Name | Transversal gates |
---|---|

Bacon-Shor code | Logical Hadamard is transversal in symmetric Bacon-Shor codes up to a qubit permutation [1] and can be implemented with teleportation [2]. Bacon-Shor codes on an \(m \times mk\) lattice admit transversal \(k\)-qubit-controlled \(Z\) gates [3]. |

Calderbank-Shor-Steane (CSS) stabilizer code | CNOT gates. Self-dual CSS codes admit transversal Clifford gates. |

Color code | Transversal CNOT can be implemented via braiding [4]. Universal transversal gates can be achieved in 3D color code using gauge fixing [5], lattice surgery [6], or code deformation [7][4]. |

Covariant code | \(G\)-covariant codes defined on a tensor product space consisting of \(n\) subsystems are equivalent to codes with a transversal gate set realizing \(G\). |

Fibonacci string-net code | A universal transversal gate set could be implemented in a folded version of this code using the techniques introduced in Ref.[8]. |

Five-qubit perfect code | Pauli gates are transversal, along with the one non-Pauli Hadmard times phase gate. |

Group GKP code | Group-GKP codes corresponding to the \(G^{k_1} \subseteq G^{ k_2} \subset G^{n}\) group construction admit \(X\)-type transversal Pauli gates representing \(G\) [9]. |

H code | Hadamard and \(TXT^{\dagger}\) gates, with the latter Clifford-equivalent to Hadamard, and where \(T=\exp(i\pi(I-Z)/8)\) is the \(\pi/8\)-rotation gate. |

Heavy-hexagon code | CNOT gates are transveral for this code. However, for most architectures, all logical gates would be implemented using lattice surgery methods. |

Holographic code | There exist holographic approximate codes with arbitrary transversal gate sets for any compact Lie group [10]. However, for sufficiently localized logical subsystems of holographic stabilizer codes, the set of transversally implementable logical operations is contained in the Clifford group [11]. |

Hypergraph product code | Hadamard (up to logical SWAP gates) and control-\(Z\) on all logical qubits [12]. |

Kitaev surface code | Transversal Pauli gates exist and are based on non-trivial loops on surface. Transversal Clifford gates can be done on folded surface codes [13]. |

Pastawski-Yoshida-Harlow-Preskill (HaPPY) code | For locality-preserving physical gates on the boundary, the set of transversally implementable logical operations in the bulk is strictly contained in the Clifford group [11]. |

Surface-17 code | Pauli gates, CNOT gate, and \(H\) gate (with relabeling). |

Triorthogonal code | Admits transversal \(T\) gates [14] and the controlled-controlled-\(Z\) gate. |

W-state code | All logical gates can be implemented transversally. The logical unitary \(U_L\) can be performed with the physical unitary \(U_L\otimes U_L\otimes\cdots\otimes U_L\), where on the physical space \(U_L\) is taken to act trivially on \(\ket\perp\), i.e., \( U_L\ket\perp = \ket\perp\). |

\([[15,1,3]]\) quantum Reed-Muller code | A transversal logical \(T^\dagger\) is implemented by applying a \(T\) gate on every qubit [15][16][17]. |

\([[4,2,2]]\) CSS code | Transversal Pauli, Hadamard, and two-qubit \(R\) gates [18]. |

## References

- [1]
- P. Aliferis and A. W. Cross, “Subsystem Fault Tolerance with the Bacon-Shor Code”, Physical Review Letters 98, (2007). DOI; quant-ph/0610063
- [2]
- X. Zhou, D. W. Leung, and I. L. Chuang, “Methodology for quantum logic gate construction”, Physical Review A 62, (2000). DOI; quant-ph/0002039
- [3]
- Theodore J. Yoder, “Universal fault-tolerant quantum computation with Bacon-Shor codes”. 1705.01686
- [4]
- A. G. Fowler, “Two-dimensional color-code quantum computation”, Physical Review A 83, (2011). DOI; 0806.4827
- [5]
- H. Bombin, “Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes”. 1311.0879
- [6]
- Andrew J. Landahl and Ciaran Ryan-Anderson, “Quantum computing by color-code lattice surgery”. 1407.5103
- [7]
- H. Bombin, “Clifford gates by code deformation”, New Journal of Physics 13, 043005 (2011). DOI
- [8]
- G. Zhu, M. Hafezi, and M. Barkeshli, “Quantum origami: Transversal gates for quantum computation and measurement of topological order”, Physical Review Research 2, (2020). DOI; 1711.05752
- [9]
- P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020). DOI; 1902.07714
- [10]
- K. Dolev et al., “Gauging the bulk: generalized gauging maps and holographic codes”, Journal of High Energy Physics 2022, (2022). DOI; 2108.11402
- [11]
- S. Cree et al., “Fault-Tolerant Logical Gates in Holographic Stabilizer Codes Are Severely Restricted”, PRX Quantum 2, (2021). DOI; 2103.13404
- [12]
- Armanda O. Quintavalle, Paul Webster, and Michael Vasmer, “Partitioning qubits in hypergraph product codes to implement logical gates”. 2204.10812
- [13]
- J. E. Moussa, “Transversal Clifford gates on folded surface codes”, Physical Review A 94, (2016). DOI; 1603.02286
- [14]
- S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012). DOI; 1209.2426
- [15]
- E. Knill, R. Laflamme, and W. Zurek, “Threshold Accuracy for Quantum Computation”. quant-ph/9610011
- [16]
- J. T. Anderson, G. Duclos-Cianci, and D. Poulin, “Fault-Tolerant Conversion between the Steane and Reed-Muller Quantum Codes”, Physical Review Letters 113, (2014). DOI; 1403.2734
- [17]
- E. T. Campbell, B. M. Terhal, and C. Vuillot, “Roads towards fault-tolerant universal quantum computation”, Nature 549, 172 (2017). DOI; 1612.07330
- [18]
- Daniel Gottesman, “Quantum fault tolerance in small experiments”. 1610.03507