| 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]. |