A Fault-Tolerant Honeycomb Memory

Craig Gidney, Michael Newman, Austin Fowler, and Michael Broughton

Google Quantum AI, Santa Barbara, California 93117, USA

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Recently, Hastings & Haah introduced a quantum memory defined on the honeycomb lattice. Remarkably, this honeycomb code assembles weight-six parity checks using only two-local measurements. The sparse connectivity and two-local measurements are desirable features for certain hardware, while the weight-six parity checks enable robust performance in the circuit model.
In this work, we quantify the robustness of logical qubits preserved by the honeycomb code using a correlated minimum-weight perfect-matching decoder. Using Monte Carlo sampling, we estimate the honeycomb code's threshold in different error models, and project how efficiently it can reach the "teraquop regime" where trillions of quantum logical operations can be executed reliably. We perform the same estimates for the rotated surface code, and find a threshold of $0.2\%-0.3\%$ for the honeycomb code compared to a threshold of $0.5\%-0.7\%$ for the surface code in a controlled-not circuit model. In a circuit model with native two-body measurements, the honeycomb code achieves a threshold of $1.5\% < p < 2.0\%$, where $p$ is the collective error rate of the two-body measurement gate - including both measurement and correlated data depolarization error processes. With such gates at a physical error rate of $10^{−3}$, we project that the honeycomb code can reach the teraquop regime with only $600$ physical qubits.

We checked how well a new quantum error correcting code works. The new code is called the honeycomb code and was found by researchers at Microsoft. We did simulations to estimate the quality and quantity of noisy qubits needed to reach error rates as low as 1 in a trillion. We compared the honeycomb code to the surface code (the current state of the art).

We used new open source tools: Stim and PyMatching. These tools allowed us to get prototype results in weeks, instead of months, even though the honeycomb code is a very recent and unusual quantum code.

We found that the honeycomb code is slightly worse than the surface code. However, the honeycomb code works on hardware with fewer connections between qubits than the surface code. Also, its performance is very good when the hardware can directly measure two-qubit parities – better than previous codes built entirely out of two-qubit parity measurements. This makes the honeycomb code interesting for quantum computer architectures with low connectivity or native parity measurements.

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► References

[1] Google Quantum AI. Exponential suppression of bit or phase errors with cyclic error correction. Nature, 595 (7867): 383, 2021. 10.1038/​s41586-021-03588-y.

[2] Dave Bacon. Operator quantum error-correcting subsystems for self-correcting quantum memories. Physical Review A, 73 (1): 012340, 2006. 10.1103/​PhysRevA.73.012340.

[3] Héctor Bombín. Topological subsystem codes. Physical review A, 81 (3): 032301, 2010. 10.1103/​PhysRevA.81.032301.

[4] Héctor Bombín and Miguel A Martin-Delgado. Optimal resources for topological two-dimensional stabilizer codes: Comparative study. Physical Review A, 76 (1): 012305, 2007. 10.1103/​PhysRevA.76.012305.

[5] Hector Bombin, Guillaume Duclos-Cianci, and David Poulin. Universal topological phase of two-dimensional stabilizer codes. New Journal of Physics, 14 (7): 073048, 2012. 10.1088/​1367-2630/​14/​7/​073048.

[6] Sergey Bravyi, Guillaume Duclos-Cianci, David Poulin, and Martin Suchara. Subsystem surface codes with three-qubit check operators. arXiv preprint arXiv:1207.1443, 2012. URL https:/​/​arxiv.org/​abs/​1207.1443.

[7] Christopher Chamberland, Aleksander Kubica, Theodore J Yoder, and Guanyu Zhu. Triangular color codes on trivalent graphs with flag qubits. New Journal of Physics, 22 (2): 023019, 2020a. https:/​/​doi.org/​10.1088/​1367-2630/​ab68fd.

[8] Christopher Chamberland, Guanyu Zhu, Theodore J Yoder, Jared B Hertzberg, and Andrew W Cross. Topological and subsystem codes on low-degree graphs with flag qubits. Physical Review X, 10 (1): 011022, 2020b. 10.1103/​PhysRevX.10.011022.

[9] Rui Chao, Michael E Beverland, Nicolas Delfosse, and Jeongwan Haah. Optimization of the surface code design for majorana-based qubits. Quantum, 4: 352, 2020. 10.22331/​q-2020-10-28-352.

[10] JM Chow, L DiCarlo, JM Gambetta, A Nunnenkamp, Lev S Bishop, L Frunzio, MH Devoret, SM Girvin, and RJ Schoelkopf. Detecting highly entangled states with a joint qubit readout. Physical Review A, 81 (6): 062325, 2010. 10.1103/​PhysRevA.81.062325.

[11] Alessandro Ciani and DP DiVincenzo. Three-qubit direct dispersive parity measurement with tunable coupling qubits. Physical Review B, 96 (21): 214511, 2017. 10.1103/​PhysRevB.96.214511.

[12] Ben Criger, Alessandro Ciani, and David P DiVincenzo. Multi-qubit joint measurements in circuit qed: stochastic master equation analysis. EPJ Quantum Technology, 3 (1): 1–21, 2016. 10.1140/​epjqt/​s40507-016-0044-6.

[13] L. DiCarlo, J. M. Chow, J. M. Gambetta, Lev S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and et al. Demonstration of two-qubit algorithms with a superconducting quantum processor. Nature, 460 (7252): 240–244, Jun 2009. ISSN 1476-4687. 10.1038/​nature08121.

[14] David P DiVincenzo and Firat Solgun. Multi-qubit parity measurement in circuit quantum electrodynamics. New Journal of Physics, 15 (7): 075001, Jul 2013. ISSN 1367-2630. 10.1088/​1367-2630/​15/​7/​075001.

[15] S. Filipp, P. Maurer, P. J. Leek, M. Baur, R. Bianchetti, J. M. Fink, M. Göppl, L. Steffen, J. M. Gambetta, A. Blais, and A. Wallraff. Two-qubit state tomography using a joint dispersive readout. Phys. Rev. Lett., 102: 200402, May 2009. 10.1103/​PhysRevLett.102.200402.

[16] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland. Surface codes: Towards practical large-scale quantum computation. Phys. Rev. A, 86: 032324, 2012. 10.1103/​PhysRevA.86.032324. arXiv:1208.0928.

[17] Austin G Fowler. Optimal complexity correction of correlated errors in the surface code. arXiv preprint arXiv:1310.0863, 2013. URL https:/​/​arxiv.org/​abs/​1310.0863.

[18] Craig Gidney. Stim: a fast stabilizer circuit simulator. Quantum, 5: 497, July 2021a. ISSN 2521-327X. 10.22331/​q-2021-07-06-497.

[19] Craig Gidney. The stim circuit file format (.stim). https:/​/​github.com/​quantumlib/​Stim/​blob/​main/​doc/​file_format_stim_circuit.md, 2021b. Accessed: 2021-08-16.

[20] Craig Gidney. The detector error model file format (.dem). https:/​/​github.com/​quantumlib/​Stim/​blob/​main/​doc/​file_format_dem_detector_error_model.md, 2021c. Accessed: 2021-08-16.

[21] Craig Gidney and Martin Ekerå. How to factor 2048 bit rsa integers in 8 hours using 20 million noisy qubits. Quantum, 5: 433, 2021. 10.22331/​q-2021-04-15-433.

[22] Luke CG Govia, Emily J Pritchett, BLT Plourde, Maxim G Vavilov, R McDermott, and Frank K Wilhelm. Scalable two-and four-qubit parity measurement with a threshold photon counter. Physical Review A, 92 (2): 022335, 2015. 10.1103/​PhysRevA.92.022335.

[23] Jeongwan Haah and Matthew B Hastings. Boundaries for the honeycomb code. arXiv preprint arXiv:2110.09545, 2021. URL https:/​/​arxiv.org/​abs/​2110.09545.

[24] Matthew B Hastings and Jeongwan Haah. Dynamically generated logical qubits. Quantum, 5: 564, 2021. 10.22331/​q-2021-10-19-564.

[25] Oscar Higgott. Pymatching: A fast implementation of the minimum-weight perfect matching decoder. arXiv preprint arXiv:2105.13082, 2021. URL https:/​/​arxiv.org/​abs/​2105.13082.

[26] Shilin Huang, Michael Newman, and Kenneth R Brown. Fault-tolerant weighted union-find decoding on the toric code. Physical Review A, 102 (1): 012419, 2020. 10.1103/​PhysRevA.102.012419.

[27] Patrick Huembeli and Simon E Nigg. Towards a heralded eigenstate-preserving measurement of multi-qubit parity in circuit qed. Physical Review A, 96 (1): 012313, 2017. 10.1103/​PhysRevA.96.012313.

[28] Thomas Häner, Samuel Jaques, Michael Naehrig, Martin Roetteler, and Mathias Soeken. Improved quantum circuits for elliptic curve discrete logarithms. In Post-Quantum Cryptography: 11th International Conference, PQCrypto 2020, Paris, France, April 15–17, 2020, Proceedings, volume 12100, page 425. Springer Nature, 2020. 10.1007/​978-3-030-44223-1_23.

[29] Joseph Kerckhoff, Luc Bouten, Andrew Silberfarb, and Hideo Mabuchi. Physical model of continuous two-qubit parity measurement in a cavity-qed network. Physical Review A, 79 (2): 024305, 2009. 10.1103/​PhysRevA.79.024305.

[30] A Yu Kitaev. Fault-tolerant quantum computation by anyons. Annals of Physics, 303 (1): 2–30, 2003. 10.1016/​S0003-4916(02)00018-0.

[31] Alexei Kitaev. Anyons in an exactly solved model and beyond. Annals of Physics, 321 (1): 2–111, 2006. 10.1016/​j.aop.2005.10.005.

[32] Christina Knapp, Michael Beverland, Dmitry I Pikulin, and Torsten Karzig. Modeling noise and error correction for majorana-based quantum computing. Quantum, 2: 88, 2018. 10.22331/​q-2018-09-03-88.

[33] David Kribs, Raymond Laflamme, and David Poulin. Unified and generalized approach to quantum error correction. Physical review letters, 94 (18): 180501, 2005. 10.1103/​PhysRevLett.94.180501.

[34] Aleksander Kubica and Michael Vasmer. Single-shot quantum error correction with the three-dimensional subsystem toric code. arXiv preprint arXiv:2106.02621, 2021. URL https:/​/​arxiv.org/​abs/​2106.02621.

[35] Kevin Lalumière, J. M. Gambetta, and Alexandre Blais. Tunable joint measurements in the dispersive regime of cavity qed. Physical Review A, 81 (4), Apr 2010. ISSN 1094-1622. 10.1103/​physreva.81.040301.

[36] Joonho Lee, Dominic W. Berry, Craig Gidney, William J. Huggins, Jarrod R. McClean, Nathan Wiebe, and Ryan Babbush. Even more efficient quantum computations of chemistry through tensor hypercontraction. PRX Quantum, 2: 030305, Jul 2021. 10.1103/​PRXQuantum.2.030305.

[37] Yi-Chan Lee, Courtney G Brell, and Steven T Flammia. Topological quantum error correction in the kitaev honeycomb model. Journal of Statistical Mechanics: Theory and Experiment, 2017 (8): 083106, 2017. 10.1088/​1742-5468/​aa7ee2.

[38] William P Livingston, Machiel S Blok, Emmanuel Flurin, Justin Dressel, Andrew N Jordan, and Irfan Siddiqi. Experimental demonstration of continuous quantum error correction. arXiv preprint arXiv:2107.11398, 2021. URL https:/​/​arxiv.org/​abs/​2107.11398.

[39] Razieh Mohseninia, Jing Yang, Irfan Siddiqi, Andrew N Jordan, and Justin Dressel. Always-on quantum error tracking with continuous parity measurements. Quantum, 4: 358, 2020. 10.22331/​q-2020-11-04-358.

[40] D. Ristè, J. G. van Leeuwen, H.-S. Ku, K. W. Lehnert, and L. DiCarlo. Initialization by measurement of a superconducting quantum bit circuit. Physical Review Letters, 109 (5), Aug 2012. ISSN 1079-7114. 10.1103/​physrevlett.109.050507.

[41] Baptiste Royer, Shruti Puri, and Alexandre Blais. Qubit parity measurement by parametric driving in circuit qed. Science advances, 4 (11): eaau1695, 2018. 10.1126/​sciadv.aau1695.

[42] Ashley M Stephens. Fault-tolerant thresholds for quantum error correction with the surface code. Physical Review A, 89 (2): 022321, 2014. 10.1103/​PhysRevA.89.022321.

[43] Martin Suchara, Sergey Bravyi, and Barbara Terhal. Constructions and noise threshold of topological subsystem codes. Journal of Physics A: Mathematical and Theoretical, 44 (15): 155301, 2011. 10.1088/​1751-8113/​44/​15/​155301.

[44] Lars Tornberg, Sh Barzanjeh, and David P DiVincenzo. Stochastic-master-equation analysis of optimized three-qubit nondemolition parity measurements. Physical Review A, 89 (3): 032314, 2014. 10.1103/​PhysRevA.89.032314.

Cited by

[1] Jeongwan Haah and Matthew B. Hastings, "Boundaries for the Honeycomb Code", arXiv:2110.09545.

[2] James R. Wootton, "Hexagonal matching codes with 2-body measurements", arXiv:2109.13308.

[3] Christophe Vuillot, "Planar Floquet Codes", arXiv:2110.05348.

[4] Andrew J. Landahl and Benjamin C. A. Morrison, "Logical Majorana fermions for fault-tolerant quantum simulation", arXiv:2110.10280.

[5] Julia Wildeboer, Thomas Iadecola, and Dominic J. Williamson, "Symmetry-Protected Infinite-Temperature Quantum Memory from Subsystem Codes", arXiv:2110.05710.

[6] Edward H. Chen, Theodore J. Yoder, Youngseok Kim, Neereja Sundaresan, Srikanth Srinivasan, Muyuan Li, Antonio D. Córcoles, Andrew W. Cross, and Maika Takita, "Calibrated decoders for experimental quantum error correction", arXiv:2110.04285.

[7] Basudha Srivastava, Anton Frisk Kockum, and Mats Granath, "The XYZ$^2$ hexagonal stabilizer code", arXiv:2112.06036.

[8] Benjamin A. Cordier, Nicolas P. D. Sawaya, Gian G. Guerreschi, and Shannon K. McWeeney, "Biology and medicine in the landscape of quantum advantages", arXiv:2112.00760.

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