Here is a list of all quantum codes that admit transversal gates. Applicable to codes living in a tensor-product space, such gates can be written as a tensor product of unitary operations, with each operation acting on its corresponding subsystem.
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].
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].
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].
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 [9]. However, for sufficiently localized logical subsystems of holographic stabilizer codes, the set of transversally implementable logical operations is contained in the Clifford group [10].
Hypergraph product code Hadamard (up to logical SWAP gates) and control-\(Z\) on all logical qubits [11].
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 [12].
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 [10].
Triorthogonal code Admits transversal \(T\) gates [13] and the controlled-controlled-\(Z\) gate.
\([[15,1,3]]\) Reed-Muller code A transversal logical \(T^\dagger\) is implemented by applying a \(T\) gate on every qubit [14][15][16].
\([[4,2,2]]\) CSS code Transversal Pauli, Hadamard, and two-qubit \(R\) gates [17].
\([[5,1,3]]\) perfect code Pauli gates are transversal.

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]
Kfir Dolev et al., “Gauging the bulk: generalized gauging maps and holographic codes”. 2108.11402
[10]
S. Cree et al., “Fault-Tolerant Logical Gates in Holographic Stabilizer Codes Are Severely Restricted”, PRX Quantum 2, (2021). DOI; 2103.13404
[11]
Armanda O. Quintavalle, Paul Webster, and Michael Vasmer, “Partitioning qubits in hypergraph product codes to implement logical gates”. 2204.10812
[12]
J. E. Moussa, “Transversal Clifford gates on folded surface codes”, Physical Review A 94, (2016). DOI; 1603.02286
[13]
S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012). DOI; 1209.2426
[14]
E. Knill, R. Laflamme, and W. Zurek, “Threshold Accuracy for Quantum Computation”. quant-ph/9610011
[15]
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
[16]
E. T. Campbell, B. M. Terhal, and C. Vuillot, “Roads towards fault-tolerant universal quantum computation”, Nature 549, 172 (2017). DOI; 1612.07330
[17]
Daniel Gottesman, “Quantum fault tolerance in small experiments”. 1610.03507