The surface code with a twist

Theodore J. Yoder1 and Isaac H. Kim2

1Department of Physics, Massachusetts Institute of Technology
2IBM, Thomas J. Watson Research Center

The surface code is one of the most successful approaches to topological quantum error-correction. It boasts the smallest known syndrome extraction circuits and correspondingly largest thresholds. Defect-based logical encodings of a new variety called twists have made it possible to implement the full Clifford group without state distillation. Here we investigate a patch-based encoding involving a modified twist. In our modified formulation, the resulting codes, called triangle codes for the shape of their planar layout, have only weight-four checks and relatively simple syndrome extraction circuits that maintain a high, near surface-code-level threshold. They also use 25% fewer physical qubits per logical qubit than the surface code. Moreover, benefiting from the twist, we can implement all Clifford gates by lattice surgery without the need for state distillation. By a surgical transformation to the surface code, we also develop a scheme of doing all Clifford gates on surface code patches in an atypical planar layout, though with less qubit efficiency than the triangle code. Finally, we remark that logical qubits encoded in triangle codes are naturally amenable to logical tomography, and the smallest triangle code can demonstrate high-pseudothreshold fault-tolerance to depolarizing noise using just 13 physical qubits.

Share

► BibTeX data

► References

[1] Sergey B Bravyi and A Yu Kitaev. Quantum codes on a lattice with boundary. quant-ph/​9811052, 1998.
arXiv:quant-ph/9811052

[2] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. Topological quantum memory. Journal of Mathematical Physics, 43 (9): 4452-4505, 2002. 10.1063/​1.1499754.
https://doi.org/10.1063/1.1499754

[3] Austin G Fowler, Matteo Mariantoni, John M Martinis, and Andrew N Cleland. Surface codes: Towards practical large-scale quantum computation. Physical Review A, 86 (3): 032324, 2012. 10.1103/​PhysRevA.86.032324.
https://doi.org/10.1103/PhysRevA.86.032324

[4] Dorit Aharonov and Lior Eldar. On the complexity of commuting local Hamiltonians, and tight conditions for topological order in such systems. In Foundations of Computer Science (FOCS), 2011 IEEE 52nd Annual Symposium on, pages 334-343. IEEE, 2011. 10.1109/​FOCS.2011.58.
https://doi.org/10.1109/FOCS.2011.58

[5] Yu Tomita and Krysta M Svore. Low-distance surface codes under realistic quantum noise. Physical Review A, 90 (6): 062320, 2014. 10.1103/​PhysRevA.90.062320.
https://doi.org/10.1103/PhysRevA.90.062320

[6] David S Wang, Austin G Fowler, and Lloyd CL Hollenberg. Surface code quantum computing with error rates over 1%. Physical Review A, 83 (2): 020302, 2011. 10.1103/​PhysRevA.83.020302.
https://doi.org/10.1103/PhysRevA.83.020302

[7] Adrian Hutter, James R Wootton, and Daniel Loss. Efficient Markov chain Monte Carlo algorithm for the surface code. Physical Review A, 89 (2): 022326, 2014. 10.1103/​PhysRevA.89.022326.
https://doi.org/10.1103/PhysRevA.89.022326

[8] Sergey Bravyi, Martin Suchara, and Alexander Vargo. Efficient algorithms for maximum likelihood decoding in the surface code. Physical Review A, 90 (3): 032326, 2014. 10.1103/​PhysRevA.90.032326.
https://doi.org/10.1103/PhysRevA.90.032326

[9] James R Wootton and Daniel Loss. High threshold error correction for the surface code. Physical Review Letters, 109 (16): 160503, 2012. 10.1103/​PhysRevLett.109.160503.
https://doi.org/10.1103/PhysRevLett.109.160503

[10] Austin G Fowler and Simon J Devitt. A bridge to lower overhead quantum computation. arXiv:1209.0510, 2012.
arXiv:1209.0510

[11] Héctor Bombín. Topological subsystem codes. Physical Review A, 81 (3): 032301, 2010a. 10.1103/​PhysRevA.81.032301.
https://doi.org/10.1103/PhysRevA.81.032301

[12] Sergey Bravyi, Guillaume Duclos-Cianci, David Poulin, and Martin Suchara. Subsystem surface codes with three-qubit check operators. Quantum Information & Computation, 13 (11-12): 963-985, 2013.

[13] Héctor Bombín and Miguel Angel Martin-Delgado. Topological quantum distillation. Physical Review Letters, 97 (18): 180501, 2006. 10.1103/​PhysRevLett.97.180501.
https://doi.org/10.1103/PhysRevLett.97.180501

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

[15] Aleksander Kubica and Michael E Beverland. Universal transversal gates with color codes: A simplified approach. Physical Review A, 91 (3): 032330, 2015. 10.1103/​PhysRevA.91.032330.
https://doi.org/10.1103/PhysRevA.91.032330

[16] Andrew J Landahl and Ciaran Ryan-Anderson. Quantum computing by color-code lattice surgery. arXiv:1407.5103, 2014.
arXiv:1407.5103

[17] Robert Raussendorf and Jim Harrington. Fault-tolerant quantum computation with high threshold in two dimensions. Physical Review Letters, 98 (19): 190504, 2007. 10.1103/​PhysRevLett.98.190504.
https://doi.org/10.1103/PhysRevLett.98.190504

[18] H Bombin and MA Martin-Delgado. Quantum measurements and gates by code deformation. Journal of Physics A: Mathematical and Theoretical, 42 (9): 095302, 2009. 10.1088/​1751-8113/​42/​9/​095302.
https://doi.org/10.1088/1751-8113/42/9/095302

[19] Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal clifford gates and noisy ancillas. Physical Review A, 71 (2): 022316, 2005. 10.1103/​PhysRevA.71.022316.
https://doi.org/10.1103/PhysRevA.71.022316

[20] Austin G Fowler, Simon J Devitt, and Cody Jones. Surface code implementation of block code state distillation. Scientific reports, 3, 2013. 10.1038/​srep01939.
https://doi.org/10.1038/srep01939

[21] Austin G Fowler, Ashley M Stephens, and Peter Groszkowski. High-threshold universal quantum computation on the surface code. Physical Review A, 80 (5): 052312, 2009. 10.1103/​PhysRevA.80.052312.
https://doi.org/10.1103/PhysRevA.80.052312

[22] Austin G Fowler. Time-optimal quantum computation. arXiv:1210.4626, 2012.
arXiv:1210.4626

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

[24] Héctor Bombín. Topological order with a twist: Ising anyons from an Abelian model. Physical Review Letters, 105 (3): 030403, 2010b. 10.1103/​PhysRevLett.105.030403.
https://doi.org/10.1103/PhysRevLett.105.030403

[25] Matthew B Hastings and A Geller. Reduced space-time and time costs using dislocation codes and arbitrary ancillas. Quantum Information & Computation, 15 (11-12): 962-986, 2015.

[26] A. R. Calderbank and Peter W. Shor. Good quantum error-correcting codes exist. Phys. Rev. A, 54: 1098-1105, 1996. 10.1103/​PhysRevA.54.1098.
https://doi.org/10.1103/PhysRevA.54.1098

[27] Andrew M Steane. Error correcting codes in quantum theory. Phys. Rev. Lett., 77 (5): 793, 1996. 10.1103/​PhysRevLett.77.793.
https://doi.org/10.1103/PhysRevLett.77.793

[28] 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. arXiv:1609.04673, 2016.
arXiv:1609.04673

[29] H Bombin and MA Martin-Delgado. Optimal resources for topological two-dimensional stabilizer codes: Comparative study. Physical Review A, 76 (1): 012305, 2007. 10.1103/​PhysRevA.76.012305.
https://doi.org/10.1103/PhysRevA.76.012305

[30] Daniel Gottesman and Isaac L Chuang. Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature, 402 (6760): 390-393, 1999. 10.1038/​46503.
https://doi.org/10.1038/46503

[31] Peter W Shor. Fault-tolerant quantum computation. In Foundations of Computer Science, 1996. Proceedings., 37th Annual Symposium on, pages 56-65. IEEE, 1996. 10.1109/​SFCS.1996.548464.
https://doi.org/10.1109/SFCS.1996.548464

[32] Peter Brooks and John Preskill. Fault-tolerant quantum computation with asymmetric Bacon-Shor codes. Physical Review A, 87 (3): 032310, 2013. 10.1103/​PhysRevA.87.032310.
https://doi.org/10.1103/PhysRevA.87.032310

[33] Jonathan E Moussa. Transversal clifford gates on folded surface codes. Physical Review A, 94 (4): 042316, 2016. 10.1103/​PhysRevA.94.042316.
https://doi.org/10.1103/PhysRevA.94.042316

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

[35] Andrew W Cross. personal communication.

[36] Panos Aliferis, Daniel Gottesman, and John Preskill. Quantum accuracy threshold for concatenated distance-3 codes. Quantum Information & Computation, 6 (2): 97-165, 2006.

[37] Krysta M Svore, Andrew W Cross, Isaac L Chuang, and Alfred V Aho. A flow-map model for analyzing pseudothresholds in fault-tolerant quantum computing. Quantum Information & Computation, 6 (3): 193-212, 2006.

[38] Daniel Gottesman. Quantum fault tolerance in small experiments. arXiv preprint arXiv:1610.03507, 2016.
arXiv:1610.03507

[39] Andrew W Cross, David P Divincenzo, and Barbara M Terhal. A comparative code study for quantum fault tolerance. Quantum Information & Computation, 9 (7): 541-572, 2009.

[40] Emanuel Knill. Quantum computing with realistically noisy devices. Nature, 434 (7029): 39-44, 2005. 10.1038/​nature03350.
https://doi.org/10.1038/nature03350

[41] Charles H Bennett, David P DiVincenzo, John A Smolin, and William K Wootters. Mixed-state entanglement and quantum error correction. Physical Review A, 54 (5): 3824, 1996. 10.1103/​PhysRevA.54.3824.
https://doi.org/10.1103/PhysRevA.54.3824

[42] Raymond Laflamme, Cesar Miquel, Juan Pablo Paz, and Wojciech Hubert Zurek. Perfect quantum error correcting code. Physical Review Letters, 77 (1): 198, 1996. 10.1103/​PhysRevLett.77.198.
https://doi.org/10.1103/PhysRevLett.77.198

[43] David P DiVincenzo and Panos Aliferis. Effective fault-tolerant quantum computation with slow measurements. Physical Review Letters, 98 (2): 020501, 2007. 10.1103/​PhysRevLett.98.020501.
https://doi.org/10.1103/PhysRevLett.98.020501

[44] Ashley M Stephens. Efficient fault-tolerant decoding of topological color codes. arXiv preprint arXiv:1402.3037, 2014.
arXiv:1402.3037

[45] Panos Aliferis and Andrew W Cross. Subsystem fault tolerance with the Bacon-Shor code. Physical Review Letters, 98 (22): 220502, 2007. 10.1103/​PhysRevLett.98.220502.
https://doi.org/10.1103/PhysRevLett.98.220502

► Cited by (beta)

[1] Daniel Litinski, Felix von Oppen, "Braiding by Majorana tracking and long-range CNOT gates with color codes", Physical Review B 96, 205413 (2017).

[2] Maika Takita, Andrew W. Cross, A. D. Córcoles, Jerry M. Chow, Jay M. Gambetta, "Experimental Demonstration of Fault-Tolerant State Preparation with Superconducting Qubits", Physical Review Letters 119, 180501 (2017).

[3] Muyuan Li, Mauricio Gutiérrez, Stanley E. David, Alonzo Hernandez, Kenneth R. Brown, "Fault tolerance with bare ancillary qubits for a [[7,1,3]] code", Physical Review A 96, 032341 (2017).

[4] Ryuji Takagi, Theodore J. Yoder, Isaac L. Chuang, "Error rates and resource overheads of encoded three-qubit gates", Physical Review A 96, 042302 (2017).

[5] Benjamin J. Brown, Katharina Laubscher, Markus S. Kesselring, James R. Wootton, "Poking Holes and Cutting Corners to Achieve Clifford Gates with the Surface Code", Physical Review X 7, 021029 (2017).

(The above data is from Crossref's cited-by service. Unfortunately not all publishers provide suitable and complete citation data so that some citing works or bibliographic details may be missing.)