Characterization of errors in a CNOT between surface code patches

Bálint Domokos; Áron Márton; János K. Asbóth

doi:10.22331/q-2024-12-27-1577

Characterization of errors in a CNOT between surface code patches

Bálint Domokos¹, Áron Márton¹, and János K. Asbóth^1,2

¹Department of Theoretical Physics, Institute of Physics, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary
²HUN-REN Wigner Research Centre for Physics, H-1525 Budapest, P.O. Box 49., Hungary

Published:	2024-12-27, volume 8, page 1577
Eprint:	arXiv:2405.05337v3
Doi:	https://doi.org/10.22331/q-2024-12-27-1577
Citation:	Quantum 8, 1577 (2024).

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Abstract

As current experiments already realize small quantum circuits on error corrected qubits, it is important to fully understand the effect of physical errors on the logical error channels of these fault-tolerant circuits. Here, we investigate a lattice-surgery-based CNOT operation between two surface code patches under phenomenological error models. (i) For two-qubit logical Pauli measurements – the elementary building block of the CNOT – we optimize the number of stabilizer measurement rounds, usually taken equal to $d$, the size (code distance) of each patch. We find that the optimal number can be greater or smaller than $d$, depending on the rate of physical and readout errors, and the separation between the code patches. (ii) We fully characterize the two-qubit logical error channel of the lattice-surgery-based CNOT. We find a symmetry of the CNOT protocol, that results in a symmetry of the logical error channel. We also find that correlations between X and Z errors on the logical level are suppressed under minimum weight decoding.

Featured image: The spacetime diagram of the CNOT has a symmetry transformation. First, time reversal; then, mirror reflection, shown here; finally, swapping the $X$ and $Z$ labels (colours), resulting in an identical spacetime diagram as initially.

Popular summary

Running practical algorithms on quantum computers requires very high precision, which can only be achieved using quantum error correction. Perhaps the most promising approach is using the so-called surface code, storing logical qubits in “patches” consisting of many physical qubits, and realizing logic gates via lattice surgery. In this work we investigated theoretically and numerically the effect of local errors (bit-flip and readout error) on a lattice-surgery-based CNOT operation. (1) We determined the logical qubit channels for noisy operations, and found that the symmetry of the CNOT operation appears in two logical error channels. (2) We also optimized , using Clifford simulation, the number of measurement rounds during a two-qubit logical Pauli measurement, which is the basic building block of the CNOT.

► BibTeX data

@article{Domokos2024characterizationof,
  doi = {10.22331/q-2024-12-27-1577},
  url = {https://doi.org/10.22331/q-2024-12-27-1577},
  title = {Characterization of errors in a {CNOT} between surface code patches},
  author = {Domokos, B{\'{a}}lint and M{\'{a}}rton, {\'{A}}ron and Asb{\'{o}}th, J{\'{a}}nos K.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1577},
  month = dec,
  year = {2024}
}

► References

[1] A.Yu. Kitaev. ``Fault-tolerant quantum computation by anyons''. Annals of Physics 303, 2–30 (2003).
https://doi.org/10.1016/s0003-4916(02)00018-0

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

[3] Rajeev Acharya et al. ``Suppressing quantum errors by scaling a surface code logical qubit''. Nature 614, 676 – 681 (2022).
https://doi.org/10.1038/s41586-022-05434-1

[4] Dolev Bluvstein, Harry Levine, Giulia Semeghini, Tout T. Wang, Sepehr Ebadi, Marcin Kalinowski, Alexander Keesling, Nishad Maskara, Hannes Pichler, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin. ``A quantum processor based on coherent transport of entangled atom arrays''. Nature 604, 451–456 (2022).
https://doi.org/10.1038/s41586-022-04592-6

[5] J. F. Marques, B. M. Varbanov, M. S. Moreira, H. Ali, N. Muthusubramanian, C. Zachariadis, F. Battistel, M. Beekman, N. Haider, W. Vlothuizen, A. Bruno, B. M. Terhal, and L. DiCarlo. ``Logical-qubit operations in an error-detecting surface code''. Nature Physics 18, 80–86 (2021).
https://doi.org/10.1038/s41567-021-01423-9

[6] Sebastian Krinner, Nathan Lacroix, Ants Remm, Agustin Di Paolo, Elie Genois, Catherine Leroux, Christoph Hellings, Stefania Lazar, Francois Swiadek, Johannes Herrmann, et al. ``Realizing repeated quantum error correction in a distance-three surface code''. Nature 605, 669–674 (2022).
https://doi.org/10.1038/s41586-022-04566-8

[7] Youwei Zhao et al. ``Realization of an error-correcting surface code with superconducting qubits''. Physical Review Letters 129 (2022).
https://doi.org/10.1103/physrevlett.129.030501

[8] Dolev Bluvstein et al. ``Logical quantum processor based on reconfigurable atom arrays''. Nature (2023).
https://doi.org/10.1038/s41586-023-06927-3

[9] Jiaxuan Zhang, Zhao-Yun Chen, Yun-Jie Wang, Bin-Han Lu, Hai-Feng Zhang, Jia-Ning Li, Peng Duan, Yu-Chun Wu, and Guo-Ping Guo. ``Demonstrating a universal logical gate set on a superconducting quantum processor'' (2024). arXiv:2405.09035.
arXiv:2405.09035

[10] Bence Hetényi and James R Wootton. ``Creating entangled logical qubits in the heavy-hex lattice with topological codes'' (2024). arXiv:2404.15989.
https://doi.org/10.1103/PRXQuantum.5.040334
arXiv:2404.15989

[11] Daniel Honciuc Menendez, Annie Ray, and Michael Vasmer. ``Implementing fault-tolerant non-clifford gates using the [[8, 3, 2]] color code'' (2023). arXiv:2309.08663.
https://doi.org/10.1103/PhysRevA.109.062438
arXiv:2309.08663

[12] Benjamin J. Brown, Katharina Laubscher, Markus S. Kesselring, and James R. Wootton. ``Poking holes and cutting corners to achieve Clifford gates with the surface code''. Physical Review X 7 (2017).
https://doi.org/10.1103/physrevx.7.021029

[13] P.W. Shor. ``Fault-tolerant quantum computation''. In Proceedings of 37th Conference on Foundations of Computer Science. Pages 56–65. (1996).
https://doi.org/10.1109/SFCS.1996.548464

[14] CWJ Beenakker, DP DiVincenzo, C Emary, and M Kindermann. ``Charge detection enables free-electron quantum computation''. Physical Review Letters 93, 020501 (2004).
https://doi.org/10.1103/physrevlett.93.020501

[15] Dominic Horsman, Austin G Fowler, Simon Devitt, and Rodney Van Meter. ``Surface code quantum computing by lattice surgery''. New Journal of Physics 14, 123011 (2012).
https://doi.org/10.1088/1367-2630/14/12/123011

[16] Daniel Litinski. ``A game of surface codes: Large-scale quantum computing with lattice surgery''. Quantum 3, 128 (2019).
https://doi.org/10.22331/q-2019-03-05-128

[17] Sergey Bravyi and Alexei Kitaev. ``Universal quantum computation with ideal Clifford gates and noisy ancillas''. Physical Review A 71 (2005).
https://doi.org/10.1103/physreva.71.022316

[18] Riddhi S. Gupta, Neereja Sundaresan, Thomas Alexander, Christopher J. Wood, Seth T. Merkel, Michael B. Healy, Marius Hillenbrand, Tomas Jochym-O’Connor, James R. Wootton, Theodore J. Yoder, Andrew W. Cross, Maika Takita, and Benjamin J. Brown. ``Encoding a magic state with beyond break-even fidelity''. Nature 625, 259–263 (2024).
https://doi.org/10.1038/s41586-023-06846-3

[19] Earl T. Campbell, Barbara M. Terhal, and Christophe Vuillot. ``Roads towards fault-tolerant universal quantum computation''. Nature 549, 172–179 (2017).
https://doi.org/10.1038/nature23460

[20] Benjamin J. Brown. ``A fault-tolerant non-Clifford gate for the surface code in two dimensions''. Science Advances 6, eaay4929 (2020).
https://doi.org/10.1126/sciadv.aay4929

[21] Katharina Laubscher, Daniel Loss, and James R. Wootton. ``Universal quantum computation in the surface code using non-Abelian islands''. Physical Review A 100 (2019).
https://doi.org/10.1103/physreva.100.012338

[22] Héctor Bombín. ``Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes''. New Journal of Physics 17, 083002 (2015).
https://doi.org/10.1088/1367-2630/17/8/083002

[23] Hyeongrak Choi, Frederic T. Chong, Dirk Englund, and Yongshan Ding. ``Fault tolerant non-Clifford state preparation for arbitrary rotations'' (2023). arXiv:2303.17380.
arXiv:2303.17380

[24] Michael E. Beverland, Aleksander Kubica, and Krysta M. Svore. ``Cost of universality: A comparative study of the overhead of state distillation and code switching with color codes''. PRX Quantum 2 (2021).
https://doi.org/10.1103/prxquantum.2.020341

[25] Daniel Litinski and Felix von Oppen. ``Lattice surgery with a twist: Simplifying Clifford gates of surface codes''. Quantum 2, 62 (2018).
https://doi.org/10.22331/q-2018-05-04-62

[26] H. Bombin and M. A. Martin-Delgado. ``Optimal resources for topological two-dimensional stabilizer codes: Comparative study''. Physical Review A 76 (2007).
https://doi.org/10.1103/physreva.76.012305

[27] Matt McEwen, Dave Bacon, and Craig Gidney. ``Relaxing hardware requirements for surface code circuits using time-dynamics''. Quantum 7, 1172 (2023).
https://doi.org/10.22331/q-2023-11-07-1172

[28] Craig Gidney. ``Stim: a fast stabilizer circuit simulator''. Quantum 5, 497 (2021).
https://doi.org/10.22331/q-2021-07-06-497

[29] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. ``Topological quantum memory''. Journal of Mathematical Physics 43, 4452–4505 (2002).
https://doi.org/10.1063/1.1499754

[30] Austin G. Fowler. ``Minimum weight perfect matching of fault-tolerant topological quantum error correction in average $o(1)$ parallel time'' (2014). arXiv:1307.1740.
arXiv:1307.1740

[31] Jack Edmonds. ``Paths, trees, and flowers''. Canadian Journal of Mathematics 17, 449–467 (1965).
https://doi.org/10.4153/CJM-1965-045-4

[32] Oscar Higgott. ``Pymatching: A python package for decoding quantum codes with minimum-weight perfect matching''. ACM Transactions on Quantum Computing 3, 1–16 (2022).
https://doi.org/10.1145/3505637

[33] Oscar Higgott and Craig Gidney. ``Sparse blossom: correcting a million errors per core second with minimum-weight matching'' (2023). arXiv:2303.15933.
arXiv:2303.15933

[34] ``Numerical data and Python codes''. https://doi.org/10.5281/zenodo.11079788.
https://doi.org/10.5281/zenodo.11079788

[35] Héctor Bombín, Chris Dawson, Ryan V. Mishmash, Naomi Nickerson, Fernando Pastawski, and Sam Roberts. ``Logical blocks for fault-tolerant topological quantum computation''. PRX Quantum 4 (2023).
https://doi.org/10.1103/prxquantum.4.020303

[36] Craig Gidney. ``Inplace access to the surface code y basis''. Quantum 8, 1310 (2024).
https://doi.org/10.22331/q-2024-04-08-1310

[37] György P. Gehér, Campbell McLauchlan, Earl T. Campbell, Alexandra E. Moylett, and Ophelia Crawford. ``Error-corrected hadamard gate simulated at the circuit level''. Quantum 8, 1394 (2024).
https://doi.org/10.22331/q-2024-07-02-1394

[38] Christopher Chamberland and Earl T. Campbell. ``Universal quantum computing with twist-free and temporally encoded lattice surgery''. PRX Quantum 3 (2022).
https://doi.org/10.1103/prxquantum.3.010331

[39] Christopher Chamberland and Earl T. Campbell. ``Circuit-level protocol and analysis for twist-based lattice surgery''. Physical Review Research 4 (2022).
https://doi.org/10.1103/physrevresearch.4.023090

[40] J. Pablo Bonilla Ataides, David K. Tuckett, Stephen D. Bartlett, Steven T. Flammia, and Benjamin J. Brown. ``The XZZX surface code''. Nature Communications 12 (2021).
https://doi.org/10.1038/s41467-021-22274-1

[41] Daniel Litinski and Felix von Oppen. ``Braiding by Majorana tracking and long-range CNOT gates with color codes''. Physical Review B 96 (2017).
https://doi.org/10.1103/physrevb.96.205413

[42] Chenyang Wang, Jim Harrington, and John Preskill. ``Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory''. Annals of Physics 303, 31–58 (2003).
https://doi.org/10.1016/s0003-4916(02)00019-2

Cited by

[1] Áron Márton, Luis Colmenarez, Lukas Bödeker, and Markus Müller, "Lattice surgery-based logical state teleportation via noisy links", Physical Review Research 7 3, 033238 (2025).

[2] Liran Shirizly, Dekel Meirom, Malcolm Carroll, and Haggai Landa, "Feedforward suppression of readout-induced faults in quantum error correction", Physical Review A 112 5, L050602 (2025).

[3] Ilya A. Simakov and Ilya S. Besedin, "Low-overhead quantum error-correction codes with a cyclic topology", Physical Review A 111 1, 012444 (2025).

[4] Ravuri Krishna, "Quantum Science Beyond the Hype: Facts, Myths, and Realistic Progress in Physics, Chemistry, and Computing", International Journal of Computational and Theoretical Chemistry 14 1, 1 (2026).

[5] Hugo Jacinto, Élie Gouzien, and Nicolas Sangouard, "Network requirements for distributed quantum computation", Physical Review Research 8 1, 013205 (2026).

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