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.

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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 LDPC (QLDPC) code There are recipes to determine transversal gates for asymmetric QLDPC codes [57].
Quantum Reed-Muller (RM) 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) [58]. Transversal \(Z\)-rotations by angles \(\pi/2^k\) acting on subcubes can implement logical multi-controlled-\(Z\) gates [58].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 \(\pi/2^{m/r}\) [59–61][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 [62].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 \(m > \nu r\) [63; 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 [64; Prop. 8].
Quantum quadratic-residue (QR) code Qubit quantum QR codes admit transversal implementations of the single-qubit Clifford group [65]. They yield a family of high-distance triorthogonal codes [65] 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 [66].
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 [30,67]. 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 [68].Fold-transversal [8,46,69] Clifford gates are transversal gates combined with qubit permutations. Some of these can be obtained from automorphism groups of the underlying classical codes [70; Thms. 2-3].Necessary and sufficient conditions for a CSS code to yield a transversal gate in the Clifford hierarchy have been formulated [73][71; Thm. 9][72; Thm. 5]. There are routines that can determine what diagonal gates in the Clifford hierarchy are realized by a code [74,75].CSS code families with asymptotic rate \(> 1/3\) and distance at \(\geq 3\) do not admit logical qubit permutations from physical permutations [76].
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 [59; 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 [37,77,78].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 [79] (see also [80,81]). Transversal gates for \(n\in\{1,2\}\) are semi-Clifford [82].No stabilizer code can implement a classical universal gate set transversally [83].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 \(Z\) [84]
Rotated surface code Fold-transversal \(S\) gate [49,85].
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 [86].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 [86].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 \(\tau\) [86].
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 [87].
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 [88].
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., \(\bar{SH}\) [89].
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 [90; 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 [91], 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 [92; 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 [93].
Valence-bond-solid (VBS) code Two classes of (approximate) VBS codes have \(SU(q)\) transversal gates [94; 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 \(\Sigma(360\phi)=3.A_6\) group gates can be realized transversally.
\(((6,2,3))\) \(\mathbb{Z}_{10}\) code The \(\mathbb{Z}_{10}\) group can be realized transversally [95]. 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 [96]. See Ref. [95] 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 [97].
\([[11,1,5]]_3\) qutrit Golay code All single-qutrit encoded Clifford gates [98].
\([[12,2,2]]\) CSS code Logical \(CS\) gate via application of physical \(T\) and \(T^{\dagger}\) gates [74].
\([[12,2,4]]\) 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 [70]. 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 [99,100].
\([[13,1,5]]\) 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 [101].Automorphism groups of the underlying classical codes can yield transversal Clifford gates when combined with qubit permutations [70; Sec. IV.A].Transversal \(CCZ\) gate [91].
\([[15,1,3]]\) quantum RM code A transversal logical \(T\) is implemented by applying a \(T^\dagger\) gate on every qubit [102–104]. This is the smallest qubit stabilizer code with a (strongly) transversal gate outside of the Clifford group [105].A subsystem version yields a transversal \(CCZ\) gate [91].
\([[16,6,4]]\) Tesseract color code Global transversal \(S\) implements a logical circuit composed of \(CZ\) and \(Z\) gates [58,58,106]
\([[17,1,5]]\) 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 \(U\) [107].
\([[2^D,D,2]]\) hypercube quantum code CZ, CCZ, and generalized \(CZ\) gates at the \((D-1)\)-st level of the Clifford hierarchy [108][109; Exam. 6.10]. CNOT and SWAP gates can be realized by qubit permutations [88].
\([[2^r-1, 2^r-2r-1, 3]]\) 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 \(-\pi/2^{r-1}\) [110,111]. These are the smallest distance-three qubit stabilizer codes with such a (strongly) transversal gate [105].
\([[2^{2r-1}-1,1,2^r-1]]\) quantum punctured RM 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 [112; Sec. 3.1]. Its \(n\)-block version houses a transversal representation of the \(n\)-qubit Clifford group [112; 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 [113].
\([[30,8,3]]\) Bring code Clifford group of four of the eight logical qubits can be done by transversal gates combined with qubit permutations [69].
\([[3k + 8, k, 2]]\) triorthogonal code The code admits a transversal \(T\) gate [90].
\([[4,2,2]]\) Four-qubit code A tensor product of Hadamard gates applies a Hadamard gate to both logical qubits, and a tensor product of \(S=\sqrt{Z}\) gates applies a \(CZ\) gate followed by a logical \(Z\) on both qubits [114] (see also [115]).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 [112; Eq. (104)]. Its \(n\)-block version is a \([[4n,2n,2]]\) code, which houses a signed permutation representation of the \(n\)-qubit Clifford group [112; 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 [116].
\([[49,1,5]]\) triorthogonal code The code admits a transversal \(T\) gate [90; Appx. B].
\([[6,4,2]]\) error-detecting code CNOT and Hadamard gates [117; Appx. B].A \(CZ\) gate implemented by transversal \(S\) and \(S^{\dagger}\) [115]; see also [97].The projector on this code commutes with the transversal action of the Clifford group [113] but the code does not house a representation of the group since diagonal gates yield zero when projected into the codespace [118].
\([[6k+2,3k,2]]\) Campbell-Howard code Quasi-transversal \(CZZ^{\otimes k}\) gates [59].
\([[7,1,3]]\) Steane code The single-qubit Clifford group [67,80].
\([[7,1,3]]\) twist-defect surface code Single-qubit Clifford group [89,115].
\([[8, 3, 3]]\) Eight-qubit Gottesman code Permutation-based gates [70; Sec. IV.D].No gates outside of the Pauli group were found in Ref. [115].
\([[8,2,2]]\) hyperbolic color code Applying transversal \(S\) and \(S^{\dagger}\), \(\sqrt{X}\), and Hadamard gates yields various logical gates [115].
\([[8,3,2]]\) Smallest interesting color code CZ gates between any two logical qubits [115] and (weakly) transversal \(CCZ\) gate [8,108,115].
\([[9,1,3]]\) 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.

References

[1]
H. Bombin and M. A. Martin-Delgado, “Topological Quantum Distillation”, Physical Review Letters 97, (2006) arXiv:quant-ph/0605138 DOI
[2]
A. J. Landahl, J. T. Anderson, and P. R. Rice, “Fault-tolerant quantum computing with color codes”, (2011) arXiv:1108.5738
[3]
T. J. Yoder, “Universal fault-tolerant quantum computation with Bacon-Shor codes”, (2017) arXiv:1705.01686
[4]
G. Zhu, S. Sikander, E. Portnoy, A. W. Cross, and B. J. Brown, “Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries”, (2024) arXiv:2310.16982
[5]
H. Bombin, “Clifford gates by code deformation”, New Journal of Physics 13, 043005 (2011) arXiv:1006.5260 DOI
[6]
A. G. Fowler, “Two-dimensional color-code quantum computation”, Physical Review A 83, (2011) arXiv:0806.4827 DOI
[7]
M. Barkeshli, P.-S. Hsin, and R. Kobayashi, “Higher-group symmetry of (3+1)D fermionic \(\mathbb{Z}_2\) gauge theory: Logical CCZ, CS, and T gates from higher symmetry”, SciPost Physics 16, (2024) arXiv:2311.05674 DOI
[8]
A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
[9]
M. Vasmer and D. E. Browne, “Three-dimensional surface codes: Transversal gates and fault-tolerant architectures”, Physical Review A 100, (2019) arXiv:1801.04255 DOI
[10]
Y.-A. Chen and S. Tata, “Higher cup products on hypercubic lattices: Application to lattice models of topological phases”, Journal of Mathematical Physics 64, (2023) arXiv:2106.05274 DOI
[11]
G. Zhu, T. Jochym-O’Connor, and A. Dua, “Topological Order, Quantum Codes, and Quantum Computation on Fractal Geometries”, PRX Quantum 3, (2022) arXiv:2108.00018 DOI
[12]
Y. Wang, Y. Wang, Y.-A. Chen, W. Zhang, T. Zhang, J. Hu, W. Chen, Y. Gu, and Z.-W. Liu, “Efficient fault-tolerant implementations of non-Clifford gates with reconfigurable atom arrays”, npj Quantum Information 10, (2024) arXiv:2312.09111 DOI
[13]
Z. Song and G. Zhu, “Magic Boundaries of 3D Color Codes”, (2024) arXiv:2404.05033
[14]
B. Yoshida, “Gapped boundaries, group cohomology and fault-tolerant logical gates”, Annals of Physics 377, 387 (2017) arXiv:1509.03626 DOI
[15]
M. Barkeshli, Y.-A. Chen, S.-J. Huang, R. Kobayashi, N. Tantivasadakarn, and G. Zhu, “Codimension-2 defects and higher symmetries in (3+1)D topological phases”, SciPost Physics 14, (2023) arXiv:2208.07367 DOI
[16]
N. P. Breuckmann, M. Davydova, J. N. Eberhardt, and N. Tantivasadakarn, “Cups and Gates I: Cohomology invariants and logical quantum operations”, (2024) arXiv:2410.16250
[17]
P. Aliferis and A. W. Cross, “Subsystem Fault Tolerance with the Bacon-Shor Code”, Physical Review Letters 98, (2007) arXiv:quant-ph/0610063 DOI
[18]
X. Zhou, D. W. Leung, and I. L. Chuang, “Methodology for quantum logic gate construction”, Physical Review A 62, (2000) arXiv:quant-ph/0002039 DOI
[19]
J. N. Eberhardt and V. Steffan, “Logical Operators and Fold-Transversal Gates of Bivariate Bicycle Codes”, (2024) arXiv:2407.03973
[20]
H. Sayginel, S. Koutsioumpas, M. Webster, A. Rajput, and Dan E Browne, “Fault-Tolerant Logical Clifford Gates from Code Automorphisms”, (2025) arXiv:2409.18175
[21]
B. Eastin and E. Knill, “Restrictions on Transversal Encoded Quantum Gate Sets”, Physical Review Letters 102, (2009) arXiv:0811.4262 DOI
[22]
N. Rengaswamy, R. Calderbank, M. Newman, and H. D. Pfister, “On Optimality of CSS Codes for Transversal T”, IEEE Journal on Selected Areas in Information Theory 1, 499 (2020) arXiv:1910.09333 DOI
[23]
A. Kubica and M. E. Beverland, “Universal transversal gates with color codes: A simplified approach”, Physical Review A 91, (2015) arXiv:1410.0069 DOI
[24]
B. J. Brown, “Color code with a logical control-\(S\) gate using transversal \(T\) rotations”, (2024) arXiv:2411.15035
[25]
S. Bravyi and A. Cross, “Doubled Color Codes”, (2015) arXiv:1509.03239
[26]
G. Zhu, M. Hafezi, and M. Barkeshli, “Quantum origami: Transversal gates for quantum computation and measurement of topological order”, Physical Review Research 2, (2020) arXiv:1711.05752 DOI
[27]
Z. He, V. Vaikuntanathan, A. Wills, and R. Y. Zhang, “Quantum Codes with Addressable and Transversal Non-Clifford Gates”, (2025) arXiv:2502.01864
[28]
A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
[29]
A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074
[30]
D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
[31]
Q. T. Nguyen, “Good binary quantum codes with transversal CCZ gate”, (2024) arXiv:2408.10140
[32]
L. Golowich and V. Guruswami, “Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates”, (2024) arXiv:2408.09254
[33]
L. Golowich and T.-C. Lin, “Quantum LDPC Codes with Transversal Non-Clifford Gates via Products of Algebraic Codes”, (2024) arXiv:2410.14662
[34]
J. Haah, “Towers of generalized divisible quantum codes”, Physical Review A 97, (2018) arXiv:1709.08658 DOI
[35]
P. Faist, S. Nezami, V. V. Albert, G. Salton, F. Pastawski, P. Hayden, and J. Preskill, “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 DOI
[36]
E. M. Rains, “Quantum codes of minimum distance two”, (1997) arXiv:quant-ph/9704043
[37]
D. Gottesman, “Stabilizer Codes and Quantum Error Correction”, (1997) arXiv:quant-ph/9705052
[38]
D. Gottesman, “Theory of fault-tolerant quantum computation”, Physical Review A 57, 127 (1998) arXiv:quant-ph/9702029 DOI
[39]
E. M. Rains, “Nonbinary quantum codes”, (1997) arXiv:quant-ph/9703048
[40]
A. Cross and D. Vandeth, “Small Binary Stabilizer Subsystem Codes”, (2025) arXiv:2501.17447
[41]
K. Dolev, V. Calvera, S. S. Cree, and D. J. Williamson, “Gauging the bulk: generalized gauging maps and holographic codes”, Journal of High Energy Physics 2022, (2022) arXiv:2108.11402 DOI
[42]
S. Cree, K. Dolev, V. Calvera, and D. J. Williamson, “Fault-Tolerant Logical Gates in Holographic Stabilizer Codes Are Severely Restricted”, PRX Quantum 2, (2021) arXiv:2103.13404 DOI
[43]
P. Webster and S. D. Bartlett, “Locality-preserving logical operators in topological stabilizer codes”, Physical Review A 97, (2018) arXiv:1709.00020 DOI
[44]
A. O. Quintavalle, P. Webster, and M. Vasmer, “Partitioning qubits in hypergraph product codes to implement logical gates”, Quantum 7, 1153 (2023) arXiv:2204.10812 DOI
[45]
Y. Hong, M. Marinelli, A. M. Kaufman, and A. Lucas, “Long-range-enhanced surface codes”, Physical Review A 110, (2024) arXiv:2309.11719 DOI
[46]
J. E. Moussa, “Transversal Clifford gates on folded surface codes”, Physical Review A 94, (2016) arXiv:1603.02286 DOI
[47]
C. Gidney, “Inplace Access to the Surface Code Y Basis”, Quantum 8, 1310 (2024) arXiv:2302.07395 DOI
[48]
C. Gidney, “Cleaner magic states with hook injection”, (2023) arXiv:2302.12292
[49]
K. H. Wan and Z. Zhong, “Pauli webs spun by transversal \(|Y\rangle\) state initialisation”, (2025) arXiv:2502.00957
[50]
D. Aasen, J. Haah, M. B. Hastings, and Z. Wang, “Geometrically Enhanced Topological Quantum Codes”, (2025) arXiv:2505.10403
[51]
T. Jochym-O’Connor and T. J. Yoder, “Four-dimensional toric code with non-Clifford transversal gates”, Physical Review Research 3, (2021) arXiv:2010.02238 DOI
[52]
F. H. E. Watson, E. T. Campbell, H. Anwar, and D. E. Browne, “Qudit color codes and gauge color codes in all spatial dimensions”, Physical Review A 92, (2015) arXiv:1503.08800 DOI
[53]
E. T. Campbell, H. Anwar, and D. E. Browne, “Magic-State Distillation in All Prime Dimensions Using Quantum Reed-Muller Codes”, Physical Review X 2, (2012) arXiv:1205.3104 DOI
[54]
E. T. Campbell, “Enhanced Fault-Tolerant Quantum Computing ind-Level Systems”, Physical Review Letters 113, (2014) arXiv:1406.3055 DOI
[55]
A. Krishna and J.-P. Tillich, “Towards Low Overhead Magic State Distillation”, Physical Review Letters 123, (2019) arXiv:1811.08461 DOI
[56]
A. Wills, M.-H. Hsieh, and H. Yamasaki, “Constant-Overhead Magic State Distillation”, (2024) arXiv:2408.07764
[57]
H. Leitch and A. Kay, “Transversal Gates for Highly Asymmetric qLDPC Codes”, (2025) arXiv:2506.15905
[58]
A. Barg, N. J. Coble, D. Hangleiter, and C. Kang, “Geometric structure and transversal logic of quantum Reed-Muller codes”, (2024) arXiv:2410.07595
[59]
E. T. Campbell and M. Howard, “Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost”, Physical Review A 95, (2017) arXiv:1606.01904 DOI
[60]
E. T. Campbell and M. Howard, “Unifying Gate Synthesis and Magic State Distillation”, Physical Review Letters 118, (2017) arXiv:1606.01906 DOI
[61]
J. Haah and M. B. Hastings, “Codes and Protocols for DistillingT, controlled-S, and Toffoli Gates”, Quantum 2, 71 (2018) arXiv:1709.02832 DOI
[62]
A. Gong and J. M. Renes, “Computation with quantum Reed-Muller codes and their mapping onto 2D atom arrays”, (2024) arXiv:2410.23263
[63]
M. B. Hastings and J. Haah, “Distillation with Sublogarithmic Overhead”, Physical Review Letters 120, (2018) arXiv:1709.03543 DOI
[64]
C. Vuillot and N. P. Breuckmann, “Quantum Pin Codes”, IEEE Transactions on Information Theory 68, 5955 (2022) arXiv:1906.11394 DOI
[65]
S. P. Jain and V. V. Albert, “Transversal Clifford and T-Gate Codes of Short Length and High Distance”, IEEE Journal on Selected Areas in Information Theory 6, 127 (2025) arXiv:2408.12752 DOI
[66]
T. R. Scruby, A. Pesah, and M. Webster, “Quantum Rainbow Codes”, (2024) arXiv:2408.13130
[67]
P. W. Shor, “Fault-tolerant quantum computation”, (1997) arXiv:quant-ph/9605011
[68]
T. Tansuwannont, Y. Takada, and K. Fujii, “Clifford gates with logical transversality for self-dual CSS codes”, (2025) arXiv:2503.19790
[69]
N. P. Breuckmann and S. Burton, “Fold-Transversal Clifford Gates for Quantum Codes”, Quantum 8, 1372 (2024) arXiv:2202.06647 DOI
[70]
M. Grassl and M. Roetteler, “Leveraging automorphisms of quantum codes for fault-tolerant quantum computation”, 2013 IEEE International Symposium on Information Theory 534 (2013) arXiv:1302.1035 DOI
[71]
J. Hu, Q. Liang, and R. Calderbank, “Designing the Quantum Channels Induced by Diagonal Gates”, Quantum 6, 802 (2022) arXiv:2109.13481 DOI
[72]
J. Hu, Q. Liang, and R. Calderbank, “Divisible Codes for Quantum Computation”, (2022) arXiv:2204.13176
[73]
E. Camps-Moreno, H. H. López, G. L. Matthews, and E. McMillon, “Toward Quantum CSS-T Codes from Sparse Matrices”, (2024) arXiv:2406.00425
[74]
M. A. Webster, A. O. Quintavalle, and S. D. Bartlett, “Transversal diagonal logical operators for stabiliser codes”, New Journal of Physics 25, 103018 (2023) arXiv:2303.15615 DOI
[75]
Webster, Mark. The XP Stabilizer Formalism. Dissertation, University of Sydney, 2023.
[76]
J. Guyot and S. Jaques, “On the Addressability Problem on CSS Codes”, (2025) arXiv:2502.13889
[77]
M. M. Wilde, “Logical operators of quantum codes”, Physical Review A 79, (2009) arXiv:0903.5256 DOI
[78]
N. Rengaswamy, R. Calderbank, H. D. Pfister, and S. Kadhe, “Synthesis of Logical Clifford Operators via Symplectic Geometry”, 2018 IEEE International Symposium on Information Theory (ISIT) 791 (2018) arXiv:1803.06987 DOI
[79]
T. Jochym-O’Connor, A. Kubica, and T. J. Yoder, “Disjointness of Stabilizer Codes and Limitations on Fault-Tolerant Logical Gates”, Physical Review X 8, (2018) arXiv:1710.07256 DOI
[80]
B. Zeng, A. Cross, and I. L. Chuang, “Transversality versus Universality for Additive Quantum Codes”, (2007) arXiv:0706.1382
[81]
J. T. Anderson and T. Jochym-O’Connor, “Classification of transversal gates in qubit stabilizer codes”, (2014) arXiv:1409.8320
[82]
B. Zeng, X. Chen, and I. L. Chuang, “Semi-Clifford operations, structure ofCkhierarchy, and gate complexity for fault-tolerant quantum computation”, Physical Review A 77, (2008) arXiv:0712.2084 DOI
[83]
M. Newman and Y. Shi, “Limitations on Transversal Computation through Quantum Homomorphic Encryption”, (2017) arXiv:1704.07798
[84]
A. G. Fowler, “Time-optimal quantum computation”, (2013) arXiv:1210.4626
[85]
Z.-H. Chen, M.-C. Chen, C.-Y. Lu, and J.-W. Pan, “Transversal Logical Clifford gates on rotated surface codes with reconfigurable neutral atom arrays”, (2024) arXiv:2412.01391
[86]
A. J. Malcolm et al., “Computing Efficiently in QLDPC Codes”, (2025) arXiv:2502.07150
[87]
R. Shen, Y. Wang, and C. Cao, “Quantum Lego and XP Stabilizer Codes”, (2023) arXiv:2310.19538
[88]
D. Hangleiter, M. Kalinowski, D. Bluvstein, M. Cain, N. Maskara, X. Gao, A. Kubica, M. D. Lukin, and M. J. Gullans, “Fault-Tolerant Compiling of Classically Hard Instantaneous Quantum Polynomial Circuits on Hypercubes”, PRX Quantum 6, (2025) arXiv:2404.19005 DOI
[89]
T. J. Yoder and I. H. Kim, “The surface code with a twist”, Quantum 1, 2 (2017) arXiv:1612.04795 DOI
[90]
S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012) arXiv:1209.2426 DOI
[91]
A. Paetznick and B. W. Reichardt, “Universal Fault-Tolerant Quantum Computation with Only Transversal Gates and Error Correction”, Physical Review Letters 111, (2013) arXiv:1304.3709 DOI
[92]
E. Kubischta and I. Teixeira, “Quantum Codes and Irreducible Products of Characters”, (2024) arXiv:2403.08999
[93]
H. G. Katzgraber, H. Bombin, R. S. Andrist, and M. A. Martin-Delgado, “Topological color codes on Union Jack lattices: a stable implementation of the whole Clifford group”, Physical Review A 81, (2010) arXiv:0910.0573 DOI
[94]
D.-S. Wang, Y.-J. Wang, N. Cao, B. Zeng, and R. Laflamme, “Theory of quasi-exact fault-tolerant quantum computing and valence-bond-solid codes”, New Journal of Physics 24, 023019 (2022) arXiv:2105.14777 DOI
[95]
C. Zhang, Z. Wu, S. Huang, and B. Zeng, “Transversal Gates in Nonadditive Quantum Codes”, (2025) arXiv:2504.20847
[96]
E. Kubischta and I. Teixeira, “Family of Quantum Codes with Exotic Transversal Gates”, Physical Review Letters 131, (2023) arXiv:2305.07023 DOI
[97]
M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
[98]
S. Prakash, “Magic state distillation with the ternary Golay code”, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, (2020) arXiv:2003.02717 DOI
[99]
E. Knill, “Quantum computing with realistically noisy devices”, Nature 434, 39 (2005) arXiv:quant-ph/0410199 DOI
[100]
A. Paetznick et al., “Demonstration of logical qubits and repeated error correction with better-than-physical error rates”, (2024) arXiv:2404.02280
[101]
R. Chao and B. W. Reichardt, “Fault-tolerant quantum computation with few qubits”, npj Quantum Information 4, (2018) arXiv:1705.05365 DOI
[102]
E. Knill, R. Laflamme, and W. Zurek, “Threshold Accuracy for Quantum Computation”, (1996) arXiv:quant-ph/9610011
[103]
J. T. Anderson, G. Duclos-Cianci, and D. Poulin, “Fault-Tolerant Conversion between the Steane and Reed-Muller Quantum Codes”, Physical Review Letters 113, (2014) arXiv:1403.2734 DOI
[104]
E. T. Campbell, B. M. Terhal, and C. Vuillot, “Roads towards fault-tolerant universal quantum computation”, Nature 549, 172 (2017) arXiv:1612.07330 DOI
[105]
S. Koutsioumpas, D. Banfield, and A. Kay, “The Smallest Code with Transversal T”, (2022) arXiv:2210.14066
[106]
N. Rengaswamy, R. Calderbank, M. Newman, and H. D. Pfister, “On Optimality of CSS Codes for Transversal T”, IEEE Journal on Selected Areas in Information Theory 1, 499 (2020) DOI
[107]
A. Paetznick and B. W. Reichardt, “Fault-tolerant ancilla preparation and noise threshold lower bounds for the 23-qubit Golay code”, (2013) arXiv:1106.2190
[108]
E. Campbell, “The smallest interesting colour code,” Online available at https://earltcampbell.com/2016/09/26/the-smallest-interesting-colour-code/ (2016), accessed on 2019-12-09.
[109]
M. A. Webster, B. J. Brown, and S. D. Bartlett, “The XP Stabiliser Formalism: a Generalisation of the Pauli Stabiliser Formalism with Arbitrary Phases”, Quantum 6, 815 (2022) arXiv:2203.00103 DOI
[110]
B. Zeng, H. Chung, A. W. Cross, and I. L. Chuang, “Local unitary versus local Clifford equivalence of stabilizer and graph states”, Physical Review A 75, (2007) arXiv:quant-ph/0611214 DOI
[111]
S. X. Cui, D. Gottesman, and A. Krishna, “Diagonal gates in the Clifford hierarchy”, Physical Review A 95, (2017) arXiv:1608.06596 DOI
[112]
H. Zhu, R. Kueng, M. Grassl, and D. Gross, “The Clifford group fails gracefully to be a unitary 4-design”, (2016) arXiv:1609.08172
[113]
L. Bittel, J. Eisert, L. Leone, A. A. Mele, and S. F. E. Oliviero, “A complete theory of the Clifford commutant”, (2025) arXiv:2504.12263
[114]
D. Gottesman, “Quantum fault tolerance in small experiments”, (2016) arXiv:1610.03507
[115]
H. Chen, M. Vasmer, N. P. Breuckmann, and E. Grant, “Automated discovery of logical gates for quantum error correction (with Supplementary (153 pages))”, Quantum Information and Computation 22, 947 (2022) arXiv:1912.10063 DOI
[116]
S. P. Jordan, E. Farhi, and P. W. Shor, “Error-correcting codes for adiabatic quantum computation”, Physical Review A 74, (2006) arXiv:quant-ph/0512170 DOI
[117]
H. Goto, “High-performance fault-tolerant quantum computing with many-hypercube codes”, Science Advances 10, (2024) arXiv:2403.16054 DOI
[118]
Victor V. Albert, private communication, 2025