Numerical Implementation of Just-In-Time Decoding in Novel Lattice Slices Through the Three-Dimensional Surface Code

T. R. Scruby; D. E. Browne; P. Webster; M. Vasmer

doi:10.22331/q-2022-05-24-721

Numerical Implementation of Just-In-Time Decoding in Novel Lattice Slices Through the Three-Dimensional Surface Code

T. R. Scruby^1,2, D. E. Browne², P. Webster³, and M. Vasmer^4,5

¹Okinawa Institute of Science and Technology, Okinawa, 904-0495, Japan
²Dept. of Physics and Astronomy, University College London, London, WC1E 6BT, UK
³Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia
⁴Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada
⁵Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Canada

Published:	2022-05-24, volume 6, page 721
Eprint:	arXiv:2012.08536v3
Doi:	https://doi.org/10.22331/q-2022-05-24-721
Citation:	Quantum 6, 721 (2022).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

We build on recent work by B. Brown (Sci. Adv. 6, eaay4929 (2020)) to develop and simulate an explicit recipe for a just-in-time decoding scheme in three 3D surface codes, which can be used to implement a transversal (non-Clifford) $\overline{CCZ}$ between three 2D surface codes in time linear in the code distance. We present a fully detailed set of bounded-height lattice slices through the 3D codes which retain the code distance and measurement-error detecting properties of the full 3D code and admit a dimension-jumping process which expands from/collapses to 2D surface codes supported on the boundaries of each slice. At each timestep of the procedure the slices agree on a common set of overlapping qubits on which $CCZ$ should be applied. We use these slices to simulate the performance of a simple JIT decoder against stochastic $X$ and measurement errors and find evidence for a threshold $p_c \sim 0.1\%$ in all three codes. We expect that this threshold could be improved by optimisation of the decoder.

Featured image: 3D spacetime (left) and 2D space (right) diagrams illustrating the relative motion of the three codes during the logical gate. Each of the three 2D codes sweeps out a 3D surface code over time and the cubic region where all three of these codes intersect supports a transversal $\overline{CCZ}$.

For an animated illustration of 3D spacetime diagram click here.

► BibTeX data

@article{Scruby2022numerical,
  doi = {10.22331/q-2022-05-24-721},
  url = {https://doi.org/10.22331/q-2022-05-24-721},
  title = {Numerical {I}mplementation of {J}ust-{I}n-{T}ime {D}ecoding in {N}ovel {L}attice {S}lices {T}hrough the {T}hree-{D}imensional {S}urface {C}ode},
  author = {Scruby, T. R. and Browne, D. E. and Webster, P. and Vasmer, M.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {721},
  month = may,
  year = {2022}
}

► References

[1] Peter W. Shor. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A, 52 (4): 2493–2496, 1995. ISSN 10502947. 10.1103/PhysRevA.52.R2493.
https://doi.org/10.1103/PhysRevA.52.R2493

[2] A. M. Steane. Error correcting codes in quantum theory. Phys. Rev. Lett., 77 (5): 793–797, 1996. ISSN 10797114. 10.1103/PhysRevLett.77.793.
https://doi.org/10.1103/PhysRevLett.77.793

[3] Earl T. Campbell, Barbara M. Terhal, and Christophe Vuillot. Roads towards fault-tolerant universal quantum computation. Nature, 549 (7671): 172–179, 2017. ISSN 14764687. 10.1038/nature23460.
https://doi.org/10.1038/nature23460

[4] A. Yu Kitaev. Fault-tolerant quantum computation by anyons. Ann. Phys., 303 (1): 2–30, 2003. ISSN 00034916. 10.1016/S0003-4916(02)00018-0.
https://doi.org/10.1016/S0003-4916(02)00018-0

[5] H. Bombín and M. A. Martin-Delgado. Topological quantum distillation. Phys. Rev. Lett., 97 (18): 180501, 2007a. 10.1103/PhysRevLett.97.180501.
https://doi.org/10.1103/PhysRevLett.97.180501

[6] Robert Raussendorf and Jim Harrington. Fault-tolerant quantum computation with high threshold in two dimensions. Phys. Rev. Lett., 98 (19): 1–4, 2007. ISSN 00319007. 10.1103/PhysRevLett.98.190504.
https://doi.org/10.1103/PhysRevLett.98.190504

[7] David S. Wang, Austin G. Fowler, and Lloyd C.L. Hollenberg. Surface code quantum computing with error rates over 1%. Phys. Rev. A, 83 (2): 2–5, 2011. ISSN 10502947. 10.1103/PhysRevA.83.020302.
https://doi.org/10.1103/PhysRevA.83.020302

[8] R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O`Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and John M. Martinis. Superconducting quantum circuits at the surface code threshold for fault tolerance. Nature, 508: 500–503, 2014. 10.1038/nature13171.
https://doi.org/10.1038/nature13171

[9] Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando G. S. L. Brandao, David A. Buell, Brian Burkett, Yu Chen, Zijun Chen, Ben Chiaro, Roberto Collins, William Courtney, Andrew Dunsworth, Edward Farhi, Brooks Foxen, Austin Fowler, Craig Gidney, Marissa Giustina, Rob Graff, Keith Guerin, Steve Habegger, Matthew P. Harrigan, Michael J. Hartmann, Alan Ho, Markus Hoffmann, Trent Huang, Travis S. Humble, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Paul V. Klimov, Sergey Knysh, Alexander Korotkov, Fedor Kostritsa, David Landhuis, Mike Lindmark, Erik Lucero, Dmitry Lyakh, Salvatore Mandrà, Jarrod R. McClean, Matthew McEwen, Anthony Megrant, Xiao Mi, Kristel Michielsen, Masoud Mohseni, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Murphy Yuezhen Niu, Eric Ostby, Andre Petukhov, John C. Platt, Chris Quintana, Eleanor G. Rieffel, Pedram Roushan, Nicholas C. Rubin, Daniel Sank, Kevin J. Satzinger, Vadim Smelyanskiy, Kevin J. Sung, Matthew D. Trevithick, Amit Vainsencher, Benjamin Villalonga, Theodore White, Z. Jamie Yao, Ping Yeh, Adam Zalcman, Hartmut Neven, and John M. Martinis. Quantum supremacy using a programmable superconducting processor. Nature, 574 (7779): 505–510, 2019. ISSN 1476-4687. 10.1038/s41586-019-1666-5.
https://doi.org/10.1038/s41586-019-1666-5

[10] Petar Jurcevic, Ali Javadi-Abhari, Lev S. Bishop, Isaac Lauer, Daniela F. Bogorin, Markus Brink, Lauren Capelluto, Oktay Günlük, Toshinari Itoko, Naoki Kanazawa, Abhinav Kandala, George A. Keefe, Kevin Krsulich, William Landers, Eric P. Lewandowski, Douglas T. McClure, Giacomo Nannicini, Adinath Narasgond, Hasan M. Nayfeh, Emily Pritchett, Mary Beth Rothwell, Srikanth Srinivasan, Neereja Sundaresan, Cindy Wang, Ken X. Wei, Christopher J. Wood, Jeng-Bang Yau, Eric J. Zhang, Oliver E. Dial, Jerry M. Chow, and Jay M. Gambetta. Demonstration of quantum volume 64 on a superconducting quantum computing system. Quantum Science and Technology, 6, 2021. ISSN 2058-9565. 10.1088/2058-9565/abe519.
https://doi.org/10.1088/2058-9565/abe519

[11] Bryan Eastin and Emanuel Knill. Restrictions on transversal encoded quantum gate sets. Phys. Rev. Lett., 102 (11): 110502, 2009. ISSN 00319007. 10.1103/PhysRevLett.102.110502.
https://doi.org/10.1103/PhysRevLett.102.110502

[12] Bei Zeng, Andrew Cross, and Isaac L. Chuang. Transversality versus universality for additive quantum codes. IEEE Trans. Inf. Theory, 57 (9): 6272–6284, 2011. ISSN 00189448. 10.1109/TIT.2011.2161917.
https://doi.org/10.1109/TIT.2011.2161917

[13] Sergey Bravyi and Robert König. Classification of topologically protected gates for local stabilizer codes. Phys. Rev. Lett., 110 (17): 170503, 2013. ISSN 00319007. 10.1103/PhysRevLett.110.170503.
https://doi.org/10.1103/PhysRevLett.110.170503

[14] Tomas Jochym-O'Connor, Aleksander Kubica, and Theodore J. Yoder. Disjointness of stabilizer codes and limitations on fault-tolerant logical gates. Phys. Rev. X, 8 (2): 021047, 2018. ISSN 21603308. 10.1103/PhysRevX.8.021047.
https://doi.org/10.1103/PhysRevX.8.021047

[15] Paul Webster, Michael Vasmer, Thomas R. Scruby, and Stephen D. Bartlett. Universal fault-tolerant quantum computing with stabilizer codes. Physical Review Research, 4 (1), 2022. 10.1103/PhysRevResearch.4.013092.
https://doi.org/10.1103/PhysRevResearch.4.013092

[16] Paul Webster and Stephen D. Bartlett. Fault-tolerant quantum gates with defects in topological stabilizer codes. Physical Review A, 102 (2), 2020. 10.1103/PhysRevA.102.022403.
https://doi.org/10.1103/PhysRevA.102.022403

[17] Benjamin J. Brown. A fault-tolerant non-Clifford gate for the surface code in two dimensions. Sci. Adv., 6 (21): eaay4929, 2020. ISSN 2375-2548. 10.1126/sciadv.aay4929.
https://doi.org/10.1126/sciadv.aay4929

[18] H. Bombín. Single-shot fault-tolerant quantum error correction. Phys. Rev. X, 5 (3): 031043, 2015. ISSN 2160-3308. 10.1103/PhysRevX.5.031043.
https://doi.org/10.1103/PhysRevX.5.031043

[19] Earl T. Campbell. A theory of single-shot error correction for adversarial noise. Quantum Sci. Technol., 4 (2): 025006, 2019. ISSN 2058-9565. 10.1088/2058-9565/aafc8f.
https://doi.org/10.1088/2058-9565/aafc8f

[20] Armanda O. Quintavalle, Michael Vasmer, Joschka Roffe, and Earl T. Campbell. Single-Shot Error Correction of Three-Dimensional Homological Product Codes. PRX Quantum, 2 (2), 2021. 10.1103/PRXQuantum.2.020340.
https://doi.org/10.1103/PRXQuantum.2.020340

[21] Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. Surface codes: Towards practical large-scale quantum computation. Phys. Rev. A, 86 (3): 032324, 2012. ISSN 1050-2947, 1094-1622. 10.1103/PhysRevA.86.032324.
https://doi.org/10.1103/PhysRevA.86.032324

[22] Hector Bombín. 2D quantum computation with 3D topological codes. 2018. 10.48550/arXiv.1810.09571.
https://doi.org/10.48550/arXiv.1810.09571

[23] H. Bombín and M. A. Martin-Delgado. Topological computation without braiding. Phys. Rev. Lett., 98 (16): 160502, 2007b. ISSN 0031-9007, 1079-7114. 10.1103/PhysRevLett.98.160502.
https://doi.org/10.1103/PhysRevLett.98.160502

[24] Aleksander Kubica and Michael E. Beverland. Universal transversal gates with color codes: A simplified approach. Phys. Rev. A, 91 (3): 032330, 2015. ISSN 10941622. 10.1103/PhysRevA.91.032330.
https://doi.org/10.1103/PhysRevA.91.032330

[25] 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. Phys. Rev. X, 7 (2): 021029, 2017. ISSN 2160-3308. 10.1103/PhysRevX.7.021029.
https://doi.org/10.1103/PhysRevX.7.021029

[26] Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal Clifford gates and noisy ancillas. Phys. Rev. A, 71 (2): 022316, 2005. ISSN 10941622. 10.1103/PhysRevA.71.022316.
https://doi.org/10.1103/PhysRevA.71.022316

[27] Daniel Litinski. Magic state distillation: Not as costly as you think. Quantum, 3: 205, 2019. ISSN 2521-327X. 10.22331/q-2019-12-02-205.
https://doi.org/10.22331/q-2019-12-02-205

[28] Aleksander Kubica, Beni Yoshida, and Fernando Pastawski. Unfolding the color code. New Journal of Physics, 17 (8): 083026, 2015. ISSN 1367-2630. 10.1088/1367-2630/17/8/083026.
https://doi.org/10.1088/1367-2630/17/8/083026

[29] Michael Vasmer and Dan E. Browne. Three-dimensional surface codes: Transversal gates and fault-tolerant architectures. Phys. Rev. A, 100 (1): 012312, 2019. ISSN 2469-9926, 2469-9934. 10.1103/PhysRevA.100.012312.
https://doi.org/10.1103/PhysRevA.100.012312

[30] H. Bombín. Dimensional jump in quantum error correction. New J. Phys., 18 (4): 043038, 2016. 10.1088/1367-2630/18/4/043038.
https://doi.org/10.1088/1367-2630/18/4/043038

[31] S. B. Bravyi and A. Yu. Kitaev. Quantum codes on a lattice with boundary. 1998. 10.48550/arXiv.quant-ph/9811052.
https://doi.org/10.48550/arXiv.quant-ph/9811052
arXiv:quant-ph/9811052

[32] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. Topological quantum memory. J. Math. Phys., 43 (9): 4452–4505, 2002. ISSN 00222488. 10.1063/1.1499754.
https://doi.org/10.1063/1.1499754

[33] Robert Raussendorf, Sergey Bravyi, and Jim Harrington. Long-range quantum entanglement in noisy cluster states. Phys. Rev. A, 71 (6): 062313, 2005. ISSN 1050-2947, 1094-1622. 10.1103/PhysRevA.71.062313.
https://doi.org/10.1103/PhysRevA.71.062313

[34] a. URL https://github.com/tRowans/JIT-supplementary-materials.
https://github.com/tRowans/JIT-supplementary-materials

[35] Jack Edmonds. Paths, trees, and flowers. Canadian J. Math., 17: 449–467, 1965. 10.4153/CJM-1965-045-4.
https://doi.org/10.4153/CJM-1965-045-4

[36] Austin G. Fowler. Minimum weight perfect matching of fault-tolerant topological quantum error correction in average O(1) parallel time. Quantum Inf. Comput., 15 (1-2): 145–158, 2014. ISSN 15337146. 10.48550/arXiv.1307.1740.
https://doi.org/10.48550/arXiv.1307.1740

[37] Aleksander Kubica and John Preskill. Cellular-automaton decoders with provable thresholds for topological codes. Phys. Rev. Lett., 123 (2): 020501, 2019. ISSN 0031-9007, 1079-7114. 10.1103/PhysRevLett.123.020501.
https://doi.org/10.1103/PhysRevLett.123.020501

[38] Michael Vasmer, Dan E. Browne, and Aleksander Kubica. Cellular automaton decoders for topological quantum codes with noisy measurements and beyond. Scientific Reports, 11 (1), 2021. ISSN 2045-2322. 10.1038/s41598-021-81138-2.
https://doi.org/10.1038/s41598-021-81138-2

[39] Nikolas P. Breuckmann, Kasper Duivenvoorden, Dominik Michels, and Barbara M. Terhal. Local decoders for the 2D and 4D toric code. Quantum Inf. Comput., 17 (3-4): 181–208, 2017. ISSN 15337146. 10.48550/arXiv.1609.00510.
https://doi.org/10.48550/arXiv.1609.00510

[40] Nikolas P. Breuckmann and Xiaotong Ni. Scalable neural network decoders for higher dimensional quantum codes. Quantum, 2: 68, 2018. ISSN 2521-327X. 10.22331/q-2018-05-24-68.
https://doi.org/10.22331/q-2018-05-24-68

[41] K. Duivenvoorden, N. P. Breuckmann, and B. M. Terhal. Renormalization group decoder for a four-dimensional toric code. IEEE Trans. Inf. Theory, 65 (4): 2545–2562, 2019. ISSN 00189448. 10.1109/TIT.2018.2879937.
https://doi.org/10.1109/TIT.2018.2879937

[42] A. B. Aloshious and P. K. Sarvepalli. Decoding toric codes on three dimensional simplical complexes. IEEE Trans. Inf. Theory (Early Access), pages 1–1, 2020. 10.1109/TIT.2020.3037042.
https://doi.org/10.1109/TIT.2020.3037042

[43] Pavel Panteleev and Gleb Kalachev. Degenerate Quantum LDPC Codes With Good Finite Length Performance. Quantum, 5, 2021. 10.22331/q-2021-11-22-585.
https://doi.org/10.22331/q-2021-11-22-585

[44] Joschka Roffe, David R. White, Simon Burton, and Earl Campbell. Decoding across the quantum low-density parity-check code landscape. Physical Review Research, 2 (4), 2020. 10.1103/PhysRevResearch.2.043423.
https://doi.org/10.1103/PhysRevResearch.2.043423

[45] Vladimir Kolmogorov. Blossom V: A new implementation of a minimum cost perfect matching algorithm. Math. Program. Comput., 1 (1): 43–67, 2009. ISSN 18672949. 10.1007/s12532-009-0002-8.
https://doi.org/10.1007/s12532-009-0002-8

[46] b. URL https://github.com/tRowans/JITdecoding-public.
https://github.com/tRowans/JITdecoding-public

[47] Alan Agresti and Brent A. Coull. Approximate is better than "exact" for interval estimation of binomial proportions. Am. Stat., 52 (2): 119–126, 1998. ISSN 00031305. 10.2307/2685469.
https://doi.org/10.2307/2685469

[48] Ashley M. Stephens. Fault-tolerant thresholds for quantum error correction with the surface code. Phys. Rev. A, 89: 022321, 2014. 10.1103/PhysRevA.89.022321.
https://doi.org/10.1103/PhysRevA.89.022321

[49] Benjamin J. Brown, Naomi H. Nickerson, and Dan E. Browne. Fault-tolerant error correction with the gauge color code. Nature Communications, 7 (1): 12302, November 2016. ISSN 2041-1723. 10.1038/ncomms12302.
https://doi.org/10.1038/ncomms12302

[50] Tomas Jochym-O'Connor and Theodore J. Yoder. Four-dimensional toric code with non-Clifford transversal gates. Physical Review Research, 3 (1), 2021. 10.1103/PhysRevResearch.3.013118.
https://doi.org/10.1103/PhysRevResearch.3.013118

Cited by

[1] Thomas R. Scruby, Michael Vasmer, and Dan E. Browne, "Non-Pauli errors in the three-dimensional surface code", Physical Review Research 4 4, 043052 (2022).

[2] 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 2, 020341 (2021).

[3] Paul Webster, Michael Vasmer, Thomas R. Scruby, and Stephen D. Bartlett, "Universal Fault-Tolerant Quantum Computing with Stabiliser Codes", arXiv:2012.05260, (2020).

[4] Paul Webster, Michael Vasmer, Thomas R. Scruby, and Stephen D. Bartlett, "Universal fault-tolerant quantum computing with stabilizer codes", Physical Review Research 4 1, 013092 (2022).

[5] Thomas R. Scruby, Michael Vasmer, and Dan E. Browne, "Non-Pauli Errors in the Three-Dimensional Surface Code", arXiv:2202.05746, (2022).

[6] Mark Webber, Vincent Elfving, Sebastian Weidt, and Winfried K. Hensinger, "The impact of hardware specifications on reaching quantum advantage in the fault tolerant regime", AVS Quantum Science 4 1, 013801 (2022).

The above citations are from Crossref's cited-by service (last updated successfully 2023-03-31 05:26:39) and SAO/NASA ADS (last updated successfully 2023-03-31 05:26:40). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.