| 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. |