Optimization of the surface code design for Majorana-based qubits

Rui Chao1, Michael E. Beverland2, Nicolas Delfosse2, and Jeongwan Haah2

1University of Southern California, Los Angeles, CA, USA
2Microsoft Quantum and Microsoft Research, Redmond, WA, USA

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The surface code is a prominent topological error-correcting code exhibiting high fault-tolerance accuracy thresholds. Conventional schemes for error correction with the surface code place qubits on a planar grid and assume native CNOT gates between the data qubits with nearest-neighbor ancilla qubits.

Here, we present surface code error-correction schemes using $\textit{only}$ Pauli measurements on single qubits and on pairs of nearest-neighbor qubits. In particular, we provide several qubit layouts that offer favorable trade-offs between qubit overhead, circuit depth and connectivity degree. We also develop minimized measurement sequences for syndrome extraction, enabling reduced logical error rates and improved fault-tolerance thresholds.

Our work applies to topologically protected qubits realized with Majorana zero modes and to similar systems in which multi-qubit Pauli measurements rather than CNOT gates are the native operations.

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[1] A. Y. Kitaev, Ann. Phys. 303, 2 (2003), quant-ph/​9707021.

[2] E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, J. Math. Phys. 43, 4452 (2002), quant-ph/​0110143.

[3] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Phys. Rev. A 86, 032324 (2012), 1208.0928.

[4] R. Raussendorf and J. Harrington, Phys. Rev. Lett. 98, 190504 (2007), quant-ph/​0610082.

[5] A. G. Fowler, A. M. Stephens, and P. Groszkowski, Phys. Rev. A 80, 052312 (2009), 0803.0272.

[6] N. Delfosse and G. Zémor, Physical Review Research 2, 033042 (2020), 1703.01517.

[7] N. Delfosse and N. H. Nickerson, 2017, 1709.06218.

[8] T. Karzig, C. Knapp, R. M. Lutchyn, P. Bonderson, M. B. Hastings, C. Nayak, J. Alicea, K. Flensberg, S. Plugge, Y. Oreg, C. M. Marcus, and M. H. Freedman, Phys. Rev. B 95, 235305 (2017), 1610.05289.

[9] C. Knapp, M. Beverland, D. I. Pikulin, and T. Karzig, Quantum 2, 88 (2018), 1806.01275.

[10] Y. Li, Physical Review Letters 117, 120403 (2016), 1512.05089.

[11] S. Plugge, L. Landau, E. Sela, A. Altland, K. Flensberg, and R. Egger, Phys. Rev. B 94, 174514 (2016), 1606.08408.

[12] D. Litinski, M. S. Kesselring, J. Eisert, and F. von Oppen, Phys. Rev. X 7, 031048 (2017), 1704.01589.

[13] A. G. Fowler, D. S. Wang, and L. C. L. Hollenberg, Quant. Info. Comput. 11, 8 (2011), 1004.0255.

[14] M. Newman, L. A. de Castro, and K. R. Brown, Quantum 4, 295 (2020), 1909.11817.

[15] A. G. Fowler, 2013, 1310.0863.

[16] N. Delfosse and J.-P. Tillich, in 2014 IEEE Int. Symp. Info. (IEEE, 2014) pp. 1071–1075, 1401.6975.

[17] S. Huang and K. R. Brown, Phys. Rev. A 101, 042312 (2020), 1911.11317.

[18] S. Huang, M. Newman, and K. R. Brown, Phys. Rev. A 102, 012419 (2020), 2004.04693.

[19] J. MacWilliams, Bell Syst. Tech. J. 42, 79 (1963).

[20] S. Bravyi and A. Vargo, Phys. Rev. A 88, 062308 (2013), 1308.6270.

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The above citations are from Crossref's cited-by service (last updated successfully 2022-11-30 04:37:24) and SAO/NASA ADS (last updated successfully 2022-11-30 04:37:25). The list may be incomplete as not all publishers provide suitable and complete citation data.