A Silicon Surface Code Architecture Resilient Against Leakage Errors

Zhenyu Cai1,2, Michael A. Fogarty1,3, Simon Schaal3, Sofia Patomäki1,3, Simon C. Benjamin1,2, and John J. L. Morton1,3,4

1Quantum Motion Technologies Ltd, Nexus, Discovery Way, Leeds, West Yorkshire, LS2 3AA, United Kingdom
2Department of Materials, University of Oxford, Oxford, OX1 3PH, United Kingdom
3London Centre for Nanotechnology, UCL, 17-19 Gordon St, London, WC1H 0AH, United Kingdom
4Dept. of Electronic and Electrical Engineering, UCL, London, WC1E 7JE, United Kingdom

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Abstract

Spin qubits in silicon quantum dots are one of the most promising building blocks for large scale quantum computers thanks to their high qubit density and compatibility with the existing semiconductor technologies. High fidelity single-qubit gates exceeding the threshold of error correction codes like the surface code have been demonstrated, while two-qubit gates have reached 98% fidelity and are improving rapidly. However, there are other types of error --- such as charge leakage and propagation --- that may occur in quantum dot arrays and which cannot be corrected by quantum error correction codes, making them potentially damaging even when their probability is small. We propose a surface code architecture for silicon quantum dot spin qubits that is robust against leakage errors by incorporating multi-electron mediator dots. Charge leakage in the qubit dots is transferred to the mediator dots via charge relaxation processes and then removed using charge reservoirs attached to the mediators. A stabiliser-check cycle, optimised for our hardware, then removes the correlations between the residual physical errors. Through simulations we obtain the surface code threshold for the charge leakage errors and show that in our architecture the damage due to charge leakage errors is reduced to a similar level to that of the usual depolarising gate noise. Spin leakage errors in our architecture are constrained to only ancilla qubits and can be removed during quantum error correction via reinitialisations of ancillae, which ensure the robustness of our architecture against spin leakage as well. Our use of an elongated mediator dots creates spaces throughout the quantum dot array for charge reservoirs, measuring devices and control gates, providing the scalability in the design.

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► References

[1] Richard P. Feynman. Simulating physics with computers. International Journal of Theoretical Physics, 21 (6-7): 467-488, June 1982. 10.1007/​BF02650179.
https:/​/​doi.org/​10.1007/​BF02650179

[2] Lov K. Grover. A Fast Quantum Mechanical Algorithm for Database Search. In Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, STOC '96, pages 212-219, New York, NY, USA, 1996. ACM. 10.1145/​237814.237866.
https:/​/​doi.org/​10.1145/​237814.237866

[3] Lov K. Grover. Quantum Mechanics Helps in Searching for a Needle in a Haystack. Physical Review Letters, 79 (2): 325-328, July 1997. 10.1103/​PhysRevLett.79.325.
https:/​/​doi.org/​10.1103/​PhysRevLett.79.325

[4] K. Temme, T. J. Osborne, K. G. Vollbrecht, D. Poulin, and F. Verstraete. Quantum Metropolis sampling. Nature, 471 (7336): 87-90, March 2011. 10.1038/​nature09770.
https:/​/​doi.org/​10.1038/​nature09770

[5] Montanaro Ashley. Quantum speedup of Monte Carlo methods. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471 (2181): 20150301, September 2015. 10.1098/​rspa.2015.0301.
https:/​/​doi.org/​10.1098/​rspa.2015.0301

[6] David S. Wang, Austin G. Fowler, and Lloyd C. L. Hollenberg. Surface code quantum computing with error rates over 1%. Physical Review A, 83 (2), February 2011. 10.1103/​PhysRevA.83.020302.
https:/​/​doi.org/​10.1103/​PhysRevA.83.020302

[7] Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. Surface codes: Towards practical large-scale quantum computation. Physical Review A, 86 (3): 032324, 2012. 10.1103/​PhysRevA.86.032324.
https:/​/​doi.org/​10.1103/​PhysRevA.86.032324

[8] Bjoern Lekitsch, Sebastian Weidt, Austin G. Fowler, Klaus Mølmer, Simon J. Devitt, Christof Wunderlich, and Winfried K. Hensinger. Blueprint for a microwave trapped ion quantum computer. Science Advances, 3 (2): e1601540, February 2017. 10.1126/​sciadv.1601540.
https:/​/​doi.org/​10.1126/​sciadv.1601540

[9] C. D. Hill, E. Peretz, S. J. Hile, M. G. House, M. Fuechsle, S. Rogge, M. Y. Simmons, and L. C. L. Hollenberg. A surface code quantum computer in silicon. Science Advances, 1 (9): e1500707-e1500707, October 2015. 10.1126/​sciadv.1500707.
https:/​/​doi.org/​10.1126/​sciadv.1500707

[10] Joe O'Gorman, Naomi H. Nickerson, Philipp Ross, John JL Morton, and Simon C. Benjamin. A silicon-based surface code quantum computer. npj Quantum Information, 2: npjqi201519, February 2016. 10.1038/​npjqi.2015.19.
https:/​/​doi.org/​10.1038/​npjqi.2015.19

[11] Joe O'Gorman and Earl T. Campbell. Quantum computation with realistic magic-state factories. Physical Review A, 95 (3), March 2017. 10.1103/​PhysRevA.95.032338.
https:/​/​doi.org/​10.1103/​PhysRevA.95.032338

[12] L. M. K. Vandersypen, H. Bluhm, J. S. Clarke, A. S. Dzurak, R. Ishihara, A. Morello, D. J. Reilly, L. R. Schreiber, and M. Veldhorst. Interfacing spin qubits in quantum dots and donors—hot, dense, and coherent. npj Quantum Information, 3 (1), December 2017. 10.1038/​s41534-017-0038-y.
https:/​/​doi.org/​10.1038/​s41534-017-0038-y

[13] M. Veldhorst, H. G. J. Eenink, C. H. Yang, and A. S. Dzurak. Silicon CMOS architecture for a spin-based quantum computer. Nature Communications, 8 (1): 1766, December 2017. 10.1038/​s41467-017-01905-6.
https:/​/​doi.org/​10.1038/​s41467-017-01905-6

[14] Ruoyu Li, Luca Petit, David P. Franke, Juan Pablo Dehollain, Jonas Helsen, Mark Steudtner, Nicole K. Thomas, Zachary R. Yoscovits, Kanwal J. Singh, Stephanie Wehner, Lieven M. K. Vandersypen, James S. Clarke, and Menno Veldhorst. A crossbar network for silicon quantum dot qubits. Science Advances, 4 (7): eaar3960, July 2018. 10.1126/​sciadv.aar3960.
https:/​/​doi.org/​10.1126/​sciadv.aar3960

[15] Brandon Buonacorsi, Zhenyu Cai, Eduardo B. Ramirez, Kyle S. Willick, Sean M. Walker, Jiahao Li, Benjamin D. Shaw, Xiaosi Xu, Simon C. Benjamin, and Jonathan Baugh. Network architecture for a topological quantum computer in silicon. Quantum Science and Technology, 4 (2): 025003, January 2019. 10.1088/​2058-9565/​aaf3c4.
https:/​/​doi.org/​10.1088/​2058-9565/​aaf3c4

[16] F. Motzoi, J. M. Gambetta, P. Rebentrost, and F. K. Wilhelm. Simple Pulses for Elimination of Leakage in Weakly Nonlinear Qubits. Physical Review Letters, 103 (11): 110501, September 2009. 10.1103/​PhysRevLett.103.110501.
https:/​/​doi.org/​10.1103/​PhysRevLett.103.110501

[17] Alejandro Ferrón and Daniel Domínguez. Intrinsic leakage of the Josephson flux qubit and breakdown of the two-level approximation for strong driving. Physical Review B, 81 (10): 104505, March 2010. 10.1103/​PhysRevB.81.104505.
https:/​/​doi.org/​10.1103/​PhysRevB.81.104505

[18] L.-M. Duan, J. I. Cirac, and P. Zoller. Geometric Manipulation of Trapped Ions for Quantum Computation. Science, 292 (5522): 1695-1697, June 2001. 10.1126/​science.1058835.
https:/​/​doi.org/​10.1126/​science.1058835

[19] H Haffner, C Roos, and R Blatt. Quantum computing with trapped ions. Physics Reports, 469 (4): 155-203, December 2008. 10.1016/​j.physrep.2008.09.003.
https:/​/​doi.org/​10.1016/​j.physrep.2008.09.003

[20] Bryan H. Fong and Stephen M. Wandzura. Universal Quantum Computation and Leakage Reduction in the 3-Qubit Decoherence Free Subsystem. February 2011. URL https:/​/​arxiv.org/​abs/​1102.2909v1.
arXiv:1102.2909v1

[21] Sebastian Mehl, Hendrik Bluhm, and David P. DiVincenzo. Fault-tolerant quantum computation for singlet-triplet qubits with leakage errors. Physical Review B, 91 (8): 085419, February 2015. 10.1103/​PhysRevB.91.085419.
https:/​/​doi.org/​10.1103/​PhysRevB.91.085419

[22] Filip K. Malinowski, Frederico Martins, Thomas B. Smith, Stephen D. Bartlett, Andrew C. Doherty, Peter D. Nissen, Saeed Fallahi, Geoffrey C. Gardner, Michael J. Manfra, Charles M. Marcus, and Ferdinand Kuemmeth. Fast spin exchange across a multielectron mediator. Nature Communications, 10 (1): 1196, March 2019. 10.1038/​s41467-019-09194-x.
https:/​/​doi.org/​10.1038/​s41467-019-09194-x

[23] Susan J. Angus, Andrew J. Ferguson, Andrew S. Dzurak, and Robert G. Clark. Gate-Defined Quantum Dots in Intrinsic Silicon. Nano Letters, 7 (7): 2051-2055, July 2007. 10.1021/​nl070949k.
https:/​/​doi.org/​10.1021/​nl070949k

[24] M. Veldhorst, J. C. C. Hwang, C. H. Yang, A. W. Leenstra, B. de Ronde, J. P. Dehollain, J. T. Muhonen, F. E. Hudson, K. M. Itoh, A. Morello, and A. S. Dzurak. An addressable quantum dot qubit with fault-tolerant control-fidelity. Nature Nanotechnology, 9 (12): 981-985, December 2014. 10.1038/​nnano.2014.216.
https:/​/​doi.org/​10.1038/​nnano.2014.216

[25] K. W. Chan, W. Huang, C. H. Yang, J. C. C. Hwang, B. Hensen, T. Tanttu, F. E. Hudson, K. M. Itoh, A. Laucht, A. Morello, and A. S. Dzurak. Assessment of a silicon quantum dot spin qubit environment via noise spectroscopy. Physical Review Applied, 10 (4), October 2018. 10.1103/​PhysRevApplied.10.044017.
https:/​/​doi.org/​10.1103/​PhysRevApplied.10.044017

[26] C. H. Yang, K. W. Chan, R. Harper, W. Huang, T. Evans, J. C. C. Hwang, B. Hensen, A. Laucht, T. Tanttu, F. E. Hudson, S. T. Flammia, K. M. Itoh, A. Morello, S. D. Bartlett, and A. S. Dzurak. Silicon qubit fidelities approaching incoherent noise limits via pulse engineering. Nature Electronics, 2 (4): 151, April 2019. 10.1038/​s41928-019-0234-1.
https:/​/​doi.org/​10.1038/​s41928-019-0234-1

[27] Erika Kawakami, Thibaut Jullien, Pasquale Scarlino, Daniel R. Ward, Donald E. Savage, Max G. Lagally, Viatcheslav V. Dobrovitski, Mark Friesen, Susan N. Coppersmith, Mark A. Eriksson, and Lieven M. K. Vandersypen. Gate fidelity and coherence of an electron spin in an Si/​SiGe quantum dot with micromagnet. Proceedings of the National Academy of Sciences, 113 (42): 11738-11743, October 2016. 10.1073/​pnas.1603251113.
https:/​/​doi.org/​10.1073/​pnas.1603251113

[28] Jun Yoneda, Kenta Takeda, Tomohiro Otsuka, Takashi Nakajima, Matthieu R. Delbecq, Giles Allison, Takumu Honda, Tetsuo Kodera, Shunri Oda, Yusuke Hoshi, Noritaka Usami, Kohei M. Itoh, and Seigo Tarucha. A quantum-dot spin qubit with coherence limited by charge noise and fidelity higher than 99.9%. Nature Nanotechnology, 13 (2): 102-106, February 2018. 10.1038/​s41565-017-0014-x.
https:/​/​doi.org/​10.1038/​s41565-017-0014-x

[29] G. Pica, B. W. Lovett, R. N. Bhatt, T. Schenkel, and S. A. Lyon. Surface code architecture for donors and dots in silicon with imprecise and nonuniform qubit couplings. Physical Review B, 93 (3), January 2016. 10.1103/​PhysRevB.93.035306.
https:/​/​doi.org/​10.1103/​PhysRevB.93.035306

[30] Cody Jones, Michael A. Fogarty, Andrea Morello, Mark F. Gyure, Andrew S. Dzurak, and Thaddeus D. Ladd. Logical Qubit in a Linear Array of Semiconductor Quantum Dots. Physical Review X, 8 (2): 021058, June 2018. 10.1103/​PhysRevX.8.021058.
https:/​/​doi.org/​10.1103/​PhysRevX.8.021058

[31] Alexei M. Tyryshkin, Shinichi Tojo, John J. L. Morton, Helge Riemann, Nikolai V. Abrosimov, Peter Becker, Hans-Joachim Pohl, Thomas Schenkel, Michael L. W. Thewalt, Kohei M. Itoh, and S. A. Lyon. Electron spin coherence exceeding seconds in high-purity silicon. Nature Materials, 11 (2): 143-147, February 2012. 10.1038/​nmat3182.
https:/​/​doi.org/​10.1038/​nmat3182

[32] Arne Laucht, Juha T. Muhonen, Fahd A. Mohiyaddin, Rachpon Kalra, Juan P. Dehollain, Solomon Freer, Fay E. Hudson, Menno Veldhorst, Rajib Rahman, Gerhard Klimeck, Kohei M. Itoh, David N. Jamieson, Jeffrey C. McCallum, Andrew S. Dzurak, and Andrea Morello. Electrically controlling single-spin qubits in a continuous microwave field. Science Advances, 1 (3): e1500022, April 2015. 10.1126/​sciadv.1500022.
https:/​/​doi.org/​10.1126/​sciadv.1500022

[33] D. Ristè, S. Poletto, M.-Z. Huang, A. Bruno, V. Vesterinen, O.-P. Saira, and L. DiCarlo. Detecting bit-flip errors in a logical qubit using stabilizer measurements. Nature Communications, 6: 6983, April 2015. 10.1038/​ncomms7983.
https:/​/​doi.org/​10.1038/​ncomms7983

[34] R. Schutjens, F. Abu Dagga, D. J. Egger, and F. K. Wilhelm. Single-qubit gates in frequency-crowded transmon systems. Physical Review A, 88 (5): 052330, November 2013. 10.1103/​PhysRevA.88.052330.
https:/​/​doi.org/​10.1103/​PhysRevA.88.052330

[35] C. H. Yang, A. Rossi, R. Ruskov, N. S. Lai, F. A. Mohiyaddin, S. Lee, C. Tahan, G. Klimeck, A. Morello, and A. S. Dzurak. Spin-valley lifetimes in a silicon quantum dot with tunable valley splitting. Nature Communications, 4: 2069, June 2013. 10.1038/​ncomms3069.
https:/​/​doi.org/​10.1038/​ncomms3069

[36] E. Kawakami, P. Scarlino, D. R. Ward, F. R. Braakman, D. E. Savage, M. G. Lagally, Mark Friesen, S. N. Coppersmith, M. A. Eriksson, and L. M. K. Vandersypen. Electrical control of a long-lived spin qubit in a Si/​SiGe quantum dot. Nature Nanotechnology, 9 (9): 666-670, September 2014. 10.1038/​nnano.2014.153.
https:/​/​doi.org/​10.1038/​nnano.2014.153

[37] R. C. C. Leon, C. H. Yang, J. C. C. Hwang, J. Camirand Lemyre, T. Tanttu, W. Huang, K. W. Chan, K. Y. Tan, F. E. Hudson, K. M. Itoh, A. Morello, A. Laucht, M. Pioro-Ladriere, A. Saraiva, and A. S. Dzurak. Coherent spin control of s-, p-, d- and f-electrons in a silicon quantum dot. arXiv:1902.01550 [cond-mat], February 2019. URL http:/​/​arxiv.org/​abs/​1902.01550.
arXiv:1902.01550

[38] Kohei M. Itoh and Hideyuki Watanabe. Isotope engineering of silicon and diamond for quantum computing and sensing applications. MRS Communications, 4 (4): 143-157, December 2014. 10.1557/​mrc.2014.32.
https:/​/​doi.org/​10.1557/​mrc.2014.32

[39] M. Veldhorst, C. H. Yang, J. C. C. Hwang, W. Huang, J. P. Dehollain, J. T. Muhonen, S. Simmons, A. Laucht, F. E. Hudson, K. M. Itoh, A. Morello, and A. S. Dzurak. A two-qubit logic gate in silicon. Nature, 526 (7573): 410-414, October 2015. 10.1038/​nature15263.
https:/​/​doi.org/​10.1038/​nature15263

[40] Wayne M. Witzel, Inès Montaño, Richard P. Muller, and Malcolm S. Carroll. Multiqubit gates protected by adiabaticity and dynamical decoupling applicable to donor qubits in silicon. Physical Review B, 92 (8): 081407, August 2015. 10.1103/​PhysRevB.92.081407.
https:/​/​doi.org/​10.1103/​PhysRevB.92.081407

[41] Navin Khaneja, Timo Reiss, Cindie Kehlet, Thomas Schulte-Herbrüggen, and Steffen J. Glaser. Optimal control of coupled spin dynamics: Design of NMR pulse sequences by gradient ascent algorithms. Journal of Magnetic Resonance, 172 (2): 296-305, February 2005. 10.1016/​j.jmr.2004.11.004.
https:/​/​doi.org/​10.1016/​j.jmr.2004.11.004

[42] C. H. Yang, W. H. Lim, N. S. Lai, A. Rossi, A. Morello, and A. S. Dzurak. Orbital and valley state spectra of a few-electron silicon quantum dot. Physical Review B, 86 (11): 115319, September 2012. 10.1103/​PhysRevB.86.115319.
https:/​/​doi.org/​10.1103/​PhysRevB.86.115319

[43] K. Ono, D. G. Austing, Y. Tokura, and S. Tarucha. Current Rectification by Pauli Exclusion in a Weakly Coupled Double Quantum Dot System. Science, 297 (5585): 1313-1317, August 2002. 10.1126/​science.1070958.
https:/​/​doi.org/​10.1126/​science.1070958

[44] J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard. Coherent Manipulation of Coupled Electron Spins in Semiconductor Quantum Dots. Science, 309 (5744): 2180-2184, September 2005. 10.1126/​science.1116955.
https:/​/​doi.org/​10.1126/​science.1116955

[45] A. C. Johnson, J. R. Petta, C. M. Marcus, M. P. Hanson, and A. C. Gossard. Singlet-triplet spin blockade and charge sensing in a few-electron double quantum dot. Physical Review B, 72 (16): 165308, October 2005a. 10.1103/​PhysRevB.72.165308.
https:/​/​doi.org/​10.1103/​PhysRevB.72.165308

[46] A. C. Betz, R. Wacquez, M. Vinet, X. Jehl, A. L. Saraiva, M. Sanquer, A. J. Ferguson, and M. F. Gonzalez-Zalba. Dispersively Detected Pauli Spin-Blockade in a Silicon Nanowire Field-Effect Transistor. Nano Letters, 15 (7): 4622-4627, July 2015. 10.1021/​acs.nanolett.5b01306.
https:/​/​doi.org/​10.1021/​acs.nanolett.5b01306

[47] Anderson West, Bas Hensen, Alexis Jouan, Tuomo Tanttu, Chih-Hwan Yang, Alessandro Rossi, M. Fernando Gonzalez-Zalba, Fay Hudson, Andrea Morello, David J. Reilly, and Andrew S. Dzurak. Gate-based single-shot readout of spins in silicon. Nature Nanotechnology, page 1, March 2019. 10.1038/​s41565-019-0400-7.
https:/​/​doi.org/​10.1038/​s41565-019-0400-7

[48] P. Pakkiam, A. V. Timofeev, M. G. House, M. R. Hogg, T. Kobayashi, M. Koch, S. Rogge, and M. Y. Simmons. Single-Shot Single-Gate rf Spin Readout in Silicon. Physical Review X, 8 (4): 041032, November 2018. 10.1103/​PhysRevX.8.041032.
https:/​/​doi.org/​10.1103/​PhysRevX.8.041032

[49] C. H. Yang, W. H. Lim, F. A. Zwanenburg, and A. S. Dzurak. Dynamically controlled charge sensing of a few-electron silicon quantum dot. AIP Advances, 1 (4): 042111, October 2011. 10.1063/​1.3654496.
https:/​/​doi.org/​10.1063/​1.3654496

[50] A. C. Johnson, J. R. Petta, J. M. Taylor, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard. Triplet–singlet spin relaxation via nuclei in a double quantum dot. Nature, 435 (7044): 925-928, June 2005b. 10.1038/​nature03815.
https:/​/​doi.org/​10.1038/​nature03815

[51] V. Srinivasa, H. Xu, and J. M. Taylor. Tunable Spin-Qubit Coupling Mediated by a Multielectron Quantum Dot. Physical Review Letters, 114 (22), June 2015. 10.1103/​PhysRevLett.114.226803.
https:/​/​doi.org/​10.1103/​PhysRevLett.114.226803

[52] Sebastian Mehl, Hendrik Bluhm, and David P. DiVincenzo. Two-qubit couplings of singlet-triplet qubits mediated by one quantum state. Physical Review B, 90 (4), July 2014. 10.1103/​PhysRevB.90.045404.
https:/​/​doi.org/​10.1103/​PhysRevB.90.045404

[53] Patrick Harvey-Collard, N. Tobias Jacobson, Martin Rudolph, Jason Dominguez, Gregory A. Ten Eyck, Joel R. Wendt, Tammy Pluym, John King Gamble, Michael P. Lilly, Michel Pioro-Ladrière, and Malcolm S. Carroll. Coherent coupling between a quantum dot and a donor in silicon. Nature Communications, 8 (1): 1029, October 2017. 10.1038/​s41467-017-01113-2.
https:/​/​doi.org/​10.1038/​s41467-017-01113-2

[54] M. D. Reed, B. M. Maune, R. W. Andrews, M. G. Borselli, K. Eng, M. P. Jura, A. A. Kiselev, T. D. Ladd, S. T. Merkel, I. Milosavljevic, E. J. Pritchett, M. T. Rakher, R. S. Ross, A. E. Schmitz, A. Smith, J. A. Wright, M. F. Gyure, and A. T. Hunter. Reduced Sensitivity to Charge Noise in Semiconductor Spin Qubits via Symmetric Operation. Physical Review Letters, 116 (11): 110402, March 2016. 10.1103/​PhysRevLett.116.110402.
https:/​/​doi.org/​10.1103/​PhysRevLett.116.110402

[55] Daniel Loss and David P. DiVincenzo. Quantum computation with quantum dots. Physical Review A, 57 (1): 120, 1998. 10.1103/​PhysRevA.57.120.
https:/​/​doi.org/​10.1103/​PhysRevA.57.120

[56] T. Meunier, V. E. Calado, and L. M. K. Vandersypen. Efficient controlled-phase gate for single-spin qubits in quantum dots. Physical Review B, 83 (12), March 2011. 10.1103/​PhysRevB.83.121403.
https:/​/​doi.org/​10.1103/​PhysRevB.83.121403

[57] T. F. Watson, S. G. J. Philips, E. Kawakami, D. R. Ward, P. Scarlino, M. Veldhorst, D. E. Savage, M. G. Lagally, Mark Friesen, S. N. Coppersmith, M. A. Eriksson, and L. M. K. Vandersypen. A programmable two-qubit quantum processor in silicon. Nature, 555 (7698): 633-637, February 2018. 10.1038/​nature25766.
https:/​/​doi.org/​10.1038/​nature25766

[58] W. Huang, C. H. Yang, K. W. Chan, T. Tanttu, B. Hensen, R. C. C. Leon, M. A. Fogarty, J. C. C. Hwang, F. E. Hudson, K. M. Itoh, A. Morello, A. Laucht, and A. S. Dzurak. Fidelity benchmarks for two-qubit gates in silicon. Nature, 569 (7757): 532, May 2019. 10.1038/​s41586-019-1197-0.
https:/​/​doi.org/​10.1038/​s41586-019-1197-0

[59] Timothy Alexander Baart, Takafumi Fujita, Christian Reichl, Werner Wegscheider, and Lieven Mark Koenraad Vandersypen. Coherent spin-exchange via a quantum mediator. Nature Nanotechnology, 12 (1): 26-30, January 2017. 10.1038/​nnano.2016.188.
https:/​/​doi.org/​10.1038/​nnano.2016.188

[60] Rifat Ferdous, Kok W. Chan, Menno Veldhorst, J. C. C. Hwang, C. H. Yang, Harshad Sahasrabudhe, Gerhard Klimeck, Andrea Morello, Andrew S. Dzurak, and Rajib Rahman. Interface-induced spin-orbit interaction in silicon quantum dots and prospects for scalability. Physical Review B, 97 (24): 241401, June 2018. 10.1103/​PhysRevB.97.241401.
https:/​/​doi.org/​10.1103/​PhysRevB.97.241401

[61] Yasuhiro Tokura, Wilfred G. van der Wiel, Toshiaki Obata, and Seigo Tarucha. Coherent Single Electron Spin Control in a Slanting Zeeman Field. Physical Review Letters, 96 (4), January 2006. 10.1103/​PhysRevLett.96.047202.
https:/​/​doi.org/​10.1103/​PhysRevLett.96.047202

[62] Andrea Corna, Léo Bourdet, Romain Maurand, Alessandro Crippa, Dharmraj Kotekar-Patil, Heorhii Bohuslavskyi, Romain Laviéville, Louis Hutin, Sylvain Barraud, Xavier Jehl, Maud Vinet, Silvano De Franceschi, Yann-Michel Niquet, and Marc Sanquer. Electrically driven electron spin resonance mediated by spin–valley–orbit coupling in a silicon quantum dot. npj Quantum Information, 4 (1): 6, February 2018. 10.1038/​s41534-018-0059-1.
https:/​/​doi.org/​10.1038/​s41534-018-0059-1

[63] Ryan M. Jock, N. Tobias Jacobson, Patrick Harvey-Collard, Andrew M. Mounce, Vanita Srinivasa, Dan R. Ward, John Anderson, Ron Manginell, Joel R. Wendt, Martin Rudolph, Tammy Pluym, John King Gamble, Andrew D. Baczewski, Wayne M. Witzel, and Malcolm S. Carroll. A silicon metal-oxide-semiconductor electron spin-orbit qubit. Nature Communications, 9 (1): 1768, May 2018. 10.1038/​s41467-018-04200-0.
https:/​/​doi.org/​10.1038/​s41467-018-04200-0

[64] Tuomo Tanttu, Bas Hensen, Kok Wai Chan, Chih Hwan Yang, Wister Wei Huang, Michael Fogarty, Fay Hudson, Kohei Itoh, Dimitrie Culcer, Arne Laucht, Andrea Morello, and Andrew Dzurak. Controlling Spin-Orbit Interactions in Silicon Quantum Dots Using Magnetic Field Direction. Physical Review X, 9 (2): 021028, May 2019. 10.1103/​PhysRevX.9.021028.
https:/​/​doi.org/​10.1103/​PhysRevX.9.021028

[65] J. M. Elzerman, R. Hanson, L. H. Willems van Beveren, B. Witkamp, L. M. K. Vandersypen, and L. P. Kouwenhoven. Single-shot read-out of an individual electron spin in a quantum dot. Nature, 430 (6998): 431-435, July 2004. 10.1038/​nature02693.
https:/​/​doi.org/​10.1038/​nature02693

[66] Patrick Harvey-Collard, Benjamin D'Anjou, Martin Rudolph, N. Tobias Jacobson, Jason Dominguez, Gregory A. Ten Eyck, Joel R. Wendt, Tammy Pluym, Michael P. Lilly, William A. Coish, Michel Pioro-Ladrière, and Malcolm S. Carroll. High-Fidelity Single-Shot Readout for a Spin Qubit via an Enhanced Latching Mechanism. Physical Review X, 8 (2): 021046, May 2018. 10.1103/​PhysRevX.8.021046.
https:/​/​doi.org/​10.1103/​PhysRevX.8.021046

[67] M. A. Fogarty, K. W. Chan, B. Hensen, W. Huang, T. Tanttu, C. H. Yang, A. Laucht, M. Veldhorst, F. E. Hudson, K. M. Itoh, D. Culcer, T. D. Ladd, A. Morello, and A. S. Dzurak. Integrated silicon qubit platform with single-spin addressability, exchange control and single-shot singlet-triplet readout. Nature Communications, 9 (1): 4370, October 2018. 10.1038/​s41467-018-06039-x.
https:/​/​doi.org/​10.1038/​s41467-018-06039-x

[68] Christopher J. Wood and Jay M. Gambetta. Quantification and characterization of leakage errors. Physical Review A, 97 (3), March 2018. 10.1103/​PhysRevA.97.032306.
https:/​/​doi.org/​10.1103/​PhysRevA.97.032306

[69] John Preskill. Fault-tolerant quantum computation. In Introduction to Quantum Computation and Information, pages 213-269. WORLD SCIENTIFIC, October 1998. 10.1142/​9789812385253_0008.
https:/​/​doi.org/​10.1142/​9789812385253_0008

[70] Daniel Gottesman. Stabilizer Codes and Quantum Error Correction. PhD thesis.

[71] Panos Aliferis and Barbara M. Terhal. Fault-tolerant Quantum Computation for Local Leakage Faults. Quantum Info. Comput., 7 (1): 139-156, January 2007. URL http:/​/​dl.acm.org/​citation.cfm?id=2011706.2011715.
http:/​/​dl.acm.org/​citation.cfm?id=2011706.2011715

[72] Austin G. Fowler. Coping with qubit leakage in topological codes. Physical Review A, 88 (4), October 2013. 10.1103/​PhysRevA.88.042308.
https:/​/​doi.org/​10.1103/​PhysRevA.88.042308

[73] M. Suchara, A. W. Cross, and J. M. Gambetta. Leakage suppression in the toric code. 2015 IEEE International Symposium on Information Theory (ISIT), pages 1119-1123, June 2015. 10.1109/​ISIT.2015.7282629.
https:/​/​doi.org/​10.1109/​ISIT.2015.7282629

[74] S. D. Barrett and C. H. W. Barnes. Double-occupation errors induced by orbital dephasing in exchange-interaction quantum gates. Physical Review B, 66 (12): 125318, September 2002. 10.1103/​PhysRevB.66.125318.
https:/​/​doi.org/​10.1103/​PhysRevB.66.125318

[75] K. Wang, C. Payette, Y. Dovzhenko, P. W. Deelman, and J. R. Petta. Charge Relaxation in a Single-Electron Si /​ SiGe Double Quantum Dot. Physical Review Letters, 111 (4), July 2013. 10.1103/​PhysRevLett.111.046801.
https:/​/​doi.org/​10.1103/​PhysRevLett.111.046801

[76] Zhenyu Cai and Simon C. Benjamin. Constructing Smaller Pauli Twirling Sets for Arbitrary Error Channels. Scientific Reports, 9 (1): 1-11, August 2019. 10.1038/​s41598-019-46722-7.
https:/​/​doi.org/​10.1038/​s41598-019-46722-7

[77] Daniel Gottesman. The Heisenberg Representation of Quantum Computers. arXiv:quant-ph/​9807006, July 1998. URL http:/​/​arxiv.org/​abs/​quant-ph/​9807006.
arXiv:quant-ph/9807006

[78] Scott Aaronson and Daniel Gottesman. Improved simulation of stabilizer circuits. Physical Review A, 70 (5): 052328, November 2004. 10.1103/​PhysRevA.70.052328.
https:/​/​doi.org/​10.1103/​PhysRevA.70.052328

[79] Michael R. Geller and Zhongyuan Zhou. Efficient error models for fault-tolerant architectures and the Pauli twirling approximation. Physical Review A, 88 (1), July 2013. 10.1103/​PhysRevA.88.012314.
https:/​/​doi.org/​10.1103/​PhysRevA.88.012314

[80] Mauricio Gutiérrez, Lukas Svec, Alexander Vargo, and Kenneth R. Brown. Approximation of realistic errors by Clifford channels and Pauli measurements. Physical Review A, 87 (3), March 2013. 10.1103/​PhysRevA.87.030302.
https:/​/​doi.org/​10.1103/​PhysRevA.87.030302

[81] Mauricio Gutiérrez and Kenneth R. Brown. Comparison of a quantum error-correction threshold for exact and approximate errors. Physical Review A, 91 (2): 022335, February 2015. 10.1103/​PhysRevA.91.022335.
https:/​/​doi.org/​10.1103/​PhysRevA.91.022335

[82] Ashley M. Stephens. Fault-tolerant thresholds for quantum error correction with the surface code. Physical Review A, 89 (2), February 2014. 10.1103/​PhysRevA.89.022321.
https:/​/​doi.org/​10.1103/​PhysRevA.89.022321

[83] David C. McKay, Christopher J. Wood, Sarah Sheldon, Jerry M. Chow, and Jay M. Gambetta. Efficient Z-Gates for Quantum Computing. Physical Review A, 96 (2), August 2017. 10.1103/​PhysRevA.96.022330.
https:/​/​doi.org/​10.1103/​PhysRevA.96.022330

[84] Guoji Zheng, Nodar Samkharadze, Marc L. Noordam, Nima Kalhor, Delphine Brousse, Amir Sammak, Giordano Scappucci, and Lieven M. K. Vandersypen. Rapid gate-based spin read-out in silicon using an on-chip resonator. Nature Nanotechnology, 14 (8): 742-746, August 2019. 10.1038/​s41565-019-0488-9.
https:/​/​doi.org/​10.1038/​s41565-019-0488-9

[85] R. Raussendorf, J. Harrington, and K. Goyal. Topological fault-tolerance in cluster state quantum computation. New Journal of Physics, 9 (6): 199, 2007. 10.1088/​1367-2630/​9/​6/​199.
https:/​/​doi.org/​10.1088/​1367-2630/​9/​6/​199

[86] J. C. C. Hwang, C. H. Yang, M. Veldhorst, N. Hendrickx, M. A. Fogarty, W. Huang, F. E. Hudson, A. Morello, and A. S. Dzurak. Impact of $g$-factors and valleys on spin qubits in a silicon double quantum dot. Physical Review B, 96 (4): 045302, July 2017. 10.1103/​PhysRevB.96.045302.
https:/​/​doi.org/​10.1103/​PhysRevB.96.045302

[87] Filip K. Malinowski, Frederico Martins, Thomas B. Smith, Stephen D. Bartlett, Andrew C. Doherty, Peter D. Nissen, Saeed Fallahi, Geoffrey C. Gardner, Michael J. Manfra, Charles M. Marcus, and Ferdinand Kuemmeth. Spin of a Multielectron Quantum Dot and Its Interaction with a Neighboring Electron. Physical Review X, 8 (1): 011045, March 2018. 10.1103/​PhysRevX.8.011045.
https:/​/​doi.org/​10.1103/​PhysRevX.8.011045

[88] Vladimir Kolmogorov. Blossom V: A new implementation of a minimum cost perfect matching algorithm. Mathematical Programming Computation, 1 (1): 43-67, July 2009. 10.1007/​s12532-009-0002-8.
https:/​/​doi.org/​10.1007/​s12532-009-0002-8

[89] https:/​/​github.com/​czydbb/​SurfaceCodeModule. URL https:/​/​github.com/​czydbb/​SurfaceCodeModule.
https:/​/​github.com/​czydbb/​SurfaceCodeModule

[90] Sergey Bravyi, Martin Suchara, and Alexander Vargo. Efficient algorithms for maximum likelihood decoding in the surface code. Physical Review A, 90 (3), September 2014. 10.1103/​PhysRevA.90.032326.
https:/​/​doi.org/​10.1103/​PhysRevA.90.032326

Cited by

[1] S. Schaal, I. Ahmed, J. A. Haigh, L. Hutin, B. Bertrand, S. Barraud, M. Vinet, C. -M. Lee, N. Stelmashenko, J. W. A. Robinson, J. Y. Qiu, S. Hacohen-Gourgy, I. Siddiqi, M. F. Gonzalez-Zalba, and J. J. L. Morton, "Fast gate-based readout of silicon quantum dots using Josephson parametric amplification", arXiv:1907.09429.

[2] Xiaosi Xu, Simon C. Benjamin, and Xiao Yuan, "Variational circuit compiler for quantum error correction", arXiv:1911.05759.

The above citations are from SAO/NASA ADS (last updated successfully 2020-01-23 08:12:21). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2020-01-23 08:12:19).