Twist-defect surface code[17]  

Also known as Surface code with a twist, Genon surface code.


A non-CSS extension of the 2D surface-code construction whose non-CSS stabilizer generators are associated with twist defects of the associated lattice. A related construction [7] doubles the number of qubits in the lattice via symplectic doubling.

For lattices with dislocations and rotational disclinations, twist-defect stabilizer generators are placed at the location of the dislocations to yield a stabilizer code whose logical dimension depends on the defects. Logical dimension is determined by the genus of the underlying surface (for closed surfaces), types of boundaries (for open surfaces), and any twist defects present.

A simple example is a surface code on a lattice with a single lattice dislocation which hosts a weight-five non-CSS twist-defect stabilizer generator [1; Fig. 2]. More generally, given a graph embedded in a 2D manifold, qubits are placed on vertices, stabilizers on faces, and twist defects are associated to odd-degree vertices.


Code properties depends on the number and size of the twist defects.


Twist-defect surface codes have negative curvature around their defects, and thus circumvent the BPT bound for codes on Euclidean lattices.


Clifford gates can be implemented via twist-based lattice surgery [8] or braiding twist defects [1,914]. State injection protocols yield arbitrary logical rotations [2].Symplectic doubles of codes yield fault-tolerant Clifford gates performed via Dehn twists [7].

Fault Tolerance

Fault-tolerant measurement of defects [2].Twisted double covers of codes yield fault-tolerant Clifford gates performed via Dehn twists [7].


Ground state of the toric code has been implemented with and without twists, and the non-Abelian braiding behavior of the twists, which realize Ising anyons, has been demonstrated [15]. Logical Clifford gates for \([[8,2,2]]\) and \([[10,2,3]]\) twist-defect surface codes realized in a trapped ion device by Quantinuum [7].



  • \([[4,2,2]]\) Four-qubit code — A small 6.6.6 color code is a \([[4,1,2]]\) subcode with three weight-three stabilizer generators [4; Fig. 7]; this code is equivalent to a twist-defect surface code on a tetrahedron inscribed in a sphere [7] via a single-qubit Clifford circuit.
  • Kitaev surface code — Twist-defect surface codes reduce to surface codes when there are no defects.
  • Rhombic dodecahedron surface code — The rhombic dodecahedron surface code is a twist-defect surface code whose degree-three vertices can be interpreted as disclination twists [16].
  • Triangular surface code
  • XZZX surface code — XZZX toric and planar codes can be treated in the general twist-defect surface code formalism [5].



H. Bombin, “Topological Order with a Twist: Ising Anyons from an Abelian Model”, Physical Review Letters 105, (2010) arXiv:1004.1838 DOI
M. B. Hastings and A. Geller, “Reduced Space-Time and Time Costs Using Dislocation Codes and Arbitrary Ancillas”, (2015) arXiv:1408.3379
T. J. Yoder and I. H. Kim, “The surface code with a twist”, Quantum 1, 2 (2017) arXiv:1612.04795 DOI
M. S. Kesselring et al., “The boundaries and twist defects of the color code and their applications to topological quantum computation”, Quantum 2, 101 (2018) arXiv:1806.02820 DOI
R. Sarkar and T. J. Yoder, “A graph-based formalism for surface codes and twists”, (2023) arXiv:2101.09349
H. Bombín et al., “Logical Blocks for Fault-Tolerant Topological Quantum Computation”, PRX Quantum 4, (2023) arXiv:2112.12160 DOI
S. Burton, E. Durso-Sabina, and N. C. Brown, “Genons, Double Covers and Fault-tolerant Clifford Gates”, (2024) arXiv:2406.09951
C. Chamberland and E. T. Campbell, “Circuit-level protocol and analysis for twist-based lattice surgery”, Physical Review Research 4, (2022) arXiv:2201.05678 DOI
H. Bombin and M. A. Martin-Delgado, “Quantum measurements and gates by code deformation”, Journal of Physics A: Mathematical and Theoretical 42, 095302 (2009) arXiv:0704.2540 DOI
A. Kitaev and L. Kong, “Models for Gapped Boundaries and Domain Walls”, Communications in Mathematical Physics 313, 351 (2012) arXiv:1104.5047 DOI
A. G. Fowler et al., “Surface codes: Towards practical large-scale quantum computation”, Physical Review A 86, (2012) arXiv:1208.0928 DOI
H. Zheng, A. Dua, and L. Jiang, “Demonstrating non-Abelian statistics of Majorana fermions using twist defects”, Physical Review B 92, (2015) arXiv:1508.04166 DOI
B. J. Brown et al., “Poking Holes and Cutting Corners to Achieve Clifford Gates with the Surface Code”, Physical Review X 7, (2017) arXiv:1609.04673 DOI
A. Benhemou, J. K. Pachos, and D. E. Browne, “Non-Abelian statistics with mixed-boundary punctures on the toric code”, Physical Review A 105, (2022) arXiv:2103.08381 DOI
S. Xu et al., “Digital Simulation of Projective Non-Abelian Anyons with 68 Superconducting Qubits”, Chinese Physics Letters 40, 060301 (2023) arXiv:2211.09802 DOI
A. J. Landahl, “The surface code on the rhombic dodecahedron”, (2020) arXiv:2010.06628
M. G. Gowda and P. K. Sarvepalli, “Quantum computation with generalized dislocation codes”, Physical Review A 102, (2020) DOI
T. D. Ellison, J. Sullivan, and A. Dua, “Floquet codes with a twist”, (2023) arXiv:2306.08027
A. J. Landahl and B. C. A. Morrison, “Logical fermions for fault-tolerant quantum simulation”, (2023) arXiv:2110.10280
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Zoo Code ID: twist_defect_surface

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“Twist-defect surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.
@incollection{eczoo_twist_defect_surface, title={Twist-defect surface code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Twist-defect surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.