Optimization of the surface code design for Majorana-based qubits

Rui Chao; Michael E. Beverland; Nicolas Delfosse; Jeongwan Haah

doi:10.22331/q-2020-10-28-352

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

Published:	2020-10-28, volume 4, page 352
Eprint:	arXiv:2007.00307v2
Doi:	https://doi.org/10.22331/q-2020-10-28-352
Citation:	Quantum 4, 352 (2020).

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Abstract

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.

Featured image: The windmill layout for implementing the distance-5 surface code with measurement-based qubits. Black nodes are data qubits; at the center of each X-type or Z-type plaquette, there is a green or yellow ancilla qubit, respectively. The measurement of a plaquette will require two-qubit measurements between the data qubits and the center ancilla (black edges), and between the center ancilla and its neighbor ancilla (blue edge).

► BibTeX data

@article{Chao2020optimizationof,
  doi = {10.22331/q-2020-10-28-352},
  url = {https://doi.org/10.22331/q-2020-10-28-352},
  title = {Optimization of the surface code design for {M}ajorana-based qubits},
  author = {Chao, Rui and Beverland, Michael E. and Delfosse, Nicolas and Haah, Jeongwan},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {352},
  month = oct,
  year = {2020}
}

► References

[1] A. Y. Kitaev, Ann. Phys. 303, 2 (2003), quant-ph/9707021.
https://doi.org/10.1016/S0003-4916(02)00018-0
arXiv:quant-ph/9707021

[2] E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, J. Math. Phys. 43, 4452 (2002), quant-ph/0110143.
https://doi.org/10.1063/1.1499754
arXiv:quant-ph/0110143

[3] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Phys. Rev. A 86, 032324 (2012), 1208.0928.
https://doi.org/10.1103/PhysRevA.86.032324
arXiv:1208.0928

[4] R. Raussendorf and J. Harrington, Phys. Rev. Lett. 98, 190504 (2007), quant-ph/0610082.
https://doi.org/10.1103/PhysRevLett.98.190504
arXiv:quant-ph/0610082

[5] A. G. Fowler, A. M. Stephens, and P. Groszkowski, Phys. Rev. A 80, 052312 (2009), 0803.0272.
https://doi.org/10.1103/PhysRevA.80.052312
arXiv:0803.0272

[6] N. Delfosse and G. Zémor, Physical Review Research 2, 033042 (2020), 1703.01517.
https://doi.org/10.1103/PhysRevResearch.2.033042
arXiv:1703.01517

[7] N. Delfosse and N. H. Nickerson, 2017, 1709.06218.
arXiv: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.
https://doi.org/10.1103/PhysRevB.95.235305
arXiv:1610.05289

[9] C. Knapp, M. Beverland, D. I. Pikulin, and T. Karzig, Quantum 2, 88 (2018), 1806.01275.
https://doi.org/10.22331/q-2018-09-03-88
arXiv:1806.01275

[10] Y. Li, Physical Review Letters 117, 120403 (2016), 1512.05089.
https://doi.org/10.1103/PhysRevLett.117.120403
arXiv:1512.05089

[11] S. Plugge, L. Landau, E. Sela, A. Altland, K. Flensberg, and R. Egger, Phys. Rev. B 94, 174514 (2016), 1606.08408.
https://doi.org/10.1103/PhysRevB.94.174514
arXiv:1606.08408

[12] D. Litinski, M. S. Kesselring, J. Eisert, and F. von Oppen, Phys. Rev. X 7, 031048 (2017), 1704.01589.
https://doi.org/10.1103/PhysRevX.7.031048
arXiv:1704.01589

[13] A. G. Fowler, D. S. Wang, and L. C. L. Hollenberg, Quant. Info. Comput. 11, 8 (2011), 1004.0255.
https://doi.org/10.26421/QIC11.1-2
arXiv:1004.0255

[14] M. Newman, L. A. de Castro, and K. R. Brown, Quantum 4, 295 (2020), 1909.11817.
https://doi.org/10.22331/q-2020-07-13-295
arXiv:1909.11817

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

[16] N. Delfosse and J.-P. Tillich, in 2014 IEEE Int. Symp. Info. (IEEE, 2014) pp. 1071–1075, 1401.6975.
https://doi.org/10.1109/ISIT.2014.6874997
arXiv:1401.6975

[17] S. Huang and K. R. Brown, Phys. Rev. A 101, 042312 (2020), 1911.11317.
https://doi.org/10.1103/PhysRevA.101.042312
arXiv:1911.11317

[18] S. Huang, M. Newman, and K. R. Brown, Phys. Rev. A 102, 012419 (2020), 2004.04693.
https://doi.org/10.1103/PhysRevA.102.012419
arXiv:2004.04693

[19] J. MacWilliams, Bell Syst. Tech. J. 42, 79 (1963).
https://doi.org/10.1002/j.1538-7305.1963.tb04003.x

[20] S. Bravyi and A. Vargo, Phys. Rev. A 88, 062308 (2013), 1308.6270.
https://doi.org/10.1103/PhysRevA.88.062308
arXiv:1308.6270

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[19] Linnea Grans-Samuelsson, Ryan V. Mishmash, David Aasen, Christina Knapp, Bela Bauer, Brad Lackey, Marcus P. da Silva, and Parsa Bonderson, "Improved Pairwise Measurement-Based Surface Code", arXiv:2310.12981, (2023).

[20] David Aasen, Jeongwan Haah, Zhi Li, and Roger S. K. Mong, "Measurement Quantum Cellular Automata and Anomalies in Floquet Codes", arXiv:2304.01277, (2023).

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