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 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].


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