Modeling noise and error correction for Majorana-based quantum computing

Christina Knapp1, Michael Beverland2, Dmitry I. Pikulin3, and Torsten Karzig3

1Department of Physics, University of California, Santa Barbara, California 93106 USA
2Station Q Quantum Architectures and Computation Group, Microsoft Research, Redmond, Washington 98052 USA
3Station Q, Microsoft Research, Santa Barbara, California 93106-6105 USA

Majorana-based quantum computing seeks to use the non-local nature of Majorana zero modes to store and manipulate quantum information in a topologically protected way. While noise is anticipated to be significantly suppressed in such systems, finite temperature and system size result in residual errors. In this work, we connect the underlying physical error processes in Majorana-based systems to the noise models used in a fault tolerance analysis. Standard qubit-based noise models built from Pauli operators do not capture leading order noise processes arising from quasiparticle poisoning events, thus it is not obvious $\textit{a priori}$ that such noise models can be usefully applied to a Majorana-based system. We develop stochastic Majorana noise models that are generalizations of the standard qubit-based models and connect the error probabilities defining these models to parameters of the physical system. Using these models, we compute pseudo-thresholds for the $d=5$ Bacon-Shor subsystem code. Our results emphasize the importance of correlated errors induced in multi-qubit measurements. Moreover, we find that for sufficiently fast quasiparticle relaxation the errors are well described by Pauli operators. This work bridges the divide between physical errors in Majorana-based quantum computing architectures and the significance of these errors in a quantum error correcting code.

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[1] A. Y. Kitaev. Fault-tolerant quantum computation by anyons. Ann. Phys., 303: 2, January 2003. 10.1016/​S0003-4916(02)00018-0.

[2] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys., 80: 1083, July 2008. 10.1103/​RevModPhys.80.1083.

[3] A. Y. Kitaev. Unpaired majorana fermions in quantum wires. Phys. Usp., 44: 131, October 2001. 10.1070/​1063-7869/​44/​10S/​S29.

[4] R. M. Lutchyn, J. D. Sau, and S. Das Sarma. Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures. Phys. Rev. Lett., 105 (7): 077001, August 2010. 10.1103/​PhysRevLett.105.077001.

[5] Y. Oreg, G. Refael, and F. von Oppen. Helical Liquids and Majorana Bound States in Quantum Wires. Phys. Rev. Lett., 105 (17): 177002, October 2010. 10.1103/​PhysRevLett.105.177002.

[6] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, and L. P. Kouwenhoven. Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices. Science, 336: 1003, 2012. 10.1126/​science.1222360.

[7] L. P. Rokhinson, X. Liu, and J. K. Furdyna. The fractional a.c. Josephson effect in a semiconductor-superconductor nanowire as a signature of Majorana particles. Nat. Phys., 8: 795, 2012. ISSN 1745-2473. 10.1038/​nphys2429.

[8] M. T. Deng, C. L. Yu, G. Y. Huang, M. Larsson, P. Caroff, and H. Q. Xu. Anomalous zero-bias conductance peak in a nb-insb nanowire-nb hybrid device. Nano Lett., 12 (12): 6414, 2012. 10.1021/​nl303758w.

[9] A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, and H. Shtrikman. Zero-bias peaks and splitting in an Al-InAs nanowire topological superconductor as a signature of Majorana fermions. Nat. Phys., 8: 887, 2012. 10.1038/​nphys2479.

[10] A. D. K. Finck, D. J. Van Harlingen, P. K. Mohseni, K. Jung, and X. Li. Anomalous Modulation of a Zero-Bias Peak in a Hybrid Nanowire-Superconductor Device. Phys. Rev. Lett., 110: 126406, Mar 2013. 10.1103/​PhysRevLett.110.126406.

[11] H. O. H. Churchill, V. Fatemi, K. Grove-Rasmussen, M. T. Deng, P. Caroff, H. Q. Xu, and C. M. Marcus. Superconductor-nanowire devices from tunneling to the multichannel regime: Zero-bias oscillations and magnetoconductance crossover. Phys. Rev. B, 87 (24): 241401, June 2013. 10.1103/​PhysRevB.87.241401.

[12] S. M. Albrecht, A. P. Higginbotham, M. Madsen, F. Kuemmeth, T. S. Jespersen, J. Nygård, P. Krogstrup, and C. M. Marcus. Exponential protection of zero modes in Majorana islands. Nature, 531: 206, March 2016. 10.1038/​nature17162.

[13] M. T. Deng, S. Vaitiekėnas, E. B. Hansen, J. Danon, M. Leijnse, K. Flensberg, J. Nygård, P. Krogstrup, and C. M. Marcus. Majorana bound state in a coupled quantum-dot hybrid-nanowire system. Science, 354 (6319): 1557, December 2016. ISSN 0036-8075, 1095-9203. 10.1126/​science.aaf3961.

[14] R. M. Lutchyn, E. P. a. M. Bakkers, L. P. Kouwenhoven, P. Krogstrup, C. M. Marcus, and Y. Oreg. Majorana zero modes in superconductor–semiconductor heterostructures. Nat. Rev. Mater., 3 (5): 52, May 2018. ISSN 2058-8437. 10.1038/​s41578-018-0003-1.

[15] L. A. Landau, S. Plugge, E. Sela, A. Altland, S. M. Albrecht, and R. Egger. Towards Realistic Implementations of a Majorana Surface Code. Phys. Rev. Lett., 116 (5): 050501, February 2016. 10.1103/​PhysRevLett.116.050501.

[16] G. Goldstein and C. Chamon. Decay rates for topological memories encoded with Majorana fermions. Phys. Rev. B, 84 (20): 205109, November 2011. 10.1103/​PhysRevB.84.205109.

[17] Jan Carl Budich, Stefan Walter, and Björn Trauzettel. Failure of protection of Majorana based qubits against decoherence. Phys. Rev. B, 85 (12): 121405, March 2012. 10.1103/​PhysRevB.85.121405.

[18] Diego Rainis and Daniel Loss. Majorana qubit decoherence by quasiparticle poisoning. Phys. Rev. B, 85 (17): 174533, May 2012. 10.1103/​PhysRevB.85.174533.

[19] L. Mazza, M. Rizzi, M. D. Lukin, and J. I. Cirac. Robustness of quantum memories based on Majorana zero modes. Phys. Rev. B, 88 (20): 205142, 2013. 10.1103/​PhysRevB.88.205142.

[20] Ying Hu, Zi Cai, Mikhail A. Baranov, and Peter Zoller. Majorana fermions in noisy Kitaev wires. Phys. Rev. B, 92 (16): 165118, 2015. 10.1103/​PhysRevB.92.165118.

[21] Fabio L. Pedrocchi, N. E. Bonesteel, and David P. DiVincenzo. Monte Carlo studies of the self-correcting properties of the Majorana quantum error correction code under braiding. Phys. Rev. B, 92 (11): 115441, 2015. 10.1103/​PhysRevB.92.115441.

[22] Matteo Ippoliti, Matteo Rizzi, Vittorio Giovannetti, and Leonardo Mazza. Quantum memories with zero-energy majorana modes and experimental constraints. Phys. Rev. A, 93: 062325, Jun 2016. 10.1103/​PhysRevA.93.062325.

[23] C. Knapp, T. Karzig, R. M. Lutchyn, and C. Nayak. Dephasing of Majorana-based qubits. Phys. Rev. B, 97 (12): 125404, March 2018. 10.1103/​PhysRevB.97.125404.

[24] S. Bravyi, B. M. Terhal, and B. Leemhuis. Majorana fermion codes. New J. Phys., 12 (8): 083039, August 2010. 10.1088/​1367-2630/​12/​8/​083039.

[25] B. M. Terhal, F. Hassler, and D. P. DiVincenzo. From majorana fermions to topological order. Phys. Rev. Lett., 108: 260504, Jun 2012. 10.1103/​PhysRevLett.108.260504.

[26] C. G. Brell, S. Burton, G. Dauphinais, S. T. Flammia, and D. Poulin. Thermalization, Error Correction, and Memory Lifetime for Ising Anyon Systems. Phys. Rev. X, 4 (3): 031058, July 2014. 10.1103/​PhysRevX.4.031058.

[27] S. Vijay, T. H. Hsieh, and L. Fu. Majorana Fermion Surface Code for Universal Quantum Computation. Phys. Rev. X, 5 (4): 041038, October 2015. 10.1103/​PhysRevX.5.041038.

[28] Ying Li. Noise threshold and resource cost of fault-tolerant quantum computing with majorana fermions in hybrid systems. Phys. Rev. Lett., 117: 120403, Sep 2016. 10.1103/​PhysRevLett.117.120403.

[29] S. Vijay and L. Fu. Physical implementation of a Majorana fermion surface code for fault-tolerant quantum computation. Phys. Scr., (1): 014002, December 2016. 10.1088/​0031-8949/​T168/​1/​014002.

[30] S. Plugge, L. A. Landau, E. Sela, A. Altland, K. Flensberg, and R. Egger. Roadmap to Majorana surface codes. Phys. Rev. B, 94 (17): 174514, November 2016. 10.1103/​PhysRevB.94.174514.

[31] Y. Li. Fault-tolerant fermionic quantum computation based on color code. Phys. Rev. A, 98 (1): 012336, July 2018. 10.1103/​PhysRevA.98.012336.

[32] Daniel Litinski, Markus S. Kesselring, Jens Eisert, and Felix von Oppen. Combining Topological Hardware and Topological Software: Color-Code Quantum Computing with Topological Superconductor Networks. Phys. Rev. X, 7 (3): 031048, September 2017. 10.1103/​PhysRevX.7.031048.

[33] D. Litinski and F. von Oppen. Braiding by Majorana tracking and long-range CNOT gates with color codes. Phys. Rev. B, 96 (20): 205413, November 2017. 10.1103/​PhysRevB.96.205413.

[34] M. B. Hastings. Small Majorana Fermion Codes. March 2017. https:/​/​​1703.00612.

[35] S. Vijay and L. Fu. Quantum Error Correction for Complex and Majorana Fermion Qubits. March 2017. https:/​/​​1703.00459.

[36] Daniel Litinski and Felix von Oppen. Quantum computing with Majorana fermion codes. Phys. Rev. B, 97 (20): 205404, May 2018. 10.1103/​PhysRevB.97.205404.

[37] Dave Bacon. Operator quantum error-correcting subsystems for self-correcting quantum memories. Phys. Rev. A, 73: 012340, Jan 2006. 10.1103/​PhysRevA.73.012340.

[38] D. W. Kribs, R. Laflamme, D. Poulin, and M. Lesosky. Operator quantum error correction. April 2005. https:/​/​​quant-ph/​0504189.

[39] P. W. Shor. Fault-tolerant quantum computation. May 1996. https:/​/​​quant-ph/​9605011.

[40] S. Plugge, A. Rasmussen, R. Egger, and K. Flensberg. Majorana box qubits. New J. Phys., 19 (1): 012001, January 2017. 10.1088/​1367-2630/​aa54e1.

[41] Torsten Karzig, Christina Knapp, Roman M. Lutchyn, Parsa Bonderson, Matthew B. Hastings, Chetan Nayak, Jason Alicea, Karsten Flensberg, Stephan Plugge, Yuval Oreg, Charles M. Marcus, and Michael H. Freedman. Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes. Phys. Rev. B, 95 (23): 235305, June 2017. 10.1103/​PhysRevB.95.235305.

[42] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, New York, NY, USA, 10th edition, 2011. ISBN 1107002176, 9781107002173. 10.1017/​CBO9780511976667.

[43] R. Raussendorf and J. Harrington. Fault-Tolerant Quantum Computation with High Threshold in Two Dimensions. Phys. Rev. Lett., 98 (19): 190504, May 2007. 10.1103/​PhysRevLett.98.190504.

[44] David S. Wang, Austin G. Fowler, and Lloyd C. L. Hollenberg. Surface code quantum computing with error rates over 1%. Phys. Rev. A, 83: 020302, Feb 2011. 10.1103/​PhysRevA.83.020302.

[45] R. M. Lutchyn and L. I. Glazman. Kinetics of quasiparticle trapping in a Cooper-pair box. Phys. Rev. B, 75 (18): 184520, May 2007. 10.1103/​PhysRevB.75.184520.

[46] A. A. Clerk, M. H. Devoret, S. M. Girvin, Florian Marquardt, and R. J. Schoelkopf. Introduction to quantum noise, measurement, and amplification. Rev. Mod. Phys., 82 (2): 1155, April 2010. 10.1103/​RevModPhys.82.1155.

[47] R. Chao and B. W. Reichardt. Fault-tolerant quantum computation with few qubits. May 2017. https:/​/​​1705.05365.

[48] Christopher Chamberland and Michael E. Beverland. Flag fault-tolerant error correction with arbitrary distance codes. Quantum, 2: 53, February 2018. ISSN 2521-327X. 10.22331/​q-2018-02-08-53.

[49] Austin G. Fowler, Ashley M. Stephens, and Peter Groszkowski. High-threshold universal quantum computation on the surface code. Phys. Rev. A, 80: 052312, Nov 2009. 10.1103/​PhysRevA.80.052312.

[50] David Poulin. Stabilizer formalism for operator quantum error correction. Phys. Rev. Lett., 95: 230504, Dec 2005. 10.1103/​PhysRevLett.95.230504.

[51] D. Gottesman. An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation. April 2009. https:/​/​​0904.2557.

[52] A. M. Steane. Active Stabilization, Quantum Computation, and Quantum State Synthesis. Phys. Rev. Lett., 78: 2252, March 1997. 10.1103/​PhysRevLett.78.2252.

[53] J. W. Harrington. Analysis of quantum error-correcting codes: Symplectic lattice codes and toric codes. PhD thesis, California Institute of Technology, 2004. https:/​/​​1747.

[54] Y. Tomita and K. M. Svore. Low-distance surface codes under realistic quantum noise. Phys. Rev. A, 90 (6): 062320, December 2014. 10.1103/​PhysRevA.90.062320.

[55] Andrew W. Cross, David P. Divincenzo, and Barbara M. Terhal. A comparative code study for quantum fault tolerance. Quantum Info. Comput., 9 (7): 541-572, July 2009. ISSN 1533-7146. http:/​/​​citation.cfm?id=2011814.2011815.

[56] F. Hassler, A. R. Akhmerov, and C. W. J. Beenakker. The top-transmon: a hybrid superconducting qubit for parity-protected quantum computation. New J. Phys., 13 (9): 095004, September 2011. 10.1088/​1367-2630/​13/​9/​095004.

[57] B. van Heck, A. R. Akhmerov, F. Hassler, M. Burrello, and C. W. J. Beenakker. Coulomb-assisted braiding of Majorana fermions in a Josephson junction array. New J. Phys., 14 (3): 035019, March 2012. 10.1088/​1367-2630/​14/​3/​035019.

[58] T. Hyart, B. van Heck, I. C. Fulga, M. Burrello, A. R. Akhmerov, and C. W. J. Beenakker. Flux-controlled quantum computation with Majorana fermions. Phys. Rev. B, 88 (3): 035121, July 2013. 10.1103/​PhysRevB.88.035121.

[59] D. Aasen, M. Hell, R. V. Mishmash, A. Higginbotham, J. Danon, M. Leijnse, T. S. Jespersen, J. A. Folk, C. M. Marcus, K. Flensberg, and J. Alicea. Milestones Toward Majorana-Based Quantum Computing. Phys. Rev. X, 6 (3): 031016, July 2016. 10.1103/​PhysRevX.6.031016.

[60] M. Cheng, V. Galitski, and S. Das Sarma. Nonadiabatic effects in the braiding of non-Abelian anyons in topological superconductors. Phys. Rev. B, 84 (10): 104529, September 2011. 10.1103/​PhysRevB.84.104529.

[61] T. Karzig, F. Pientka, G. Refael, and F. von Oppen. Shortcuts to non-Abelian braiding. Phys. Rev. B, 91 (20): 201102, May 2015. 10.1103/​PhysRevB.91.201102.

[62] M. S. Scheurer and A. Shnirman. Nonadiabatic processes in Majorana qubit systems. Phys. Rev. B, 88 (6): 064515, August 2013. 10.1103/​PhysRevB.88.064515.

[63] M. Hell, J. Danon, K. Flensberg, and M. Leijnse. Time scales for Majorana manipulation using Coulomb blockade in gate-controlled superconducting nanowires. Phys. Rev. B, 94 (3): 035424, July 2016. 10.1103/​PhysRevB.94.035424.

[64] Christina Knapp, Michael Zaletel, Dong E. Liu, Meng Cheng, Parsa Bonderson, and Chetan Nayak. The nature and correction of diabatic errors in anyon braiding. Phys. Rev. X, 6: 041003, Oct 2016. 10.1103/​PhysRevX.6.041003.

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