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 |
|---|---|
| 2D color code | CNOT gate because the code is CSS.Hadamard gates for any qubit geometry which yields a self-dual CSS code.Transversal \(S\) gate [1,2]. |
| 3D Bacon-Shor code | Logical \(CCZ\) gates on three codeblocks of different orientations [3]. |
| 3D color code | Transversal action of \(T\) gates on color codes on general 3-manifolds realizes a \(CCZ\) gate on three logical qubits and is related to a topological invariant that is called the triple intersection number; this gate is related to the fact that this code admits a cup product structure [4].Transversal \(S\) gate on color codes on general 3-manifolds corresponds to a higher-form symmetry [4].Universal transversal gates can be achieved using lattice surgery or code deformation [5,6]. |
| 3D fermionic surface code | \(CCZ\) and \(CS\) gates can be obtained for the fermionic 3D surface code on certain manifolds by circuits that can be interpreted as moving and spreading lattice realizations of Kitaev chain and \(p+ip\) defects [7]. |
| 3D surface code | Transversal CZ and \(CCZ\) gates [4,8–13], with CZ gates formulated in terms of the slant product [14,15] or cup product [16] structures. |
| Bacon-Shor code | Logical Hadamard is transversal in symmetric Bacon-Shor codes up to a qubit permutation [17] and can be implemented with teleportation [18].Bacon-Shor codes on an \(m \times m^k\) lattice admit transversal \(k\)-qubit-controlled \(Z\) gates [3]. |
| Ball color code | The 3D ball codes on duals of the truncated octahedron, truncated cuboctahedron, and truncated icosidodecahedron have transveral \(CCZ\) gates. |
| Binary dihedral PI code | Binary dihedral group gates can be realized transversally, which include subgroups of any level of the Clifford hierarchy and subgroups which cannot be realized by any qubit stabilizer code. |
| Bivariate bicycle (BB) code | Logical Pauli operators and fold-transversal gates studied in Ref. [19,20]. |
| Block quantum code | Eastin-Knill theorem: Transversal gatesare 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 \(n\). For subsets of size one, gates are sometimes called strongly transversal the single-subsystem unitaries are identical and weakly transversal otherwise. A universal gate set for a finite-dimensional block quantum code cannot be transversal for any code that detects single-block errors due to the Eastin-Knill theorem [21]. |
| CSS-T code | A physical transversal \(T\) gate is either the identity (up to a global phase) or a logical gate [22]. |
| Capped color code (CCC) | Capped color codes in H (T) form admit a transversal Hadamard (T) gate. |
| Color code | Some color codes on \(D\)-dimensional lattices can transversally implement a gate at the \((D-1)\)st level of the Clifford hierarchy in the form of a \(Z\)-rotation by angle \(-\pi/2^D\) [23; Fig. 3]. |
| Covariant block quantum code | \(G\)-covariant codes defined on a tensor product space consisting of \(n\) subsystems are equivalent to codes with a transversal gate set realizing \(G\). |
| Cubic honeycomb color code | A code family on a truncated cube with particular boundary conditions admits a transversal control-\(S\) gate via physical \(T\) and \(T^{\dagger}\) gates [24]. |
| Doubled color code | Doubled color codes are triply even, so they yield a transversal \(T\) gate [25]. 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 [26]. |
| Galois-qudit RS code | There exists an order \([[n,\Theta(n),\Theta(n)]]_{n^2}\) punctured RS code family that admits transversal \(CCZ\) gates for any three logical qubits [27]. This code can be treated as a qubit code by decomposing each Galois qudit into a Kronecker product of several qubits; see [28,30–32][29; Sec. 5.3]. This yields a qubit code family that is asymptotically good up to poly-logarithmic factors [27]. |
| Galois-qudit expander code | Hypergraph products of expander codes with RS inner codes yield \([[n,k\geq n^{1-\epsilon},d\geq n^{1/r}/\text{poly}(\log n)]]_q\) QLDPC Galois-qudit quantum expander codes with transversal \(C^{r-1} Z\) gates [33]. |
| Generalized quantum divisible code | A level-\(\nu\) generalized quantum divisible code admits a diagonal transversal gate at the \(\nu\)th level of the Clifford hierarchy [34; Lemma V.3]. |
| 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 \(G\) [35]. |
| Heavy-hexagon code | CNOT gates are transveral for this code. However, for most architectures, all logical gates would be implemented using lattice surgery methods. |
| Hermitian qubit code | All code automorphisms lie in the Clifford group [36; Corr. 16], so transversal physical gates implement only Clifford logical gates.Transversal \(SH\) and \(HS\) "facet" gates (a.k.a. \(M_3\) gates) which cyclically permute Paulis as \(X \to Y\), \(Y \to Z\), and \(Z \to X\) [37; Sec. 8.2].The three-block transversal gate mapping each physical \(X \to XYZ\) and each \(Z \to ZXY\) implements a logical gate [38][39; Exam. 2].A qubit stabilizer code is Hermitian if and only if a transversal \(R\) gate leaves the stabilizer group invariant [40]. |
| Holographic tensor-network code | There exist holographic approximate codes with arbitrary transversal gate sets for any compact Lie group [41]. However, for sufficiently localized logical subsystems of holographic stabilizer codes, the set of transversally implementable logical operations is contained in the Clifford group [42]. |
| Homological code | Locality preserving operations can be determined for stacks of homoogical codes in any dimension [43]. |
| Homological rotor code | All generalized Pauli gates are realized transversally. |
| Honeycomb (6.6.6) color code | CNOT gate because the code is CSS.Hadamard gates for any qubit geometry which yields a self-dual CSS code.Transversal \(S\) gate [1,2]. |
| Hypergraph product (HGP) code | Hadamard (up to logical SWAP gates) and control-\(Z\) on all logical qubits [44].Patch-transversal gates inherited from the automorphism group of the underlying classical codes [45; Appx. D]. |
| Kitaev surface code | Transversal Clifford gates can be done on folded surface codes [46].Fold-transversal initialization of the \(|Y\rangle\) logical state [46–49]. |
| Lattice stabilizer code | Fold-transversal logical gates can be obtained from crystalline symmetries [50]. |
| Loop toric code | Only logical Clifford gates can be implemented transversally when defined on a hypercubic lattice [51]. |
| Modular-qudit color code | Some modular-qudit color codes on \(D\)-dimensional lattices can transversally implement a gate at the \((D-1)\)st level of the qudit Clifford hierarchy [52]. |
| 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 [42]. |
| Prime-qudit RM code | An odd-prime-qudit CSS code family constructed from first-order punctured GRM codes transversally implements a diagonal gate at any level of the qudit Clifford hierarchy [53,54]. |
| Prime-qudit triorthogonal code | Admits a transversal gate from the third level of the qudit Clifford hierarchy [55]. |
| Quantum AG code | There exist three asymptotically good code families [31,32,56] that admit a diagonal transversal gate at the third level of the Clifford hierarchy.There exists an asymptotically good code family that admits three-Galois-qudit non-Clifford gates for any three logical qubits [27]. |
| Quantum Reed-Muller code | Stabilizer generators are Pauli strings can be defined as acting on subsets of qubits corresponding to subcubes of the Hamming \(n\)-cube (a.k.a. Boolean hypercube) [57]. Transversal \(Z\)-rotations by angles \(\pi/2^k\) acting on subcubes can implement logical multi-controlled-\(Z\) gates [57].The \([[2^m,{m \choose r}, 2^{\min(r,m-r)}]]\) family, where \(r\) divides \(m\), admits diagonal gates in the form of \(Z\)-rotations by angle \(\pi/2^{m/r}\) [58–60][22; Exam. 8 and Thm. 19]. Of these, the sub-family for \(m=2r\) admits logical Clifford group gates via permutations, transversal gates, and fold-transversal gates [61].The family constructed out of shortened RM codes with parameters \([[\sum_{i=w+1}^m \binom{m}{i}, \sum_{i=0}^{w} \binom{m}{i}, \sum_{i=w+1}^{r+1} \binom{r+1}{i}]]\) for integers \(m > 2r\) and \(r > w \geq 0\) admits a transversal gate at the \(\nu\)th level in the hierarchy whenever \(m > \nu r\) [62; Thm. 1]. |
| Quantum divisible code | Doubly even codes can yield a transversal \(S\) gate, while triply even codes yield a logical \(T\) gate for odd \(n\) via physical action of \(T\) gates on each qubit [25].If the \(X\)-type stabilizers of a CSS code form an \(\nu\)-even classical code, and if all \(X\)-type logicals are \((\nu-1)\)-even, then the code admits a diagonal transversal gate in the \(\nu\)th level of the Clifford hierarchy [63; Prop. 8]. |
| Quantum quadratic-residue (QR) code | Qubit quantum QR codes admit transversal implementations of the single-qubit Clifford group [64]. They yield a family of high-distance triorthogonal codes [64] via the doubling transformation [25]; such codes admit transversal implementations of the \(T\) gate. |
| Quantum rainbow code | Hypergraph products of color codes yield quantum rainbow codes with growing distance and transversal gates in the Clifford hierarchy [65]. |
| Qubit CSS code | Transversal Pauli gates since qubit CSS codes are qubit stabilizer codes.Transversal CNOT gates preserve the logical subspace, up to \(X\)-type Paulis, iff a qubit stabilizer code is CSS [30,66]. The Paulis are necessary for when the code is stabilized by stabilizers with a minus in front of them, e.g., \(-XXXX\) and \(ZZZZ\).Self-dual CSS codes admit a transversal Hadamard gate. There are criteria for when such codes realize logical gates from tensor products of \(S\) and \(S^{\dagger}\) gates [67].Fold-transversal [8,46,68] Clifford gates are transversal gates combined with qubit permutations. Some of these can be obtained from automorphism groups of the underlying classical codes [69; Thms. 2-3].Necessary and sufficient conditions for a CSS code to yield a transversal gate in the Clifford hierarchy have been formulated [72][70; Thm. 9][71; Thm. 5]. There are routines that can determine what diagonal gates in the Clifford hierarchy are realized by a code [73,74].CSS code families with asymptotic rate \(> 1/3\) and distance at \(\geq 3\) do not admit logical qubit permutations from physical permutations [75]. |
| Qubit code | A qubit code is \(U\)-quasi-transversal if it can realize the logical gate \(U\) in the third level of the Clifford hierarchy using the physical gate \(C T^{\otimes n}\), where \(C\) is some Clifford gate [58; Def. 4]. |
| Qubit stabilizer code | All stabilizer codes realize Pauli transformations transversally; for a single logical qubit, these realize a dicyclic subgroup of \(SU(2)\). Several algorithms exist for finding logical Pauli operators [37,76,77].The four-block transversal gate mapping each \(X \to IXXX\) and each \(Z \to IZZZ\) implements the same logical gate on all qubits [37][39; Exam. 1].Transversal logical gates are in a finite level of the Clifford hierarchy, which is shown using stabilizer disjointness [78] (see also [79,80]). Transversal gates for \(n\in\{1,2\}\) are semi-Clifford [81].No stabilizer code can implement a classical universal gate set transversally [82].Fold-transversal gates have been extended from qubit CSS codes to qubit stabilizer codes, and there is an algorithm to determine them from the stabilizer group [20].Computation can be sped up substantially for codes that admit transversal measurements of logical \(X\) and \(Z\) [83] |
| Rotated surface code | Fold-transversal \(S\) gate [49,84]. |
| 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. |
| Square-octagon (4.8.8) color code | CNOT gate because the code is CSS.Hadamard gates for any qubit geometry which yields a self-dual CSS code.Transversal \(S\) gate [1,2].Single-qubit Clifford and CNOT gates between qubits encoded in holes in the lattice can be implemented via braiding [6]. |
| Subset-Sum-Linear-Programming (SS-LP) code | SS-LP codes are optimized to admit diagonal gates transversally and include \(((7,2,3))\) codes that realize the \(\mathsf{BD}_{16}\) and \(\mathsf{BD}_{32}\) groups transversally, yielding \(T\) and \(\sqrt{T}\) gates, respectively. Larger codes include an \(((8,2,3))\) code that transversally realizes \(\mathsf{BD}_{64}\). |
| Subsystem Hypergraph Product Simplex (SHYPS) code | Cross-block transversal CNOT gates \(\prod_{i=1}^n CNOT_{i, n+(\sigma_1\otimes\sigma_2)(i)}\) for \(\sigma_1, \sigma_2\) authomorphisms of the classical simplex code. These operators implement a generating set of cross-block logical CNOT gates.In-block phase-type fold-transversal gates with \(SP_{2k}(\mathbb{F}_2)\) representation \(\begin{bmatrix} I & (\sigma \otimes \sigma^T)\tau \\ 0 & 1\end{bmatrix}\) for \(\sigma\) an authomorphism of the classical simplex code, and \(\tau\) a self-inverse permutation which acts like \(\tau (e_i \otimes e_j) = e_j \otimes e_i\) for the canonical basis \(\{e_i\}\) of \(GF(2)^\sqrt{n}\). These operators implement a generating set of logical in-block diagonal gates [85].Cross-block phase-type fold-transversal gates \(\prod_{i=1}^n CZ_{i, n+(\sigma_1\otimes\sigma_2)\tau(i)}\) for \(\sigma_1, \sigma_2\) authomorphisms of the classical simplex code and \(\tau\) as above. These operators implement a generating set of logical cross-block diagonal gates [85].Fold-transversal Hadamard gate \(H^{\otimes n} \tau \), with \(\tau\) as above. Implements logical Hadamard-SWAP operator \(H^{\otimes k} \tau_k \), with \(\tau_k\) defined analogously to \(\tau\) [85]. |
| Tensor-network code | The quantum Lego framework yields an \([[8,1,2]]\) stabilizer code admits a transversal logical \(T\) gate that originates from that of a trivial (distance-one) \([[7,1]]\) code. This code, in turn, is obtained from the \([[15,1,3]]\) code [86]. |
| Tetrahedral color code | A \([[5d^3-12d^2+16,3,d]]\) close relative of this code admits a logical \(CCZ\) gate via single-qubit rotations [87]. |
| Toric code | Transversal logical Pauli gates correspond to Pauli strings on non-trivial loops of the torus. |
| Triangular surface code | Triangle codes admit transversal order-three single-qubit Clifford gates, e.g., \(\bar{SH}\) [88]. |
| Triorthogonal code | Transversal action of \(T\) gates on all qubits, followed by a particular pattern of \(CZ\) and \(S\) gates, will realize a logical \(T\) gate [89; Lemma 2]. When an additional condition on logical-\(X\) operators is satisfied, the \(CZ\) and \(S\) gates are no longer necessary [22; Thm. 14].Triorthogonality is necessary but not sufficient for physical transversal \(T\) gates on each qubit to realize the identity logical gate [22; Thm. 12].Certain codes realize controlled-controlled-\(Z\) gates [90], realized via physical \(CCZ\) gates on three code blocks. |
| Truncated trihexagonal (4.6.12) color code | CNOT gate because the code is CSS.Hadamard gates for any qubit geometry which yields a self-dual CSS code.Transversal \(S\) gate [1,2]. |
| Twisted \(1\)-group code | All gates in the underlying twisted \(1\)-group. See [91; Table II] for other notable groups including the sporadic groups. |
| Two-block group-algebra (2BGA) codes | Logical Pauli operators and fold-transversal gates have been studied [19,20]. |
| Union-Jack color code | Transversal logical Clifford gates [92]. |
| Valence-bond-solid (VBS) code | Two classes of (approximate) VBS codes have \(SU(q)\) transversal gates [93; Tab. III]. |
| 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., \( U_L\ket\perp = \ket\perp\). |
| \(((5,3,2))_3\) qutrit code | \(\Sigma(360\phi)\) group gates can be realized transversally. |
| \(((6,2,3))\) \(\mathbb{Z}_{10}\) code | The \(\mathbb{Z}_{10}\) group can be realized transversally [94]. This is the smallest distance-three code supporting a transversal gate outside of the Clifford group. |
| \(((7,2,3))\) Pollatsek-Ruskai code | Binary icosahedral group \(2I\) gates can be realized transversally [95]. See Ref. [94] for other non-PI codes realizing \(2I\) gates transversally. |
| \((1,3)\) 4D toric code | Logical \(CCCZ\) gate on a hyper-diamond lattice [51]. |
| \([[10,1,2]]\) CSS code | Logical \(T\) gate via application of physical \(T\), \(T^{\dagger}\), and \(CCZ\) gates [96]. |
| \([[11,1,5]]_3\) qutrit Golay code | All single-qutrit encoded Clifford gates [97]. |
| \([[12,2,2]]\) CSS code | Logical \(CS\) gate via application of physical \(T\) and \(T^{\dagger}\) gates [73]. |
| \([[12,2,4]]\) carbon code | Two-block CNOT gates are transversal because the code is CSS.Automorphism groups of the underlying classical codes can yield transversal Clifford gates when combined with qubit permutations [69]. In particular, logical Hadamard is realized by a transversal physical Hadamard followed by a qubit permutation, and a logical one-block CNOT is implemented by a qubit permutation [98,99]. |
| \([[13,1,5]]\) cyclic code | No non-Pauli transversal gates. |
| \([[144,12,12]]\) gross code | Logical Pauli operators and fold-transversal gates studied in Ref. [20]. |
| \([[15, 7, 3]]\) quantum Hamming code | CNOT gate because it is a CSS code.Single-qubit Clifford operations applied transversally yield the corresponding Clifford gates on one of the logical qubits [100].Automorphism groups of the underlying classical codes can yield transversal Clifford gates when combined with qubit permutations [69; Sec. IV.A].Transversal \(CCZ\) gate [90]. |
| \([[15,1,3]]\) quantum Reed-Muller code | A transversal logical \(T\) is implemented by applying a \(T^\dagger\) gate on every qubit [101–103]. This is the smallest qubit stabilizer code with a (strongly) transversal gate outside of the Clifford group [104].A subsystem version yields a transversal \(CCZ\) gate [90]. |
| \([[16,6,4]]\) Tesseract color code | Global transversal \(S\) implements a logical circuit composed of \(CZ\) and \(Z\) gates [57,57,105] |
| \([[17,1,5]]\) 4.8.8 color code | Transversal implementation of the logical Clifford group. |
| \([[23, 1, 7]]\) Quantum Golay code | Single-qubit Clifford group by choosing \(\overline{U}=U^{\otimes 23}\) for every Clifford unitary \(U\) [106]. |
| \([[2^D,D,2]]\) hypercube quantum code | CZ, CCZ, and generalized \(CZ\) gates at the \((D-1)\)-st level of the Clifford hierarchy [107][108; Exam. 6.10]. CNOT and SWAP gates can be realized by qubit permutations [87]. |
| \([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code | Pauli, Hadamard, and CNOT gates. |
| \([[2^r-1,1,3]]\) simplex code | Each code transversally implements a diagonal gate at the \((r-1)\)st level of the Clifford hierarchy in the form of a \(Z\)-rotation by angle \(-\pi/2^{r-1}\) [109,110]. These are the smallest distance-three qubit stabilizer codes with such a (strongly) transversal gate [104]. |
| \([[2^{2r-1}-1,1,2^r-1]]\) 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 [38; Sec. VII].For \(2m\) being a multiple of four, this code houses a transversal representation of the single-qubit Clifford group [111; Sec. 3.1]. Its \(n\)-block version houses a transversal representation of the \(n\)-qubit Clifford group [111; Secs. 3.1 and 3.4]. The commutant of transversal representations of the Clifford group consists of qubit permutations and projections onto the \([[2m,2m-2,2]]\) error-detecting code [112]. |
| \([[30,8,3]]\) Bring code | Clifford group of four of the eight logical qubits can be done by transversal gates combined with qubit permutations [68]. |
| \([[3k + 8, k, 2]]\) triorthogonal code | The code admits a transversal \(T\) gate [89]. |
| \([[4,2,2]]\) Four-qubit code | Transversal Pauli, Hadamard, and two-qubit \(S=\sqrt{Z}\) [113] (see also [114]) as well as transversal inter-block CNOT gates since the code is CSS.This code is the only four-qubit subspace to house a transversal representation of the Clifford group but not of the single-qubit unitary group [111; Eq. (104)]. Its \(n\)-block version is a \([[4n,2n,2]]\) code, which houses a signed permutation representation of the \(n\)-qubit Clifford group [111; Sec. 3.1 and Appx. B].A transversal \(CZ\) gate is realized by the rotation \(\sqrt{Z}\otimes\sqrt{Z}^{\dagger}\otimes\sqrt{Z}^{\dagger}\otimes\sqrt{Z}\).Adding \(XYZI\) to the stabilizer group produces a \([[4,1,2]]\) subcode that admits weight-two transversal logical Pauli operations [115]. |
| \([[49,1,5]]\) triorthogonal code | The code admits a transversal \(T\) gate [89; Appx. B]. |
| \([[6,4,2]]\) error-detecting code | CNOT and Hadamard gates [116; Appx. B].A \(CZ\) gate implemented by transversal \(S\) and \(S^{\dagger}\) [114]; see also [96].The projector on this code commutes with the transversal action of the Clifford group [112] but the code does not house a representation of the group since diagonal gates yield zero when projected into the codespace [117]. |
| \([[6k+2,3k,2]]\) Campbell-Howard code | Quasi-transversal \(CZZ^{\otimes k}\) gates [58]. |
| \([[7,1,3]]\) Steane code | The single-qubit Clifford group [66,79]. |
| \([[7,1,3]]\) twist-defect surface code | Single-qubit Clifford group [88,114]. |
| \([[8, 3, 3]]\) Eight-qubit Gottesman code | Permutation-based gates [69; Sec. IV.D].No gates outside of the Pauli group were found in Ref. [114]. |
| \([[8,2,2]]\) hyperbolic color code | Applying transversal \(S\) and \(S^{\dagger}\), \(\sqrt{X}\), and Hadamard gates yields various logical gates [114]. |
| \([[8,3,2]]\) Smallest interesting color code | CZ gates between any two logical qubits [114] and (weakly) transversal \(CCZ\) gate [8,107,114]. |
| \([[9,1,3]]\) Surface-17 code | Pauli gates, CNOT gate, and \(H\) gate (with relabeling). |
| \([[k+4,k,2]]\) 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. |
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