wire networks


The synthesis of a quantum computer remains an ongoing challenge in modern physics. Whereas decoherence stymies most approaches, topological quantum computation schemes evade decoherence at the hardware level by storing quantum information non-locally. Here we establish that a key operation—braiding of non-Abelian anyons—can be implemented using one-dimensional semiconducting wires. Such wires can be driven into a topological phase supporting long-sought particles known as Majorana fermions that can encode topological qubits. We show that in wire networks, Majorana fermions can be meaningfully braided by simply adjusting gate voltages, and that they exhibit non-Abelian statistics like vortices in a p+i p superconductor. We propose experimental set-ups that enable probing of the Majorana fusion rules and the efficient exchange of arbitrary numbers of Majorana fermions. This work should open a new direction in topological quantum computation that benefits from physical transparency and experimental feasibility.

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Figure 1: Majorana fermions appear at the ends of a 1D ‘spinless’ p-wave superconductor, which can be experimentally realized in semiconducting wires21,22.
Figure 2: Applying a ‘keyboard’ of individually tunable gates to the wire allows local control of which regions are topological (dark blue) and non-topological (light blue), and hence manipulate Majorana fermions while maintaining the bulk gap.
Figure 3: A T-junction provides the simplest wire network that enables meaningful adiabatic exchange of Majorana fermions.
Figure 4: Experimental set-ups that allow the probing of non-Abelian statistics and Majorana-fermion fusion rules.

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  1. Kitaev, A. Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003).

    Article  ADS  MathSciNet  Google Scholar 

  2. Freedman, M. H. P/NP, and the quantum field computer. Proc. Natl Acad. Sci. USA 95, 98–101 (1998).

    Article  ADS  MathSciNet  Google Scholar 

  3. Freedman, M. H., Kitaev, A., Larsen, M. J. & Wang, Z. Topological quantum computation. Bull. Am. Math. Soc. 40, 31–38 (2003).

    Article  MathSciNet  Google Scholar 

  4. Das Sarma, S., Freedman, M. & Nayak, C. Topologically protected qubits from a possible non-Abelian fractional quantum Hall state. Phys. Rev. Lett. 94, 166802 (2005).

    Article  ADS  Google Scholar 

  5. Bonderson, P., Freedman, M. & Nayak, C. Measurement-only topological quantum computation. Phys. Rev. Lett. 101, 010501 (2008).

    Article  ADS  MathSciNet  Google Scholar 

  6. Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008).

    Article  ADS  MathSciNet  Google Scholar 

  7. Leinaas, J. M. & Myrheim, J. On the theory of identical particles. Nuovo Cimento Soc. Ital. Fis. B 37B, 1–23 (1977).

    Article  ADS  Google Scholar 

  8. Fredenhagen, K., Rehren, K. H. & Schroer, B. Superselection sectors with braid group statistics and exchange algebras. Commun. Math. Phys. 125, 201–226 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  9. Fröhlich, J. & Gabbiani, F. Braid statistics in local quantum theory. Rev. Math. Phys. 2, 251–353 (1990).

    MathSciNet  MATH  Google Scholar 

  10. Read, N. & Green, D. Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect. Phys. Rev. B 61, 10267–10297 (2000).

    Article  ADS  Google Scholar 

  11. Ivanov, D. A. Non-Abelian statistics of half-quantum vortices in p-wave superconductors. Phys. Rev. Lett. 86, 268–271 (2001).

    Article  ADS  Google Scholar 

  12. Moore, G. & Read, N. Nonabelions in the fractional quantum Hall effect. Nucl. Phys. B 360, 362–396 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  13. Fu, L. & Kane, C. L. Superconducting proximity effect and Majorana fermions at the surface of a topological insulator. Phys. Rev. Lett. 100, 096407 (2008).

    Article  ADS  Google Scholar 

  14. Linder, J., Tanaka, Y., Yokoyama, T., Sudbø, A. & Nagaosa, N. Unconventional superconductivity on a topological insulator. Phys. Rev. Lett. 104, 067001 (2010).

    Article  ADS  Google Scholar 

  15. Sau, J. D., Lutchyn, R. M., Tewari, S. & Das Sarma, S. Generic new platform for topological quantum computation using semiconductor heterostructures. Phys. Rev. Lett. 104, 040502 (2010).

    Article  ADS  Google Scholar 

  16. Alicea, J. Majorana fermions in a tunable semiconductor device. Phys. Rev. B 81, 125318 (2010).

    Article  ADS  Google Scholar 

  17. Sato, M. & Fujimoto, S. Topological phases of noncentrosymmetric superconductors: Edge states, Majorana fermions, and non-Abelian statistics. Phys. Rev. B 79, 094504 (2009).

    Article  ADS  Google Scholar 

  18. Lee, P. A. Proposal for creating a spin-polarized p x+ip y state and Majorana fermions. Preprint at (2009).

  19. Ghosh, P., Sau, J. D., Tewari, S. & Das Sarma, S. Non-Abelian topological order in noncentrosymmetric superconductors with broken time-reversal symmetry. Phys. Rev. B 82, 184525 (2010).

    Article  ADS  Google Scholar 

  20. Qi, X-L., Hughes, T. L. & Zhang, S-C. Chiral topological superconductor from the quantum Hall state. Phys. Rev. B 82, 184516 (2010).

    Article  ADS  Google Scholar 

  21. Lutchyn, R. M., Sau, J. D. & Das Sarma, S. Majorana fermions and a topological phase transition in semiconductor–superconductor heterostructures. Phys. Rev. Lett. 105, 077001 (2010).

    Article  ADS  Google Scholar 

  22. Oreg, Y., Refael, G. & von Oppen, F. Helical liquids and Majorana bound states in quantum wires. Phys. Rev. Lett. 105, 177002 (2010).

    Article  ADS  Google Scholar 

  23. Kitaev, A. Y. Unpaired Majorana fermions in quantum wires. Phys.-Usp. 44, 131–136 (2001).

    Article  ADS  Google Scholar 

  24. Wimmer, M., Akhmerov, A. R., Medvedyeva, M. V., Tworzydło, J. & Beenakker, C. W. J. Majorana bound states without vortices in topological superconductors with electrostatic defects. Phys. Rev. Lett. 105, 046803 (2010).

    Article  ADS  Google Scholar 

  25. Fu, L. & Kane, C. L. Josephson current and noise at a superconductor/quantum-spin-Hall-insulator/superconductor junction. Phys. Rev. B 79, 161408(R) (2009).

    Article  ADS  Google Scholar 

  26. Hassler, F., Akhmerov, A. R., Hou, C-Y. & Beenakker, C. W. J. Anyonic interferometry without anyons: How a flux qubit can read out a topological qubit. New J. Phys. 12, 125002 (2010).

    Article  ADS  Google Scholar 

  27. Bravyi, S. & Kitaev, A. Universal quantum computation with ideal Clifford gates and noisy ancillas. Phys. Rev. A 71, 022316 (2005).

    Article  ADS  MathSciNet  Google Scholar 

  28. Freedman, M., Nayak, C. & Walker, K. Towards universal topological quantum computation in the ν=5/2 fractional quantum Hall state. Phys. Rev. B 73, 245307 (2006).

    Article  ADS  Google Scholar 

  29. Bonderson, P., Das Sarma, S., Freedman, M. & Nayak, C. A blueprint for a topologically fault-tolerant quantum computer. Preprint at (2010).

  30. Bonderson, P., Clarke, D. J., Nayak, C. & Shtengel, K. Implementing arbitrary phase gates with Ising anyons. Phys. Rev. Lett. 104, 180505 (2010).

    Article  ADS  Google Scholar 

  31. Dresselhaus, G. Spin–orbit coupling effects in zinc blende structures. Phys. Rev. 100, 580–586 (1955).

    Article  ADS  Google Scholar 

  32. Bychkov, Y. A. & Rashba, E. I. Oscillatory effects and the magnetic susceptibility of carriers in inversion layers. J. Phys. C 17, 6039–6045 (1984).

    Article  ADS  Google Scholar 

  33. Winkler, R. Spin–Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems (Springer, 2003).

    Book  Google Scholar 

  34. Stern, A., von Oppen, F. & Mariani, E. Geometric phases and quantum entanglement as building blocks for non-Abelian quasiparticle statistics. Phys. Rev. B 70, 205338 (2004).

    Article  ADS  Google Scholar 

  35. Doh, Y-J., van Dam, J. A., Roest, A. L., Bakkers, E. P. A. M., Kouwenhoven, L. P. & De Franceschi, S. Tunable supercurrent through semiconductor nanowires. Science 309, 272–275 (2005).

    Article  ADS  Google Scholar 

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We have benefited greatly from stimulating conversations with P. Bonderson, S. Das Sarma, L. Fidkowski, E. Henriksen, A. Kitaev, P. Lee, X. Qi and A. Stern. We also gratefully acknowledge support from the Lee A. DuBridge Foundation, ISF, BSF, DIP and SPP 1285 grants, Packard and Sloan fellowships, the Institute for Quantum Information under NSF grants PHY-0456720 and PHY-0803371, and the National Science Foundation through grant DMR-0529399.

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All authors contributed to the inception of the ideas in the manuscript, design of networks and proposed experimental setups, and proof of non-Abelian statistics.

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Correspondence to Jason Alicea.

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Alicea, J., Oreg, Y., Refael, G. et al. Non-Abelian statistics and topological quantum information processing in 1D wire networks. Nature Phys 7, 412–417 (2011).

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