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].
Calderbank-Shor-Steane (CSS) stabilizer code All CSS codes admit transversal Pauli and CNOT gates. Self-dual CSS codes admit a transversal Hadamard, completing the Clifford group. A CSS code is doubly even (triply even) if all $$X$$-type stabilizer generators have weight divisible by two (three); such codes yield a transversal $$S$$ ($$T$$) gate [4].
Color code Transversal CNOT can be implemented via braiding [5]. Universal transversal gates can be achieved in 3D color code using gauge fixing [6], lattice surgery [7], or code deformation [8][5].
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 $$G$$.
Doubled color code Doubled color codes are triply even, so they yield a transversal $$T$$ gate [4]. Using gauge fixing, the codes admit a Clifford + $$T$$ transversal gate set.
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.[9].
Five-qubit perfect code Pauli gates are transversal, along with a non-Pauli Hadamard-phase gate $$SH$$ and three-qubit Clifford operation $$M_3$$ [10].
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 $$G$$ [11].
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 [12]. However, for sufficiently localized logical subsystems of holographic stabilizer codes, the set of transversally implementable logical operations is contained in the Clifford group [13].
Hypergraph product code Hadamard (up to logical SWAP gates) and control-$$Z$$ on all logical qubits [14].
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 [15].
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 [13].
Qubit stabilizer code Transversal logical gates are in a finite level of the Clifford hierarchy [16] (see also [17][18]). Transversal gates for $$n\in\{1,2\}$$ are semi-Clifford [19].
Single-spin code When the physical Hilbert space is thought of a collective spin, logical gates for spin codes have the form $$U^{\otimes N}$$, where $$U$$ is a local rotation on the physical system.
Surface-17 code Pauli gates, CNOT gate, and $$H$$ gate (with relabeling).
Triorthogonal code Admits transversal $$T$$ gates [20] and the controlled-controlled-$$Z$$ gate [21].
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., $$U_L\ket\perp = \ket\perp$$.
$$[[15, 7, 3]]$$ Hamming-based CSS code Single-qubit Clifford operations applied transversally yield the corresponding Clifford gates on one of the logical qubits [22]. CNOT gate because it is a CSS code. Transversal CCZ gate [21].
$$[[15,1,3]]$$ quantum Reed-Muller code This code is the smallest qubit stabilizer code with a transversal gate outside of the Clifford group [23]. A transversal logical $$T^\dagger$$ is implemented by applying a $$T$$ gate on every qubit [24][25][26]. A subsystem version yields a transversal $$CCZ$$ gate [21]. The code fails to have a transversal Hadamard gate; otherwise, it would vioalate the Eastin-Knill theorem.
$$[[2^r-1, 1, 3]]$$ quantum Reed-Muller code $$Z$$-rotation by angle $$-\pi/2^{r-1}$$ [27].
$$[[2^{2r-1}-1,1,2^r-1]]$$ quantum punctured Reed-Muller code All single-qubit Clifford gates.
$$[[2m,2m-2,2]]$$ error-detecting code Transveral CNOT gates can be performed by first teleporting qubits into different code blocks [28].
$$[[4,2,2]]$$ CSS code Transversal Pauli, Hadamard, and two-qubit $$R$$ gates [29].
$$[[7,1,3]]$$ Steane code All single-qubit Clifford gates [30][17].
$$[[8,3,2]]$$ code CZ gates between any two logical qubits [31] and CCZ gate [32][33][31].
$$[[k+4,k,2]]$$ 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.

## 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]
Sergey Bravyi and Andrew Cross, “Doubled Color Codes”. 1509.03239
[5]
A. G. Fowler, “Two-dimensional color-code quantum computation”, Physical Review A 83, (2011). DOI; 0806.4827
[6]
H. Bombin, “Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes”. 1311.0879
[7]
Andrew J. Landahl and Ciaran Ryan-Anderson, “Quantum computing by color-code lattice surgery”. 1407.5103
[8]
H. Bombin, “Clifford gates by code deformation”, New Journal of Physics 13, 043005 (2011). DOI
[9]
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
[10]
Daniel Gottesman, “Stabilizer Codes and Quantum Error Correction”. quant-ph/9705052
[11]
P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020). DOI; 1902.07714
[12]
K. Dolev et al., “Gauging the bulk: generalized gauging maps and holographic codes”, Journal of High Energy Physics 2022, (2022). DOI; 2108.11402
[13]
S. Cree et al., “Fault-Tolerant Logical Gates in Holographic Stabilizer Codes Are Severely Restricted”, PRX Quantum 2, (2021). DOI; 2103.13404
[14]
Armanda O. Quintavalle, Paul Webster, and Michael Vasmer, “Partitioning qubits in hypergraph product codes to implement logical gates”. 2204.10812
[15]
J. E. Moussa, “Transversal Clifford gates on folded surface codes”, Physical Review A 94, (2016). DOI; 1603.02286
[16]
T. Jochym-O’Connor, A. Kubica, and T. J. Yoder, “Disjointness of Stabilizer Codes and Limitations on Fault-Tolerant Logical Gates”, Physical Review X 8, (2018). DOI; 1710.07256
[17]
Bei Zeng, Andrew Cross, and Isaac L. Chuang, “Transversality versus Universality for Additive Quantum Codes”. 0706.1382
[18]
Jonas T. Anderson and Tomas Jochym-O'Connor, “Classification of transversal gates in qubit stabilizer codes”. 1409.8320
[19]
B. Zeng, X. Chen, and I. L. Chuang, “Semi-Clifford operations, structure of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="script">C</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:math>hierarchy, and gate complexity for fault-tolerant quantum computation”, Physical Review A 77, (2008). DOI; 0712.2084
[20]
S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012). DOI; 1209.2426
[21]
A. Paetznick and B. W. Reichardt, “Universal Fault-Tolerant Quantum Computation with Only Transversal Gates and Error Correction”, Physical Review Letters 111, (2013). DOI; 1304.3709
[22]
R. Chao and B. W. Reichardt, “Fault-tolerant quantum computation with few qubits”, npj Quantum Information 4, (2018). DOI; 1705.05365
[23]
Stergios Koutsioumpas, Darren Banfield, and Alastair Kay, “The Smallest Code with Transversal T”. 2210.14066
[24]
E. Knill, R. Laflamme, and W. Zurek, “Threshold Accuracy for Quantum Computation”. quant-ph/9610011
[25]
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
[26]
E. T. Campbell, B. M. Terhal, and C. Vuillot, “Roads towards fault-tolerant universal quantum computation”, Nature 549, 172 (2017). DOI; 1612.07330
[27]
B. Zeng et al., “Local unitary versus local Clifford equivalence of stabilizer and graph states”, Physical Review A 75, (2007). DOI; quant-ph/0611214
[28]
D. Gottesman, “Theory of fault-tolerant quantum computation”, Physical Review A 57, 127 (1998). DOI; quant-ph/9702029
[29]
Daniel Gottesman, “Quantum fault tolerance in small experiments”. 1610.03507
[30]
Peter W. Shor, “Fault-tolerant quantum computation”. quant-ph/9605011
[31]
H. Chen et al., “Automated discovery of logical gates for quantum error correction (with Supplementary (153 pages))”, Quantum Information and Computation 22, 947 (2022). DOI; 1912.10063
[32]
A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015). DOI; 1503.02065
[33]
E. Campbell, “The smallest interesting colour code,” Online available at https://earltcampbell.com/2016/09/26/the-smallest-interesting-colour-code/ (2016), accessed on 2019-12-09.