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The germanium quantum information route

Abstract

In the effort to develop disruptive quantum technologies, germanium is emerging as a versatile material to realize devices capable of encoding, processing and transmitting quantum information. These devices leverage the special properties of holes in germanium, such as their inherently strong spin–orbit coupling and their ability to host superconducting pairing correlations. In this Review, we start by introducing the physics of holes in low-dimensional germanium structures, providing key insights from a theoretical perspective. We then examine the materials-science progress underpinning germanium-based planar heterostructures and nanowires. We go on to review the most significant experimental results demonstrating key building blocks for quantum technology, such as an electrically driven universal quantum gate set with spin qubits in quantum dots and superconductor–semiconductor devices for hybrid quantum systems. We conclude by identifying the most promising avenues towards scalable quantum information processing in germanium-based systems.

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Fig. 1: Quantum confined holes in germanium.
Fig. 2: Planar Ge/SiGe heterostructures.
Fig. 3: Ge-based nanowires.
Fig. 4: Quantum dots and qubits in core–shell nanowires, hut wires and planar systems.
Fig. 5: State-of-the-art coherence times and single-qubit gate fidelities for core–shell nanowires, hut wires and planar heterostructures.
Fig. 6: Superconductor–semiconductor hybrids in Ge nanowires and planar systems.
Fig. 7: Ge-based quantum technology.

References

  1. Pillarisetty, R. Academic and industry research progress in germanium nanodevices. Nature 479, 324–328 (2011).

    CAS  Article  Google Scholar 

  2. Kamata, Y. High-k/Ge MOSFETs for future nanoelectronics. Mater. Today 11, 30–38 (2008).

    Article  Google Scholar 

  3. Loss, D. & DiVincenzo, D. P. Quantum computation with quantum dots. Phys. Rev. A 57, 120–126 (1998).

    CAS  Article  Google Scholar 

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

    Article  Google Scholar 

  5. Larsen, T. et al. Semiconductor-nanowire-based superconducting qubit. Phys. Rev. Lett. 115, 127001 (2015).

    CAS  Article  Google Scholar 

  6. Dimoulas, A., Tsipas, P., Sotiropoulos, A. & Evangelou, E. K. Fermi-level pinning and charge neutrality level in germanium. Appl. Phys. Lett. 89, 252110 (2006).

    Article  CAS  Google Scholar 

  7. Pillarisetty, R. et al. in Proceedings of the 2018 IEEE International Electron Devices Meeting (IEDM) 6.3.1–6.3.4 (IEEE, 2018).

  8. Dobbie, A. et al. Ultra-high hole mobility exceeding one million in a strained germanium quantum well. Appl. Phys. Lett. 101, 172108 (2012).

    Article  CAS  Google Scholar 

  9. Sammak, A. et al. Shallow and undoped germanium quantum wells: a playground for spin and hybrid quantum technology. Adv. Funct. Mater. 29, 1807613 (2019).

    Article  CAS  Google Scholar 

  10. Gao, F. et al. Site-controlled uniform Ge/Si hut wires with electrically tunable spin–orbit coupling. Adv. Mater. 32, 1906523 (2020).

    CAS  Article  Google Scholar 

  11. Hu, Y. et al. A Ge/Si heterostructure nanowire-based double quantum dot with integrated charge sensor. Nat. Nanotechnol. 2, 622–625 (2007).

    CAS  Article  Google Scholar 

  12. Hu, Y., Kuemmeth, F., Lieber, C. M. & Marcus, C. M. Hole spin relaxation in Ge–Si core–shell nanowire qubits. Nat. Nanotechnol. 7, 47–50 (2012).

    CAS  Article  Google Scholar 

  13. Ares, N. et al. Nature of tunable hole g factors in quantum dots. Phys. Rev. Lett. 110, 046602 (2013).

    CAS  Article  Google Scholar 

  14. Watzinger, H. et al. Heavy-hole states in germanium hut wires. Nano Lett. 16, 6879–6885 (2016).

    CAS  Article  Google Scholar 

  15. Watzinger, H. et al. A germanium hole spin qubit. Nat. Commun. 9, 3902 (2018).

    Article  CAS  Google Scholar 

  16. Xu, G. et al. Dipole coupling of a hole double quantum dot in germanium hut wire to a microwave resonator. New J. Phys. 22, 083068 (2020).

    CAS  Article  Google Scholar 

  17. Li, Y. et al. Coupling a germanium hut wire hole quantum dot to a superconducting microwave resonator. Nano Lett. 18, 2091–2097 (2018).

    CAS  Article  Google Scholar 

  18. Hendrickx, N. W. et al. Gate-controlled quantum dots and superconductivity in planar germanium. Nat. Commun. 9, 2835 (2018).

    CAS  Article  Google Scholar 

  19. Hendrickx, N., Franke, D., Sammak, A., Scappucci, G. & Veldhorst, M. Fast two-qubit logic with holes in germanium. Nature 577, 487–491 (2020).

    CAS  Article  Google Scholar 

  20. Hendrickx, N. W. et al. A four-qubit germanium quantum processor. Preprint at arXiv http://arxiv.org/abs/2009.04268 (2020).

  21. Hendrickx, N. W. et al. A single-hole spin qubit. Nat. Commun. 11, 3478 (2020).

    CAS  Article  Google Scholar 

  22. Xiang, J., Vidan, A., Tinkham, M., Westervelt, R. M. & Lieber, C. M. Ge/Si nanowire mesoscopic Josephson junctions. Nat. Nanotechnol. 1, 208–213 (2006).

    CAS  Article  Google Scholar 

  23. Ridderbos, J. et al. Josephson effect in a few-hole quantum dot. Adv. Mater. 30, 1802257 (2018).

    Article  CAS  Google Scholar 

  24. Vigneau, F. et al. Germanium quantum-well Josephson field-effect transistors and interferometers. Nano Lett. 19, 1023–1027 (2019).

    CAS  Article  Google Scholar 

  25. Petta, J. R. et al. Coherent manipulation of coupled electron spins in semiconductor quantum dots. Science 309, 2180–2184 (2005).

    CAS  Article  Google Scholar 

  26. Koppens, F. H. L. et al. Driven coherent oscillations of a single electron spin in a quantum dot. Nature 442, 766–771 (2006).

    CAS  Article  Google Scholar 

  27. Nowack, K. C., Koppens, F. H. L., Nazarov, Y. V. & Vandersypen, L. M. K. Coherent control of a single electron spin with electric fields. Science 318, 1430–1433 (2007).

    CAS  Article  Google Scholar 

  28. Kloeffel, C. & Loss, D. Prospects for spin-based quantum computing in quantum dots. Annu. Rev. Condens. Matter Phys. 4, 51–81 (2013).

    CAS  Article  Google Scholar 

  29. Bulaev, D. V. & Loss, D. Spin relaxation and decoherence of holes in quantum dots. Phys. Rev. Lett. 95, 076805 (2005).

    Article  CAS  Google Scholar 

  30. Bulaev, D. V. & Loss, D. Electric dipole spin resonance for heavy holes in quantum dots. Phys. Rev. Lett. 98, 097202 (2007).

    Article  CAS  Google Scholar 

  31. Fischer, J., Coish, W. A., Bulaev, D. V. & Loss, D. Spin decoherence of a heavy hole coupled to nuclear spins in a quantum dot. Phys. Rev. B 78, 155329 (2008).

    Article  CAS  Google Scholar 

  32. Trif, M., Simon, P. & Loss, D. Relaxation of hole spins in quantum dots via two-phonon processes. Phys. Rev. Lett. 103, 106601 (2009).

    Article  CAS  Google Scholar 

  33. Heiss, D. et al. Observation of extremely slow hole spin relaxation in self-assembled quantum dots. Phys. Rev. B 76, 241306 (2007).

    Article  CAS  Google Scholar 

  34. Gerardot, B. D. et al. Optical pumping of a single hole spin in a quantum dot. Nature 451, 441–444 (2008).

    CAS  Article  Google Scholar 

  35. Brunner, D. et al. A coherent single-hole spin in a semiconductor. Science 325, 70–72 (2009).

    CAS  Article  Google Scholar 

  36. Warburton, R. J. Single spins in self-assembled quantum dots. Nat. Mater. 12, 483–493 (2013).

    CAS  Article  Google Scholar 

  37. Nolting, W. & Ramakanth, A. Quantum Theory of Magnetism (Springer, 2009).

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

  39. Luttinger, J. M. & Kohn, W. Motion of electrons and holes in perturbed periodic fields. Phys. Rev. 97, 869–883 (1955).

    CAS  Article  Google Scholar 

  40. Luttinger, J. M. Quantum theory of cyclotron resonance in semiconductors: general theory. Phys. Rev. 102, 1030–1041 (1956).

    CAS  Article  Google Scholar 

  41. Winkler, R., Culcer, D., Papadakis, S. J., Habib, B. & Shayegan, M. Spin orientation of holes in quantum wells. Semicond. Sci. Technol. 23, 114017 (2008).

    Article  CAS  Google Scholar 

  42. Lawaetz, P. Valence-band parameters in cubic semiconductors. Phys. Rev. B 4, 3460–3467 (1971).

    Article  Google Scholar 

  43. Terrazos, L. A. et al. Theory of hole-spin qubits in strained germanium quantum dots. Preprint at arXiv http://arxiv.org/abs/1803.10320 (2020).

  44. Bir, G. L. & Pikus, G. E. Symmetry and Strain-Induced Effects in Semiconductors (Wiley, 1974).

  45. van Kesteren, H. W., Cosman, E. C., van der Poel, W. A. J. A. & Foxon, C. T. Fine structure of excitons in type-II GaAs/AlAs quantum wells. Phys. Rev. B 41, 5283–5292 (1990).

    Article  Google Scholar 

  46. Lodari, M. et al. Light effective hole mass in undoped Ge/SiGe quantum wells. Phys. Rev. B 100, 041304 (2019).

    CAS  Article  Google Scholar 

  47. Sercel, P. C. & Vahala, K. J. Analytical formalism for determining quantum-wire and quantum-dot band structure in the multiband envelope-function approximation. Phys. Rev. B 42, 3690–3710 (1990).

    CAS  Article  Google Scholar 

  48. Harada, Y., Kita, T., Wada, O. & Ando, H. Anisotropic magneto-optical effects in one-dimensional diluted magnetic semiconductors. Phys. Rev. B 74, 245323 (2006).

    Article  CAS  Google Scholar 

  49. Csontos, D., Brusheim, P., Zülicke, U. & Xu, H. Q. Spin-\(\tfrac{3}{2}\) physics of semiconductor hole nanowires: Valence-band mixing and tunable interplay between bulk-material and orbital bound-state spin splittings. Phys. Rev. B 79, 155323 (2009).

    Article  CAS  Google Scholar 

  50. Kloeffel, C., Trif, M. & Loss, D. Strong spin-orbit interaction and helical hole states in Ge/Si nanowires. Phys. Rev. B 84, 195314 (2011).

    Article  CAS  Google Scholar 

  51. Kloeffel, C., Rančić, M. J. & Loss, D. Direct Rashba spin-orbit interaction in Si and Ge nanowires with different growth directions. Phys. Rev. B 97, 235422 (2018).

    CAS  Article  Google Scholar 

  52. Menéndez, J., Singh, R. & Drucker, J. Theory of strain effects on the Raman spectrum of Si-Ge core-shell nanowires. Ann. Phys. 523, 145–156 (2011).

    Article  CAS  Google Scholar 

  53. Kloeffel, C., Trif, M. & Loss, D. Acoustic phonons and strain in core/shell nanowires. Phys. Rev. B 90, 115419 (2014).

    Article  CAS  Google Scholar 

  54. Kloeffel, C., Trif, M., Stano, P. & Loss, D. Circuit QED with hole-spin qubits in Ge/Si nanowire quantum dots. Phys. Rev. B 88, 241405 (2013).

    Article  CAS  Google Scholar 

  55. Nigg, S. E., Fuhrer, A. & Loss, D. Superconducting grid-bus surface code architecture for hole-spin qubits. Phys. Rev. Lett. 118, 147701 (2017).

    Article  Google Scholar 

  56. Maier, F., Meng, T. & Loss, D. Strongly interacting holes in Ge/Si nanowires. Phys. Rev. B 90, 155437 (2014).

    Article  CAS  Google Scholar 

  57. Mao, L., Gong, M., Dumitrescu, E., Tewari, S. & Zhang, C. Hole-doped semiconductor nanowire on top of an s-wave superconductor: a new and experimentally accessible system for Majorana fermions. Phys. Rev. Lett. 108, 177001 (2012).

    Article  CAS  Google Scholar 

  58. Maier, F., Klinovaja, J. & Loss, D. Majorana fermions in Ge/Si hole nanowires. Phys. Rev. B 90, 195421 (2014).

    Article  CAS  Google Scholar 

  59. Ivchenko, E. L., Kaminski, A. Y. & Rössler, U. Heavy-light hole mixing at zinc-blende (001) interfaces under normal incidence. Phys. Rev. B 54, 5852–5859 (1996).

    CAS  Article  Google Scholar 

  60. Vervoort, L., Ferreira, R. & Voisin, P. Effects of interface asymmetry on hole subband degeneracies and spin-relaxation rates in quantum wells. Phys. Rev. B 56, R12744–R12747 (1997).

    CAS  Article  Google Scholar 

  61. Guettler, T. et al. Optical polarization relaxation in InxGa1−xAs-based quantum wells: Evidence of the interface symmetry-reduction effect. Phys. Rev. B 58, R10179–R10182 (1998).

    CAS  Article  Google Scholar 

  62. Vervoort, L., Ferreira, R. & Voisin, P. Spin-splitting of the subbands of InGaAs-InP and other ‘no common atom’ quantum wells. Semicond. Sci. Technol. 14, 227–230 (1999).

    CAS  Article  Google Scholar 

  63. Olesberg, J. T. et al. Interface contributions to spin relaxation in a short-period InAs/GaSb superlattice. Phys. Rev. B 64, 201301 (2001).

    Article  CAS  Google Scholar 

  64. Hall, K. C. et al. Spin relaxation in (110) and (001) InAs/GaSb superlattices. Phys. Rev. B 68, 115311 (2003).

    Article  CAS  Google Scholar 

  65. Golub, L. E. & Ivchenko, E. L. Spin splitting in symmetrical SiGe quantum wells. Phys. Rev. B 69, 115333 (2004).

    Article  CAS  Google Scholar 

  66. Nestoklon, M. O., Ivchenko, E. L., Jancu, J.-M. & Voisin, P. Electric field effect on electron spin splitting in SiGe/Si quantum wells. Phys. Rev. B 77, 155328 (2008).

    Article  CAS  Google Scholar 

  67. Prada, M., Klimeck, G. & Joynt, R. Spin–orbit splittings in Si/SiGe quantum wells: from ideal Si membranes to realistic heterostructures. New J. Phys. 13, 013009 (2011).

    Article  CAS  Google Scholar 

  68. Furthmeier, S. et al. Enhanced spin–orbit coupling in core/shell nanowires. Nat. Commun. 7, 12413 (2016).

    CAS  Article  Google Scholar 

  69. Wojcik, P., Bertoni, A. & Goldoni, G. Enhanced Rashba spin-orbit coupling in core-shell nanowires by the interfacial effect. Appl. Phys. Lett. 114, 073102 (2019).

    Article  CAS  Google Scholar 

  70. Hao, X.-J. et al. Strong and tunable spin-orbit coupling of one-dimensional holes in Ge/Si core/shell nanowires. Nano Lett. 10, 2956–2960 (2010).

    CAS  Article  Google Scholar 

  71. Higginbotham, A. P. et al. Antilocalization of coulomb blockade in a Ge/Si nanowire. Phys. Rev. Lett. 112, 216806 (2014).

    Article  CAS  Google Scholar 

  72. Brauns, M., Ridderbos, J., Li, A., Bakkers, E. P. A. M. & Zwanenburg, F. A. Electric-field dependent g-factor anisotropy in Ge-Si core-shell nanowire quantum dots. Phys. Rev. B 93, 121408 (2016).

    Article  CAS  Google Scholar 

  73. Wang, R., Deacon, R. S., Yao, J., Lieber, C. M. & Ishibashi, K. Electrical modulation of weak-antilocalization and spin–orbit interaction in dual gated Ge/Si core/shell nanowires. Semicond. Sci. Technol. 32, 094002 (2017).

    Article  CAS  Google Scholar 

  74. Sun, J. et al. Helical hole state in multiple conduction modes in Ge/Si core/shell nanowire. Nano Lett. 18, 6144–6149 (2018).

    CAS  Article  Google Scholar 

  75. de Vries, F. K. et al. Spin–orbit interaction and induced superconductivity in a one-dimensional hole gas. Nano Lett. 18, 6483–6488 (2018).

    Article  CAS  Google Scholar 

  76. Golovach, V. N., Borhani, M. & Loss, D. Electric-dipole-induced spin resonance in quantum dots. Phys. Rev. B 74, 165319 (2006).

    Article  CAS  Google Scholar 

  77. Stano, P. et al. g-factor of electrons in gate-defined quantum dots in a strong in-plane magnetic field. Phys. Rev. B 98, 195314 (2018).

    CAS  Article  Google Scholar 

  78. Camenzind, L. C. et al. Hyperfine-phonon spin relaxation in a single-electron GaAs quantum dot. Nat. Commun. 9, 3454 (2018).

    Article  CAS  Google Scholar 

  79. Stano, P. et al. Orbital effects of a strong in-plane magnetic field on a gate-defined quantum dot. Phys. Rev. B 99, 085308 (2019).

    CAS  Article  Google Scholar 

  80. Winkler, R. Rashba spin splitting in two-dimensional electron and hole systems. Phys. Rev. B 62, 4245–4248 (2000).

    CAS  Article  Google Scholar 

  81. Chesi, S., Giuliani, G. F., Rokhinson, L. P., Pfeiffer, L. N. & West, K. W. Anomalous spin-resolved point-contact transmission of holes due to cubic Rashba spin-orbit coupling. Phys. Rev. Lett. 106, 236601 (2011).

    Article  CAS  Google Scholar 

  82. Nichele, F. et al. Characterization of spin-orbit interactions of GaAs heavy holes using a quantum point contact. Phys. Rev. Lett. 113, 046801 (2014).

    Article  CAS  Google Scholar 

  83. Nichele, F. et al. Spin-orbit splitting and effective masses in p-type GaAs two-dimensional hole gases. Phys. Rev. B 89, 081306 (2014).

    Article  CAS  Google Scholar 

  84. Miserev, D. S. & Sushkov, O. P. Dimensional reduction of the Luttinger Hamiltonian and g-factors of holes in symmetric two-dimensional semiconductor heterostructures. Phys. Rev. B 95, 085431 (2017).

    Article  Google Scholar 

  85. Srinivasan, A. et al. Detection and control of spin-orbit interactions in a GaAs hole quantum point contact. Phys. Rev. Lett. 118, 146801 (2017).

    CAS  Article  Google Scholar 

  86. Hung, J.-T., Marcellina, E., Wang, B., Hamilton, A. R. & Culcer, D. Spin blockade in hole quantum dots: Tuning exchange electrically and probing Zeeman interactions. Phys. Rev. B 95, 195316 (2017).

    Article  Google Scholar 

  87. Marcellina, E., Hamilton, A. R., Winkler, R. & Culcer, D. Spin-orbit interactions in inversion-asymmetric two-dimensional hole systems: a variational analysis. Phys. Rev. B 95, 075305 (2017).

    Article  Google Scholar 

  88. Liu, H., Marcellina, E., Hamilton, A. R. & Culcer, D. Strong spin-orbit contribution to the hall coefficient of two-dimensional hole systems. Phys. Rev. Lett. 121, 087701 (2018).

    Article  Google Scholar 

  89. Mizokuchi, R. et al. Hole weak anti-localization in a strained-Ge surface quantum well. Appl. Phys. Lett. 111, 063102 (2017).

    Article  CAS  Google Scholar 

  90. Moriya, R. et al. Cubic Rashba spin-orbit interaction of a two-dimensional hole gas in a strained-Ge/SiGe quantum well. Phys. Rev. Lett. 113, 086601 (2014).

    CAS  Article  Google Scholar 

  91. Chou, C.-T. et al. Weak anti-localization of two-dimensional holes in germanium beyond the diffusive regime. Nanoscale 10, 20559–20564 (2018).

    CAS  Article  Google Scholar 

  92. Nenashev, A. V., Dvurechenskii, A. V. & Zinovieva, A. F. Wave functions and g factor of holes in Ge/Si quantum dots. Phys. Rev. B 67, 205301 (2003).

    Article  CAS  Google Scholar 

  93. Pryor, C. E. & Flatté, M. E. Landé g factors and orbital momentum quenching in semiconductor quantum dots. Phys. Rev. Lett. 96, 026804 (2006).

    Article  CAS  Google Scholar 

  94. van Bree, J. et al. Anisotropy of electron and hole g tensors of quantum dots: an intuitive picture based on spin-correlated orbital currents. Phys. Rev. B 93, 035311 (2016).

    Article  CAS  Google Scholar 

  95. Katsaros, G. et al. Hybrid superconductor–semiconductor devices made from self-assembled SiGe nanocrystals on silicon. Nat. Nanotechnol. 5, 458–464 (2010).

    CAS  Article  Google Scholar 

  96. Wimbauer, T., Oettinger, K., Efros, A. L., Meyer, B. K. & Brugger, H. Zeeman splitting of the excitonic recombination in InxGa1−xAs/GaAs single quantum wells. Phys. Rev. B 50, 8889–8892 (1994).

    CAS  Article  Google Scholar 

  97. Durnev, M. V. et al. Magnetic field induced valence band mixing in [111] grown semiconductor quantum dots. Phys. Rev. B 87, 085315 (2013).

    Article  CAS  Google Scholar 

  98. Drichko, I. L. et al. In-plane magnetic field effect on hole cyclotron mass and gz factor in high-mobility SiGe/Ge/SiGe structures. Phys. Rev. B 90, 125436 (2014).

    Article  CAS  Google Scholar 

  99. Simion, G. E. & Lyanda-Geller, Y. B. Magnetic field spectral crossings of Luttinger holes in quantum wells. Phys. Rev. B 90, 195410 (2014).

    Article  CAS  Google Scholar 

  100. Pingenot, J., Pryor, C. E. & Flatté, M. E. Electric-field manipulation of the Landé g tensor of a hole in an In0.5Ga0.5As/GaAs self-assembled quantum dot. Phys. Rev. B 84, 195403 (2011).

    Article  CAS  Google Scholar 

  101. Maier, F., Kloeffel, C. & Loss, D. Tunable g factor and phonon-mediated hole spin relaxation in Ge/Si nanowire quantum dots. Phys. Rev. B 87, 161305 (2013).

    Article  CAS  Google Scholar 

  102. Ares, N. et al. SiGe quantum dots for fast hole spin Rabi oscillations. Appl. Phys. Lett. 103, 263113 (2013).

    Article  CAS  Google Scholar 

  103. Marcellina, E. et al. Electrical control of the Zeeman spin splitting in two-dimensional hole systems. Phys. Rev. Lett. 121, 077701 (2018).

    CAS  Article  Google Scholar 

  104. Crippa, A. et al. Electrical spin driving by g-matrix modulation in spin-orbit qubits. Phys. Rev. Lett. 120, 137702 (2018).

    CAS  Article  Google Scholar 

  105. Venitucci, B., Bourdet, L., Pouzada, D. & Niquet, Y.-M. Electrical manipulation of semiconductor spin qubits within the g-matrix formalism. Phys. Rev. B 98, 155319 (2018).

    CAS  Article  Google Scholar 

  106. DiVincenzo, D. P. The physical implementation of quantum computation. Fortschr. Phys. 48, 771–783 (2000).

    Article  Google Scholar 

  107. Itoh, K. et al. High purity isotopically enriched 70Ge and 74Ge single crystals: isotope separation, growth, and properties. J. Mater. Res. 8, 1341–1347 (1993).

    CAS  Article  Google Scholar 

  108. Asen-Palmer, M. et al. Thermal conductivity of germanium crystals with different isotopic compositions. Phys. Rev. B 56, 9431–9447 (1997).

    CAS  Article  Google Scholar 

  109. Becker, P., Pohl, H.-J., Riemann, H. & Abrosimov, N. Enrichment of silicon for a better kilogram. Phys. Status Solidi A 207, 49–66 (2010).

    CAS  Article  Google Scholar 

  110. Tyryshkin, A. M. et al. Electron spin coherence exceeding seconds in high-purity silicon. Nat. Mater. 11, 143–147 (2012).

    CAS  Article  Google Scholar 

  111. Veldhorst, M. et al. An addressable quantum dot qubit with fault-tolerant control-fidelity. Nat. Nanotechnol. 9, 981–985 (2014).

    CAS  Article  Google Scholar 

  112. Muhonen, J. T. et al. Storing quantum information for 30 seconds in a nanoelectronic device. Nat. Nanotechnol. 9, 986–991 (2014).

    CAS  Article  Google Scholar 

  113. Sigillito, A. J. et al. Electron spin coherence of shallow donors in natural and isotopically enriched germanium. Phys. Rev. Lett. 115, 247601 (2015).

    CAS  Article  Google Scholar 

  114. Fischer, J. & Loss, D. Hybridization and spin decoherence in heavy-hole quantum dots. Phys. Rev. Lett. 105, 266603 (2010).

    Article  CAS  Google Scholar 

  115. Maier, F. & Loss, D. Effect of strain on hyperfine-induced hole-spin decoherence in quantum dots. Phys. Rev. B 85, 195323 (2012).

    Article  CAS  Google Scholar 

  116. Zwanenburg, F. A. et al. Silicon quantum electronics. Rev. Mod. Phys. 85, 961–1019 (2013).

    CAS  Article  Google Scholar 

  117. Vandersypen, L. M. K. & Eriksson, M. A. Quantum computing with semiconductor spins. Phys. Today 72, 38–45 (2019).

    CAS  Article  Google Scholar 

  118. Burkard, G., Loss, D. & DiVincenzo, D. P. Coupled quantum dots as quantum gates. Phys. Rev. B 59, 2070–2078 (1999).

    CAS  Article  Google Scholar 

  119. Weiss, K. M., Elzerman, J. M., Delley, Y. L., Miguel-Sanchez, J. & Imamoglu, A. Coherent two-electron spin qubits in an optically active pair of coupled InGaAs quantum dots. Phys. Rev. Lett. 109, 107401 (2012).

    CAS  Article  Google Scholar 

  120. Chesi, S. et al. Single-spin manipulation in a double quantum dot in the field of a micromagnet. Phys. Rev. B 90, 235311 (2014).

    Article  CAS  Google Scholar 

  121. Wong, C. H., Eriksson, M. A., Coppersmith, S. N. & Friesen, M. High-fidelity singlet-triplet S-T_ qubits in inhomogeneous magnetic fields. Phys. Rev. B 92, 045403 (2015).

    Article  CAS  Google Scholar 

  122. Reed, M. D. et al. Reduced sensitivity to charge noise in semiconductor spin qubits via symmetric operation. Phys. Rev. Lett. 116, 110402 (2016).

    CAS  Article  Google Scholar 

  123. Martins, F. et al. Noise suppression using symmetric exchange gates in spin qubits. Phys. Rev. Lett. 116, 116801 (2016).

    Article  CAS  Google Scholar 

  124. Abadillo-Uriel, J. C., Eriksson, M. A., Coppersmith, S. N. & Friesen, M. Enhancing the dipolar coupling of a S-T0 qubit with a transverse sweet spot. Nat. Commun. 10, 5641 (2019).

    CAS  Article  Google Scholar 

  125. Khaetskii, A. V. & Nazarov, Y. V. Spin-flip transitions between Zeeman sublevels in semiconductor quantum dots. Phys. Rev. B 64, 125316 (2001).

    Article  CAS  Google Scholar 

  126. Golovach, V. N., Khaetskii, A. & Loss, D. Phonon-induced decay of the electron spin in quantum dots. Phys. Rev. Lett. 93, 016601 (2004).

    Article  CAS  Google Scholar 

  127. Stano, P. & Fabian, J. Theory of phonon-induced spin relaxation in laterally coupled quantum dots. Phys. Rev. Lett. 96, 186602 (2006).

    Article  CAS  Google Scholar 

  128. Kornich, V., Kloeffel, C. & Loss, D. Phonon-assisted relaxation and decoherence of singlet-triplet qubits in Si/SiGe quantum dots. Quantum 2, 70 (2018).

    Article  Google Scholar 

  129. Li, J., Venitucci, B. & Niquet, Y.-M. Hole-phonon interactions in quantum dots: Effects of phonon confinement and encapsulation materials on spin-orbit qubits. Phys. Rev. B 102, 075415 (2020).

    CAS  Article  Google Scholar 

  130. People, R. & Bean, J. C. Band alignments of coherently strained GeSi/Si heterostructures on 001 GeSi substrates. Appl. Phys. Lett. 48, 538–540 (1986).

    CAS  Article  Google Scholar 

  131. People, R. Indirect band gap and band alignment for coherently strained bulk alloys on germanium (001) substrates. Phys. Rev. B 34, 2508–2510 (1986).

    CAS  Article  Google Scholar 

  132. Virgilio, M. & Grosso, G. Type-I alignment and direct fundamental gap in SiGe based heterostructures. J. Phys. Condens. Matter 18, 1021–1031 (2006).

    CAS  Article  Google Scholar 

  133. Giorgioni, A. et al. Strong confinement-induced engineering of the g factor and lifetime of conduction electron spins in Ge quantum wells. Nat. Commun. 7, 13886 (2016).

    CAS  Article  Google Scholar 

  134. Paul, D. The progress towards terahertz quantum cascade lasers on silicon substrates. Laser Photonics Rev. 4, 610–632 (2010).

    CAS  Article  Google Scholar 

  135. Matthews, J. W. & Blakeslee, A. E. Defects in epitaxial multilayers: III. Preparation of almost perfect multilayers. J. Cryst. Growth 32, 265–273 (1976).

    CAS  Article  Google Scholar 

  136. Wagner, G. R. & Janocko, M. A. Observation of a two-dimensional hole gas in boron-doped Si0.5Ge0.5/Ge heterostructures. Appl. Phys. Lett. 54, 66–68 (1989).

    CAS  Article  Google Scholar 

  137. Murakami, E., Etoh, H., Nakagawa, K. & Miyao, M. High hole mobility in modulation-doped and strain-controlled p-Si0.5Ge0.5/Ge/Si1−xGex heterostructures fabricated using molecular beam epitaxy. Jpn. J. Appl. Phys. 29, L1059–L1061 (1990).

    CAS  Article  Google Scholar 

  138. Murakami, E., Nakagawa, K., Nishida, A. & Miyao, M. Strain-controlled Si-Ge modulation-doped FET with ultrahigh hole mobility. IEEE Electron Device Lett. 12, 71–73 (1991).

    CAS  Article  Google Scholar 

  139. Xie, Y. H. et al. Very high mobility two-dimensional hole gas in Si/GexSi1−x/Ge structures grown by molecular beam epitaxy. Appl. Phys. Lett. 63, 2263–2264 (1993).

    CAS  Article  Google Scholar 

  140. Schäffler, F. High-mobility Si and Ge structures. Semicond. Sci. Technol. 12, 1515–1549 (1997).

    Article  Google Scholar 

  141. Lee, M. L., Fitzgerald, E. A., Bulsara, M. T., Currie, M. T. & Lochtefeld, A. Strained Si, SiGe, and Ge channels for high-mobility metal-oxide-semiconductor field-effect transistors. J. Appl. Phys. 97, 011101 (2005).

    Article  CAS  Google Scholar 

  142. Isella, G. et al. Low-energy plasma-enhanced chemical vapor deposition for strained Si and Ge heterostructures and devices. Solid State Electron. 48, 1317–1323 (2004).

    CAS  Article  Google Scholar 

  143. Känel, H. V., Kummer, M., Isella, G., Müller, E. & Hackbarth, T. Very high hole mobilities in modulation-doped Ge quantum wells grown by low-energy plasma enhanced chemical vapor deposition. Appl. Phys. Lett. 80, 2922–2924 (2002).

    Article  CAS  Google Scholar 

  144. Rössner, B., Chrastina, D., Isella, G. & von Känel, H. Scattering mechanisms in high-mobility strained Ge channels. Appl. Phys. Lett. 84, 3058–3060 (2004).

    Article  CAS  Google Scholar 

  145. Shah, V. A. et al. Reverse graded relaxed buffers for high Ge content SiGe virtual substrates. Appl. Phys. Lett. 93, 192103 (2008).

    Article  CAS  Google Scholar 

  146. Shah, V. A., Dobbie, A., Myronov, M. & Leadley, D. R. Reverse graded SiGe/Ge/Si buffers for high-composition virtual substrates. J. Appl. Phys. 107, 064304 (2010).

    Article  CAS  Google Scholar 

  147. Colace, L. et al. Metal–semiconductor–metal near-infrared light detector based on epitaxial Ge/Si. Appl. Phys. Lett. 72, 3175–3177 (1998).

    CAS  Article  Google Scholar 

  148. Gunn, L. C. III, Capellini, G., Rattier, M. J. & Pinguet, T. J. Methods of incorporating germanium within CMOS process. US Patent 6,887,773 (2005).

  149. Hartmann, J. M. et al. Reduced pressure–chemical vapor deposition of Ge thick layers on Si(001) for 1.3–1.55-μm photodetection. J. Appl. Phys. 95, 5905–5913 (2004).

    CAS  Article  Google Scholar 

  150. Lu, T. M. et al. Enhancement-mode buried strained silicon channel quantum dot with tunable lateral geometry. Appl. Phys. Lett. 99, 043101 (2011).

    Article  CAS  Google Scholar 

  151. Borselli, M. G. et al. Pauli spin blockade in undoped Si/SiGe two-electron double quantum dots. Appl. Phys. Lett. 99, 063109 (2011).

    Article  CAS  Google Scholar 

  152. Maune, B. M. et al. Coherent singlet-triplet oscillations in a silicon-based double quantum dot. Nature 481, 344–347 (2012).

    CAS  Article  Google Scholar 

  153. Su, Y.-H., Chuang, Y., Liu, C.-Y., Li, J.-Y. & Lu, T.-M. Effects of surface tunneling of two-dimensional hole gases in undoped Ge/GeSi heterostructures. Phys. Rev. Mater. 1, 044601 (2017).

    Article  Google Scholar 

  154. Lodari, M. et al. Low percolation density and charge noise with holes in germanium. Mater. Quantum Technol. https://doi.org/10.1088/2633-4356/abcd82 (in the press).

  155. Rößner, B., Isella, G. & Känel, H. V. Effective mass in remotely doped Ge quantum wells. Appl. Phys. Lett. 82, 754–756 (2003).

    Article  CAS  Google Scholar 

  156. Irisawa, T. et al. Hole density dependence of effective mass, mobility and transport time in strained Ge channel modulation-doped heterostructures. Appl. Phys. Lett. 82, 1425–1427 (2003).

    CAS  Article  Google Scholar 

  157. Sawano, K. et al. Magnetotransport properties of Ge channels with extremely high compressive strain. Appl. Phys. Lett. 89, 162103 (2006).

    Article  CAS  Google Scholar 

  158. Sawano, K. et al. Strain dependence of hole effective mass and scattering mechanism in strained Ge channel structures. Appl. Phys. Lett. 95, 122109 (2009).

    Article  CAS  Google Scholar 

  159. Foronda, J., Morrison, C., Halpin, J. E., Rhead, S. D. & Myronov, M. Weak antilocalization of high mobility holes in a strained Germanium quantum well heterostructure. J. Phys. Condens. Matter 27, 022201 (2015).

    CAS  Article  Google Scholar 

  160. Hassan, A. H. A. et al. Anisotropy in the hole mobility measured along the [110] and [1–10] orientations in a strained Ge quantum well. Appl. Phys. Lett. 104, 132108 (2014).

    Article  CAS  Google Scholar 

  161. Morrison, C. et al. Observation of Rashba zero-field spin splitting in a strained germanium 2D hole gas. Appl. Phys. Lett. 105, 182401 (2014).

    Article  CAS  Google Scholar 

  162. Failla, M., Myronov, M., Morrison, C., Leadley, D. R. & Lloyd-Hughes, J. Narrow heavy-hole cyclotron resonances split by the cubic Rashba spin-orbit interaction in strained germanium quantum wells. Phys. Rev. B 92, 045303 (2015).

    Article  CAS  Google Scholar 

  163. Shi, Q., Zudov, M. A., Morrison, C. & Myronov, M. Spinless composite fermions in an ultrahigh-quality strained Ge quantum well. Phys. Rev. B 91, 241303 (2015).

    Article  CAS  Google Scholar 

  164. Morrison, C., Casteleiro, C., Leadley, D. R. & Myronov, M. Complex quantum transport in a modulation doped strained Ge quantum well heterostructure with a high mobility 2D hole gas. Appl. Phys. Lett. 109, 102103 (2016).

    Article  CAS  Google Scholar 

  165. Holmes, S. N. et al. Spin-splitting in p-type Ge devices. J. Appl. Phys. 120, 085702 (2016).

    Article  CAS  Google Scholar 

  166. Mironov, O. et al. Fractional quantum hall states in a Ge quantum well. Phys. Rev. Lett. 116, 176802 (2016).

    CAS  Article  Google Scholar 

  167. Morrison, C. & Myronov, M. Electronic transport anisotropy of 2D carriers in biaxial compressive strained germanium. Appl. Phys. Lett. 111, 192103 (2017).

    Article  CAS  Google Scholar 

  168. Drichko, I. L. et al. Effective g factor of 2D holes in strained Ge quantum wells. J. Appl. Phys. 123, 165703 (2018).

    Article  CAS  Google Scholar 

  169. Berkutov, I. B., Andrievskii, V. V., Kolesnichenko, Y. A. & Mironov, O. A. Quantum effects in a germanium quantum well with ultrahigh mobility of charge carrier. Low Temp. Phys. 45, 1202–1208 (2019).

    CAS  Article  Google Scholar 

  170. Laroche, D. et al. Magneto-transport analysis of an ultra-low-density two-dimensional hole gas in an undoped strained Ge/SiGe heterostructure. Appl. Phys. Lett. 108, 233504 (2016).

    Article  CAS  Google Scholar 

  171. Lu, T. M. et al. Density-controlled quantum Hall ferromagnetic transition in a two-dimensional hole system. Sci. Rep. 7, 2468 (2017).

    CAS  Article  Google Scholar 

  172. Lu, T. M. et al. Effective g factor of low-density two-dimensional holes in a Ge quantum well. Appl. Phys. Lett. 111, 102108 (2017).

    Article  CAS  Google Scholar 

  173. Hardy, W. J. et al. Single and double hole quantum dots in strained Ge/SiGe quantum wells. Nanotechnology 30, 215202 (2019).

    CAS  Article  Google Scholar 

  174. Failla, M. et al. Terahertz quantum Hall effect for spin-split heavy-hole gases in strained Ge quantum wells. New J. Phys. 18, 113036 (2016).

    Article  CAS  Google Scholar 

  175. Gul, Y. et al. Quantum ballistic transport in strained epitaxial germanium. Appl. Phys. Lett. 111, 233512 (2017).

    Article  CAS  Google Scholar 

  176. Mizokuchi, R., Maurand, R., Vigneau, F., Myronov, M. & De Franceschi, S. Ballistic one-dimensional holes with strong g-factor anisotropy in germanium. Nano Lett. 18, 4861–4865 (2018).

    CAS  Article  Google Scholar 

  177. Gul, Y., Holmes, S. N., Myronov, M., Kumar, S. & Pepper, M. Self-organised fractional quantisation in a hole quantum wire. J. Phys. Condens. Matter 30, 09LT01 (2018).

    CAS  Article  Google Scholar 

  178. Lauhon, L. J., Gudiksen, M. S., Wang, D. & Lieber, C. M. Epitaxial core–shell and core–multishell nanowire heterostructures. Nature 420, 57–61 (2002).

    CAS  Article  Google Scholar 

  179. Morales, A. M. & Lieber, C. M. A laser ablation method for the synthesis of crystalline semiconductor nanowires. Science 279, 208–211 (1998).

    CAS  Article  Google Scholar 

  180. Woodruff, J. H., Ratchford, J. B., Goldthorpe, I. A., McIntyre, P. C. & Chidsey, C. E. D. Vertically oriented germanium nanowires grown from gold colloids on silicon substrates and subsequent gold removal. Nano Lett. 7, 1637–1642 (2007).

    CAS  Article  Google Scholar 

  181. Dayeh, S. A. & Picraux, S. T. Direct observation of nanoscale size effects in Ge semiconductor nanowire growth. Nano Lett. 10, 4032–4039 (2010).

    CAS  Article  Google Scholar 

  182. Tian, B., Xie, P., Kempa, T. J., Bell, D. C. & Lieber, C. M. Single-crystalline kinked semiconductor nanowire superstructures. Nat. Nanotechnol. 4, 824–829 (2009).

    CAS  Article  Google Scholar 

  183. Goldthorpe, I. A., Marshall, A. F. & McIntyre, P. C. Inhibiting strain-induced surface roughening: dislocation-free Ge/Si and Ge/SiGe core–shell nanowires. Nano Lett. 9, 3715–3719 (2009).

    CAS  Article  Google Scholar 

  184. Conesa-Boj, S. et al. Boosting hole mobility in coherently strained [110]-oriented Ge–Si core–shell nanowires. Nano Lett. 17, 2259–2264 (2017).

    CAS  Article  Google Scholar 

  185. Dillen, D. C., Kim, K., Liu, E.-S. & Tutuc, E. Radial modulation doping in core–shell nanowires. Nat. Nanotechnol. 9, 116–120 (2014).

    CAS  Article  Google Scholar 

  186. Sistani, M. et al. Highly transparent contacts to the 1D hole gas in ultrascaled Ge/Si core/shell nanowires. ACS Nano 13, 14145–14151 (2019).

    CAS  Article  Google Scholar 

  187. Lu, W., Xiang, J., Timko, B. P., Wu, Y. & Lieber, C. M. One-dimensional hole gas in germanium/silicon nanowire heterostructures. Proc. Natl Acad. Sci. USA 102, 10046–10051 (2005).

    CAS  Article  Google Scholar 

  188. Kotekar-Patil, D., Nguyen, B.-M., Yoo, J., Dayeh, S. A. & Frolov, S. M. Quasiballistic quantum transport through Ge/Si core/shell nanowires. Nanotechnology 28, 385204 (2017).

    CAS  Article  Google Scholar 

  189. Zhang, X., Jevasuwan, W., Sugimoto, Y. & Fukata, N. Controlling catalyst-free formation and hole gas accumulation by fabricating Si/Ge core-shell and Si/Ge/Si core-double shell nanowires. ACS Nano 13, 13403–13412 (2019).

    CAS  Article  Google Scholar 

  190. Tersoff, J. & Tromp, R. M. Shape transition in growth of strained islands: spontaneous formation of quantum wires. Phys. Rev. Lett. 70, 2782–2785 (1993).

    CAS  Article  Google Scholar 

  191. Mo, Y.-W., Savage, D. E., Swartzentruber, B. S. & Lagally, M. G. Kinetic pathway in Stranski-Krastanov growth of Ge on Si(001). Phys. Rev. Lett. 65, 1020–1023 (1990).

    CAS  Article  Google Scholar 

  192. McKay, M. R., Venables, J. A. & Drucker, J. Kinetically suppressed Ostwald ripening of Ge/Si(100) hut clusters. Phys. Rev. Lett. 101, 216104 (2008).

    Article  CAS  Google Scholar 

  193. Zhang, J. J. et al. Monolithic growth of ultrathin Ge nanowires on Si(001). Phys. Rev. Lett. 109, 085502 (2012).

    CAS  Article  Google Scholar 

  194. Tersoff, J. & LeGoues, F. K. Competing relaxation mechanisms in strained layers. Phys. Rev. Lett. 72, 3570–3573 (1994).

    CAS  Article  Google Scholar 

  195. Watzinger, H., Glaser, M., Zhang, J. J., Daruka, I. & Schäffler, F. Influence of composition and substrate miscut on the evolution of {105}-terminated in-plane Si1−xGex quantum wires on Si(001). APL Mater. 2, 076102 (2014).

    Article  CAS  Google Scholar 

  196. Brauns, M. et al. Highly tuneable hole quantum dots in Ge-Si core-shell nanowires. Appl. Phys. Lett. 109, 143113 (2016).

    Article  CAS  Google Scholar 

  197. Froning, F. N. M. et al. Single, double, and triple quantum dots in Ge/Si nanowires. Appl. Phys. Lett. 113, 073102 (2018).

    Article  CAS  Google Scholar 

  198. Roddaro, S. et al. Spin states of holes in Ge/Si nanowire quantum dots. Phys. Rev. Lett. 101, 186802 (2008).

    CAS  Article  Google Scholar 

  199. Brauns, M. et al. Anisotropic Pauli spin blockade in hole quantum dots. Phys. Rev. B 94, 041411 (2016).

    Article  CAS  Google Scholar 

  200. Zarassi, A. et al. Magnetic field evolution of spin blockade in Ge/Si nanowire double quantum dots. Phys. Rev. B 95, 155416 (2017).

    Article  Google Scholar 

  201. Vukušic´, L. et al. Single-shot readout of hole spins in Ge. Nano Lett. 18, 7141–7145 (2018).

    Article  CAS  Google Scholar 

  202. Vukušic´, L., Kukucˇka, J., Watzinger, H. & Katsaros, G. Fast hole tunneling times in germanium hut wires probed by single-shot reflectometry. Nano Lett. 17, 5706–5710 (2017).

    Article  CAS  Google Scholar 

  203. Hanson, R., Kouwenhoven, L. P., Petta, J. R., Tarucha, S. & Vandersypen, L. M. K. Spins in few-electron quantum dots. Rev. Mod. Phys. 79, 1217–1265 (2007).

    CAS  Article  Google Scholar 

  204. Lawrie, W. I. L. et al. Quantum dot arrays in silicon and germanium. Appl. Phys. Lett. 116, 080501 (2020).

    CAS  Article  Google Scholar 

  205. Hofmann, A. et al. Assessing the potential of Ge/SiGe quantum dots as hosts for singlet-triplet qubits. Preprint at arXiv https://arxiv.org/abs/1910.05841 (2019).

  206. Higginbotham, A. P. et al. Hole spin coherence in a Ge/Si heterostructure nanowire. Nano Lett. 14, 3582–3586 (2014).

    CAS  Article  Google Scholar 

  207. Vandersypen, L. M. K. et al. Interfacing spin qubits in quantum dots and donors — hot, dense, and coherent. NPJ Quantum Inf. 3, 34 (2017).

    Article  Google Scholar 

  208. Lawrie, W. I. L. et al. Spin relaxation benchmarks and individual qubit addressability for holes in quantum dots. Nano Lett. 20, 7237–7242 (2017).

    Article  CAS  Google Scholar 

  209. Hutin, L. et al. in Proceedings of the 2018 48th European Solid-State Device Research Conference (ESSDERC) 12–17 (IEEE, 2018).

  210. Froning, F. N. M. et al. Ultrafast hole spin qubit with gate-tunable spin-orbit switch. Preprint at arXiv http://arxiv.org/abs/2006.11175 (2020).

  211. Knill, E. et al. Randomized benchmarking of quantum gates. Phys. Rev. A 77, 012307 (2008).

    Article  CAS  Google Scholar 

  212. Bertrand, B. et al. Quantum manipulation of two-electron spin states in isolated double quantum dots. Phys. Rev. Lett. 115, 096801 (2015).

    Article  CAS  Google Scholar 

  213. De Franceschi, S., Kouwenhoven, L., Schönenberger, C. & Wernsdorfer, W. Hybrid superconductor–quantum dot devices. Nat. Nanotechnol. 5, 703–711 (2010).

    Article  CAS  Google Scholar 

  214. Lutchyn, R. M. et al. Majorana zero modes in superconductor–semiconductor heterostructures. Nat. Rev. Mater. 3, 52–68 (2018).

    CAS  Article  Google Scholar 

  215. Clark, T. D., Prance, R. J. & Grassie, A. D. C. Feasibility of hybrid Josephson field effect transistors. J. Appl. Phys. 51, 2736 (1980).

    CAS  Article  Google Scholar 

  216. de Lange, G. et al. Realization of microwave quantum circuits using hybrid superconducting-semiconducting nanowire Josephson elements. Phys. Rev. Lett. 115, 127002 (2015).

    Article  CAS  Google Scholar 

  217. Casparis, L. et al. Superconducting gatemon qubit based on a proximitized two-dimensional electron gas. Nat. Nanotechnol. 13, 915–919 (2018).

    CAS  Article  Google Scholar 

  218. Sau, J. D. & Sarma, S. D. Realizing a robust practical Majorana chain in a quantum-dot-superconductor linear array. Nat. Commun. 3, 964 (2012).

    Article  CAS  Google Scholar 

  219. Ridderbos, J. et al. Multiple Andreev reflections and Shapiro steps in a Ge-Si nanowire Josephson junction. Phys. Rev. Mater. 3, 084803 (2019).

    CAS  Article  Google Scholar 

  220. Ridderbos, J. et al. Hard superconducting gap and diffusion-induced superconductors in Ge–Si nanowires. Nano Lett. 20, 122–130 (2020).

    CAS  Article  Google Scholar 

  221. Hendrickx, N. W. et al. Ballistic supercurrent discretization and micrometer-long Josephson coupling in germanium. Phys. Rev. B 99, 075435 (2019).

    CAS  Article  Google Scholar 

  222. De Franceschi, S. et al. Andreev reflection in Si-engineered Al/InGaAs hybrid junctions. Appl. Phys. Lett. 73, 3890–3892 (1998).

    Article  Google Scholar 

  223. Krogstrup, P. et al. Epitaxy of semiconductor–superconductor nanowires. Nat. Mater. 14, 400–406 (2015).

    CAS  Article  Google Scholar 

  224. Kral, S. et al. Abrupt Schottky junctions in Al/Ge nanowire heterostructures. Nano Lett. 15, 4783–4787 (2015).

    CAS  Article  Google Scholar 

  225. El Hajraoui, K. et al. In situ transmission electron microscopy analysis of aluminum–germanium nanowire solid-state reaction. Nano Lett. 19, 2897–2904 (2019).

    CAS  Article  Google Scholar 

  226. Franke, D. P., Clarke, J. S., Vandersypen, L. M. K. & Veldhorst, M. Rent’s rule and extensibility in quantum computing. Microprocess. Microsyst. 67, 1–7 (2019).

    Article  Google Scholar 

  227. Taylor, J. M. et al. Fault-tolerant architecture for quantum computation using electrically controlled semiconductor spins. Nat. Phys. 1, 177–183 (2005).

    CAS  Article  Google Scholar 

  228. Veldhorst, M., Eenink, H. G. J., Yang, C. H. & Dzurak, A. S. Silicon CMOS architecture for a spin-based quantum computer. Nat. Commun. 8, 1766 (2017).

    CAS  Article  Google Scholar 

  229. Li, R. et al. A crossbar network for silicon quantum dot qubits. Sci. Adv. 4, eaar3960 (2018).

    Article  CAS  Google Scholar 

  230. Hill, C. D. et al. A surface code quantum computer in silicon. Sci. Adv. 1, e1500707 (2015).

    Article  CAS  Google Scholar 

  231. Nickerson, N. H., Li, Y. & Benjamin, S. C. Topological quantum computing with a very noisy network and local error rates approaching one percent. Nat. Commun. 4, 1756 (2013).

    Article  CAS  Google Scholar 

  232. Veldhorst, M. et al. A two-qubit logic gate in silicon. Nature 526, 410–414 (2015).

    CAS  Article  Google Scholar 

  233. Watson, T. F. et al. A programmable two-qubit quantum processor in silicon. Nature 555, 633–637 (2018).

    CAS  Article  Google Scholar 

  234. Zheng, G. et al. Rapid gate-based spin read-out in silicon using an on-chip resonator. Nat. Nanotechnol. 14, 742–746 (2019).

    CAS  Article  Google Scholar 

  235. West, A. et al. Gate-based single-shot readout of spins in silicon. Nat. Nanotechnol. 14, 437–441 (2019).

    CAS  Article  Google Scholar 

  236. Urdampilleta, M. et al. Gate-based high fidelity spin readout in a CMOS device. Nat. Nanotechnol. 14, 737–741 (2019).

    CAS  Article  Google Scholar 

  237. Crippa, A. et al. Gate-reflectometry dispersive readout and coherent control of a spin qubit in silicon. Nat. Commun. 10, 2776 (2019).

    CAS  Article  Google Scholar 

  238. Samkharadze, N. et al. Strong spin-photon coupling in silicon. Science 359, 1123–1127 (2018).

    CAS  Article  Google Scholar 

  239. Mi, X. et al. A coherent spin–photon interface in silicon. Nature 555, 599–603 (2018).

    CAS  Article  Google Scholar 

  240. Borjans, F., Croot, X. G., Mi, X., Gullans, M. J. & Petta, J. R. Resonant microwave-mediated interactions between distant electron spins. Nature 577, 195–198 (2020).

    CAS  Article  Google Scholar 

  241. Fowler, A. G., Mariantoni, M., Martinis, J. M. & Cleland, A. N. Surface codes: Towards practical large-scale quantum computation. Phys. Rev. A 86, 032324 (2012).

    Article  CAS  Google Scholar 

  242. Itoh, K. M. & Watanabe, H. Isotope engineering of silicon and diamond for quantum computing and sensing applications. MRS Commun. 4, 143–157 (2014).

    CAS  Article  Google Scholar 

  243. Hu, X., Liu, Y.-x. & Nori, F. Strong coupling of a spin qubit to a superconducting stripline cavity. Phys. Rev. B 86, 035314 (2012).

    Article  CAS  Google Scholar 

  244. Sigrist, M. & Ueda, K. Phenomenological theory of unconventional superconductivity. Rev. Mod. Phys. 63, 239–311 (1991).

    CAS  Article  Google Scholar 

  245. 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  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  247. 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).

    CAS  Article  Google Scholar 

  248. 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).

    CAS  Article  Google Scholar 

  249. Mao, L., Shi, J., Niu, Q. & Zhang, C. Superconducting phase with a chiral f-wave pairing symmetry and Majorana fermions induced in a hole-doped semiconductor. Phys. Rev. Lett. 106, 157003 (2011).

    Article  CAS  Google Scholar 

  250. Leijnse, M. & Flensberg, K. Quantum information transfer between topological and spin qubit systems. Phys. Rev. Lett. 107, 210502 (2011).

    Article  CAS  Google Scholar 

  251. Leijnse, M. & Flensberg, K. Hybrid topological-spin qubit systems for two-qubit-spin gates. Phys. Rev. B 86, 104511 (2012).

    Article  CAS  Google Scholar 

  252. Hoffman, S., Schrade, C., Klinovaja, J. & Loss, D. Universal quantum computation with hybrid spin-Majorana qubits. Phys. Rev. B 94, 045316 (2016).

    Article  Google Scholar 

  253. Rancˇic´, M. J., Hoffman, S., Schrade, C., Klinovaja, J. & Loss, D. Entangling spins in double quantum dots and Majorana bound states. Phys. Rev. B 99, 165306 (2019).

    Article  Google Scholar 

  254. Choi, M.-S., Bruder, C. & Loss, D. Spin-dependent Josephson current through double quantum dots and measurement of entangled electron states. Phys. Rev. B 62, 13569–13572 (2000).

    CAS  Article  Google Scholar 

  255. Leijnse, M. & Flensberg, K. Coupling spin qubits via superconductors. Phys. Rev. Lett. 111, 060501 (2013).

    Article  CAS  Google Scholar 

  256. Hassler, F., Catelani, G. & Bluhm, H. Exchange interaction of two spin qubits mediated by a superconductor. Phys. Rev. B 92, 235401 (2015).

    Article  CAS  Google Scholar 

  257. Wang, K. et al. Ultrafast operations of a hole spin qubit in Ge quantum dot. Preprint at arXiv http://arxiv.org/abs/2006.12340 (2020).

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Acknowledgements

G.S., M.V. and F.A.Z. acknowledge financial support from the Netherlands Organization for Scientific Research (NWO). F.A.Z., D.L. and G.K. acknowledge funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 862046. G.K. acknowledges funding from FP7 ERC Starting Grant 335497, FWF Y 715-N30 and FWF P-30207. S.D.F. acknowledges support from the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 81050 and from the Agence Nationale de la Recherche through the TOPONANO and QSPIN projects. J.-J.Z. acknowledges support from the National Key R&D Program of China (grant no. 2016YFA0301701) and Strategic Priority Research Program of the Chinese Academy of Sciences (grant no. XDB30000000). D.L. and C.K. acknowledge the Swiss National Science Foundation and NCCR QSIT.

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Scappucci, G., Kloeffel, C., Zwanenburg, F.A. et al. The germanium quantum information route. Nat Rev Mater 6, 926–943 (2021). https://doi.org/10.1038/s41578-020-00262-z

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