1 Häffner, H., Roos, C. F. & Blatt, R. Quantum computing with trapped ions. Phys. Rep. 469, 155–203 (2008)

2 Ballance, C. J., Harty, T. P., Linke, N. M., Sepiol, M. A. & Lucas, D. M. High-fidelity quantum logic gates using trapped-ion hyperfine qubits. Phys. Rev. Lett. 117, 060504 (2016)

3 Devoret, M. H. & Schoelkopf, R. J. Superconducting circuits for quantum information: an outlook. Science 339, 1169–1174 (2013)

4 Córcoles, A. D. et al. Demonstration of a quantum error detection code using a square lattice of four superconducting qubits. Nat. Commun. 6, 6979 (2015)

5 Gambetta, J. M., Chow, J. M. & Steffen M. Building logical qubits in a superconducting quantum computing system. Quantum Inf. 3, 2 (2017)

6 Kelly, J. et al. State preservation by repetitive error detection in a superconducting quantum circuit. Nature 519, 66–69 (2015)

7 Ristè, D. et al. Detecting bit-flip errors in a logical qubit using stabilizer measurements. Nat. Commun. 6, 6983 (2015)

8 Terhal, B. M. Quantum error correction for quantum memories. Rev. Mod. Phys. 87, 307–346 (2015)

9 Gottesman, D. Stabilizer Codes and Quantum Error Correction. PhD thesis, Californian Institute of Technology (1997)

10 Aaronson, S. & Gottesman, D. Improved simulation of stabilizer circuits. Phys. Rev. A 70, 052328 (2004)

11 Chen, X., Chung, H., Cross, A. W., Zeng, B. & Chuang, I. L. Subsystem stabilizer codes cannot have a universal set of transversal gates for even one encoded qudit. Phys. Rev. A 78, 012353 (2008)

12 Eastin, B. & Knill, E. Restrictions on transversal encoded quantum gate sets. Phys. Rev. Lett. 102, 110502 (2009)

13 Dennis, E., Kitaev, A., Landahl, A. & Preskill, J. Topological quantum memory. J. Math. Phys. 43, 4452–4505 (2002). Seminal paper on using the surface code as a quantum memory

14 Raussendorf, R. & Harrington, J. Fault-tolerant quantum computation with high threshold in two dimensions. Phys. Rev. Lett. 98, 190504 (2007). Presented a planar 0.75%-threshold surface code architecture that realizes universal logic by combining the topological execution of the CNOT gate and magic-state distillation

15 Fowler, A. G., Stephens, A. M. & Groszkowski, P. High-threshold universal quantum computation on the surface code. Phys. Rev. A 80, 052312 (2009)

16 Fowler, A., Mariantoni, M., Martinis, J. M. & Cleland, A. N. Surface codes: towards practical large-scale quantum computation. Phys. Rev. A 86, 032324 (2012)

17 Raussendorf, R., Harrington, J. & Goyal, K. Topological fault-tolerance in cluster state quantum computation. New J. Phys. 9, 199 (2007)

18 Bombín, H. Topological order with a twist: Ising anyons from an Abelian model. Phys. Rev. Lett. 105, 030403 (2010)

19 Hastings, M. B. & Geller, A. Reduced space-time and time costs using dislocation codes and arbitrary ancillas. Quantum Inf. Comput. 15, 962–986 (2015)

20 Horsman, C., Fowler, A. G. & Devitt, S. & Van Meter, R. Surface code quantum computing by lattice surgery. New J. Phys. 14, 123011 (2012)

21 Brown, B. J., Laubscher, K., Kesselring, M. S. & Wootton, J. R. Poking holes and cutting corners to achieve Clifford gates with the surface code. Phys. Rev. X 7, 021029 (2017)

22 Aliferis, P. Level Reduction and the Quantum Threshold Theorem. PhD thesis, California Institute of Technology (2007)

23 Jones, N. C. et al. Layered architecture for quantum computing. Phys. Rev. X 2, 031007 (2012)

24 Shor, P. W. Fault-tolerant quantum computation. In 37th Annual Symposium on Foundations of Computer Science, FOCS ‘96 56–65 (IEEE, 1996). Presented theoretical schemes for realizing fault-tolerant universal quantum computation for the first time, using quantum error correcting codes

25 Knill, E., Laflamme, R. & Zurek, W. Threshold accuracy for quantum computation. Preprint at https://arxiv.org/abs/quant-ph/9610011 (1996)

26 Knill, E., Laflamme, R. & Zurek, W. Resilient quantum computation. Science 279, 342–345 (1998)

27 Bravyi, S. & Kitaev, A. Universal quantum computation with ideal Clifford gates and noisy ancillas. Phys. Rev. A 71, 022316 (2005). Introduced magic-state distillation as an efficient high-threshold way of getting from Clifford circuits to universality

28 Fowler, A. G., Devitt, S. J. & Jones, C. Surface code implementation of block code state distillation. Sci. Rep. 3, 1939 (2013)

29 O’Gorman, J. & Campbell, E. T. Quantum computation with realistic magic state factories. Phys. Rev. A 95, 032338 (2017)

30 Kitaev, A. Yu ., Shen, A. H. & Vyalyi, M. N. Classical and Quantum Computation (American Mathematical Society, 2002)

31 Ross, N. J. & Selinger, P. Optimal ancilla-free Clifford + T approximation of z-rotations. Quantum Inf. Comput. 16, 901–953 (2016)

32 Bocharov, A., Roetteler, M. & Svore, K. M. Efficient synthesis of probabilistic quantum circuits with fallback. Phys. Rev. A 91, 052317 (2015)

33 Amy, M., Maslov, D., Mosca, M. & Roetteler, M. A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits. IEEE Trans. Comput. Aided Des. Integrated Circ. Syst. 32, 818–830 (2013)

34 Cesare, C. Topological Code Architectures for Quantum Computation. PhD thesis, Univ. New Mexico (2014)

35 Duclos-Cianci, G. & Poulin, D. Reducing the quantum-computing overhead with complex gate distillation. Phys. Rev. A 91, 042315 (2015)

36 Campbell, E. T. & O’Gorman, J. An efficient magic state approach to small angle rotations. Quant. Sci. Tech 1, 015007 (2016)

37 Eastin, B. Distilling one-qubit magic states into Toffoli states. Phys. Rev. A 87, 032321 (2013)

38 Jones, C. Low-overhead constructions for the fault-tolerant Toffoli gate. Phys. Rev. A 87, 022328 (2013)

39 Campbell, E. T. & Howard, M. Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost. Phys. Rev. A 95, 022316 (2017)

40 Paler, A., Devitt, S. J. & Fowler, A. G. Synthesis of arbitrary quantum circuits to topological assembly. Sci. Rep. 6, 30600 (2016)

41 Bombín, H. & Martin-Delgado, M. A. Topological quantum distillation. Phys. Rev. Lett. 97, 180501 (2006)

42 Bombín, H. & Martin-Delgado, M. A. Topological computation without braiding. Phys. Rev. Lett. 98, 160502 (2007). Introduced 3D colour codes with a transversal T gate

43 Bombín, H. & Martin-Delgado, M. A. Homological error correction: classical and quantum codes. J. Math. Phys. 48, 052105 (2007)

44 Katzgraber, H. G., Bombin, H., Andrist, R. S. & Martin-Delgado, M. A. Topological color codes on union jack lattices: a stable implementation of the whole clifford group. Phys. Rev. A 81, 012319 (2010)

45 Kubica, A. & Beverland, M. E. Universal transversal gates with color codes: a simplified approach. Phys. Rev. A 91, 032330 (2015)

46 Bombín, H. Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes. New J. Phys. 17, 083002 (2015)

47 Kubica, A., Yoshida, B. & Pastawski, F. Unfolding the color code. New J. Phys. 17, 083026 (2015)

48 Landahl, A. J. & Ryan-Anderson, C. Quantum Computing by Color-Code Lattice Surgery. Report SAND2014-15911J, https://arxiv.org/abs/1407.5103 (Sandia National Laboratories, 2014)

49 Paetznick, A. & Reichardt, B. W. Universal fault-tolerant quantum computation with only transversal gates and error correction. Phys. Rev. Lett. 111, 090505 (2013). Showed that a universal set of transversal gates can be realized using gauge fixing

50 Bombín, H. Single-shot fault-tolerant quantum error correction. Phys. Rev. X 5, 031043 (2015)

51 Bombín, H. Dimensional jump in quantum error correction. New J. Phys. 18, 043038 (2016)

52 Delfosse, N. Decoding color codes by projection onto surface codes. Phys. Rev. A 89, 012317 (2014)

53 Beverland, M. Toward Realizable Quantum Computers. PhD thesis, California Institute of Technology (2016)

54 Criger, B. & Terhal, B. M. Noise thresholds for the [[4, 2, 2]]-concatenated toric code. Quantum Inf. Comput. 16, 1261–1281 (2016)

55 Cross, A. W., DiVincenzo, D. P. & Terhal, B. M. A comparative code study for quantum fault tolerance. Quantum Inf. Comput. 9, 541–572 (2009)

56 Paetznick, A. & Reichardt, B. W. Fault-tolerant ancilla preparation and noise threshold lower bounds for the 23-qubit Golay code. Quantum Inf. Comput. 12, 1034–1080 (2012)

57 Jochym-O’Connor, T. & Laflamme, R. Using concatenated quantum codes for universal fault-tolerant quantum gates. Phys. Rev. Lett. 112, 010505 (2014)

58 Chamberland, C., Jochym-O’Connor, T. & Laflamme, R. Overhead analysis of universal concatenated quantum codes. Phys. Rev. A 95, 022313 (2017)

59 Yoder, Th. J., Takagi, R. & Chuang, I. L. Universal fault-tolerant gates on concatenated stabilizer codes. Phys. Rev. X 6, 031039 (2016)

60 Bravyi, S. & Cross, A. Doubled color codes. Preprint at https://arxiv.org/abs/1509.03239 (2015)

61 Jones, C., Brooks, P. & Harrington, J. Gauge color codes in two dimensions. Phys. Rev. A 93, 052332 (2016)

62 Jochym-O’Connor, T. & Bartlett, S. D. Stacked codes: universal fault-tolerant quantum computation in a two-dimensional layout. Phys. Rev. A 93, 022323 (2016)

63 Nikahd, E., Sedighi, M. & Zamani, M. S. Non-uniform code concatenation for universal fault-tolerant quantum computing. Preprint at https://arxiv.org/abs/1605.07007 (2016)

64 Bravyi, S. & Koenig, R. Classification of topologically protected gates for local stabilizer codes. Phys. Rev. Lett. 110, 170503 (2013). Proved that the available transversal gates are constrained by the dimension of a topological code

65 Bravyi, S., Poulin, D. & Terhal, B. M. Tradeoffs for reliable quantum information storage in 2D systems. Phys. Rev. Lett. 104, 050503 (2010)

66 Breuckmann, N. P. & Terhal, B. M. Constructions and noise threshold of hyperbolic surface codes. IEEE Trans. Inf. Theory 62, 3731–3744 (2016)

67 Delfosse, N. Tradeoffs for reliable quantum information storage in surface codes and color codes. In Proc. IEEE International Symposium on Information Theory 917–921 (IEEE, 2013)

68 Bravyi, S. & Terhal, B. M. A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes. New J. Phys. 11, 043029 (2009)

69 Tillich, J.-P. & Zémor, G. Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength. IEEE Trans. Inf. Theory 60, 1193–1202 (2014)

70 Freedman, M. H. & Hastings, M. B. Quantum systems on non-k-hyperfinite complexes: a generalization of classical statistical mechanics on expander graphs. Quantum Inf. Comput. 14, 144–180 (2014)

71 Monroe, C. et al. Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects. Phys. Rev. A 89, 022317 (2014)

72 Nickerson, N. H., Fitzsimons, J. F. & Benjamin, S. C. Freely scalable quantum technologies using cells of 5-to-50 qubits with very lossy and noisy photonic links. Phys. Rev. X 4, 041041 (2014)

73 Meier, A. M., Eastin, B. & Knill, E. Magic-state distillation with the four-qubit code. Quantum Inf. Comput. 13, 195–209 (2013)

74 Bravyi, S. & Haah, J. Magic-state distillation with low overhead. Phys. Rev. A 86, 052329 (2012)

75 Jones, C. Multilevel distillation of magic states for quantum computing. Phys. Rev. A 87, 042305 (2013)

76 Wang, C., Harrington, J. & Preskill, J. Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory. Ann. Phys. 303, 31–58 (2003)

77 Brown, B. J., Nickerson, N. H. & Browne, D. E. Fault-tolerant error correction with the gauge color code. Nat. Commun. 7, 12302 (2016)

78 Breuckmann, N. P., Duivenvoorden, K., Michels, D. & Terhal, B. M. Local decoders for the 2D and 4D toric code. Quantum Inf. Comput. 17, 181–208 (2017)