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]. |
| Block quantum code | Transversal gates are logical gates on block codes that can be realized as tensor products of unitary operations acting on subsets of subsystems whose size is independent of \(n\). When the subsets are of size one and the single-subsystem unitaries are identical, then the gates are sometimes called strongly transversal. |
| Clifford code | Discrete subgroups of \(SU(2)\) can be realized transversally. |
| 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 [4,7]. |
| 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 [8]. 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]. These realize the \(2T\) binary tetrahedral subgroup of \(SU(2)\). |
| 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]. |
| Homological rotor code | All generalized Pauli gates are realized transversally. |
| Hypergraph product (HGP) 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 CSS 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 [8]. |
| Qubit stabilizer code | All stabilizer codes realize Pauli transformations transversally; for a single logical qubit, these a realize dicyclic subgroup of \(SU(2)\). More generally, transversal logical gates are in a finite level of the Clifford hierarchy, which is shown using stabilizer disjointness [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., \( U_L\ket\perp = \ket\perp\). |
| \(((7,2,3))\) permutation-invariant code | Binary icosahedral group \(2I\) gates can be realized transversally [22]. |
| \([[15, 7, 3]]\) Hamming-based CSS code | Single-qubit Clifford operations applied transversally yield the corresponding Clifford gates on one of the logical qubits [23]. 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 [24]. A transversal logical \(T^\dagger\) is implemented by applying a \(T\) gate on every qubit [25–27]. A subsystem version yields a transversal \(CCZ\) gate [21]. The code fails to have a transversal Hadamard gate; otherwise, it would violate the Eastin-Knill theorem. |
| \([[2^r-1, 1, 3]]\) quantum Reed-Muller code | \(Z\)-rotation by angle \(-\pi/2^{r-1}\) [28]. These are the smallest qubit stabilizer codes with such a gate [24]. |
| \([[2^r-1, 2^r-2r-1, 3]]\) Hamming-based CSS code | Pauli, Hadamard, and CNOT gates. |
| \([[2^{2r-1}-1,1,2^r-1]]\) 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 [29]. |
| \([[4,2,2]]\) CSS code | Transversal Pauli, Hadamard, and two-qubit \(R\) [30]. A transversal \(CZ\) gate is realized by the rotation \(\sqrt{Z}\otimes\sqrt{Z}^{\dagger}\otimes\sqrt{Z}^{\dagger}\otimes\sqrt{Z}\). |
| \([[7,1,3]]\) Steane code | All single-qubit Clifford gates, which realize the \(2O\) binary octahedral subgroup of \(SU(2)\) [17,31]. |
| \([[8,3,2]]\) code | CZ gates between any two logical qubits [32] and CCZ gate [32–34]. |
| \([[k+4,k,2]]\) 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. |