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 color code Universal transversal gates can be achieved using lattice surgery or code deformation [3,4].
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 [5].
3D surface code Transversal CZ and CCZ gates [6,7].
Bacon-Shor code Logical Hadamard is transversal in symmetric Bacon-Shor codes up to a qubit permutation [8] and can be implemented with teleportation [9].Bacon-Shor codes on an \(m \times mk\) lattice admit transversal \(k\)-qubit-controlled \(Z\) gates [10].
Ball color code The 3D ball codes on duals of the truncated octahedron, truncated cuboctahedron, and truncated icosidodecahedron have transveral \(CCZ\) gates.
Binary dihedral permutation-invariant 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.
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.
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 \(-\pi/2^D\) [11; Fig. 3].
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 [12]. 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 [13].
Five-qubit perfect code Pauli gates are transversal, along with a non-Pauli Hadamard-phase gate \(SH\) and three-qubit Clifford operation \(M_3\) [14,15]. These realize the \(2T\) binary tetrahedral subgroup of \(SU(2)\).The code does not admit any non-Clifford transversal gates [16]; in particular, see [17] for the case of collective \(Z\) rotations.All transversal gates can be interpreted as monodromies under a particular notion of parallel transport [18; Exam. 6.4.2].
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 [19; 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 \(G\) [20].
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 The three-block transversal gate mapping each \(X \to XYZ\) and each \(Z \to ZXY\) implements a logical gate [14][21; Exam. 2].
Holographic code There exist holographic approximate codes with arbitrary transversal gate sets for any compact Lie group [22]. However, for sufficiently localized logical subsystems of holographic stabilizer codes, the set of transversally implementable logical operations is contained in the Clifford group [23].
Homological code Locality preserving operations can be determined for stacks of homoogical codes in any dimension [24].
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 [25].
Kitaev surface code For the toric code, transversal logical Pauli gates correspond to Pauli strings on non-trivial loops of the torus.Transversal Clifford gates can be done on folded surface codes [26].
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 [27].
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 [23].
Prime-qudit RM code An odd-prime-qudit CSS code family constructed from first-order punctured RM codes transversally implements a diagonal gate at any level of the qudit Clifford hierarchy [28].
Quantum Reed-Muller code The \([[2^m,{m \choose r}, 2^r]]\) family, where \(r\) divides \(m\), admits diagonal gates in the form of \(Z\)-rotations by angle \(\pi/2^{m/r}\) [30–32][29; Exam. 8].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 \(m > \nu r\) [33; Thm. 1].
Quantum divisible code Doubly-even codes can yield a transversal \(S\) gate, while triply-even codes yield a transversal \(T\) gate for odd \(n\) [12].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 [34; Prop. 8].Necessary and sufficient conditions for a CSS code to yield a gate in the Clifford hierarchy have been formulated [35; Thm. 9][36; Thm. 5].
Qubit CSS code All CSS codes admit transversal Pauli and CNOT gates [37]. Self-dual CSS codes admit a transversal Hadamard gate.Fold-transversal [38] Clifford gates are transversal gates combined with qubit permutations. Some of these can be obtained from automorphism groups of the underlying classical codes [39; Thms. 2-3].Realizing transversal gates outside of the Clifford group requires certain higher-order (i.e., non-quadratic) constraints to be satisfied on the code [40] (see quantum divisible codes). There are routines that can determine what diagonal gates in the Clifford hierarchy are realized by a code [41,42].
Qubit Golay code All encoded Clifford gates by choosing \(\overline{U}=U^{\otimes 23}\) for every Clifford unitary \(U\) [43].
Qubit stabilizer code All stabilizer codes realize Pauli transformations transversally; for a single logical qubit, these a realize dicyclic subgroup of \(SU(2)\). Several algorithms exist for finding logical Pauli operators [15,44,45].The four-block transversal gate mapping each \(X \to IXXX\) and each \(Z \to IZZZ\) implements the same logical gate on all qubits [15].Transversal logical gates are in a finite level of the Clifford hierarchy, which is shown using stabilizer disjointness [46] (see also [47,48]). Transversal gates for \(n\in\{1,2\}\) are semi-Clifford [49].No stabilizer code can implement a classical universal gate set transversally [50].
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 [4].
Surface-17 code Pauli gates, CNOT gate, and \(H\) gate (with relabeling).
Triangular surface code Triangle codes admit transversal order-three single-qubit Clifford gates, e.g., \(\bar{SH}\) [51].
Triorthogonal code Admits transversal \(T\) gates [40] and the controlled-controlled-\(Z\) gate [52]. The \(T\) gates are realized via physical \(T\) gates on each qubit [29].Triorthogonality is necessary for physical \(T\) gates on each qubit to realize the identity logical gate [29; Thm. 12].
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 [53; Table II] for other notable groups including the sporadic groups.
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.
\(((7,2,3))\) Pollatsek-Ruskai code Binary icosahedral group \(2I\) gates can be realized transversally [54].
\([[10,1,2]]\) CSS code Logical \(T\) gate via application of \(T\), \(T^{\dagger}\), and \(CCZ\) [55].
\([[11,1,5]]_3\) qutrit Golay code All single-qutrit encoded Clifford gates [56].
\([[12,2,4]]\) 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 [39]. 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 [57].
\([[15, 7, 3]]\) 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 [58].Automorphism groups of the underlying classical codes can yield transversal Clifford gates when combined with qubit permutations [39; Sec. IV.A].Transversal CCZ gate [52].
\([[15,1,3]]\) quantum Reed-Muller code This is the smallest qubit stabilizer code with a (strongly) transversal gate outside of the Clifford group [59].A transversal logical \(T^\dagger\) is implemented by applying a \(T\) gate on every qubit [60–62].A subsystem version yields a transversal \(CCZ\) gate [52].
\([[2^D,D,2]]\) hypercube code CZ, CCZ, CNOT, and generalized \(CZ\) gates at the \((D-1)\)-st level of the Clifford hierarchy [63; Exam. 6.10].
\([[2^r-1, 1, 3]]\) 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 \(-\pi/2^{r-1}\) [64,65]. These are the smallest qubit stabilizer codes with such a (strongly) transversal gate [59].
\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming 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 [14].
\([[30,8,3]]\) Bring code Clifford group of four of the eight logical qubits can be done by transversal gates combined with qubit permutations [38].
\([[3k + 8, k, 2]]\) triorthogonal code The code admits a transversal \(T\) gate [40].
\([[4,2,2]]\) CSS code Transversal Pauli, Hadamard, and two-qubit \(S=\sqrt{Z}\) [66] (see also [67]).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 [68].
\([[49,1,5]]\) triorthogonal code The code admits a transversal \(T\) gate [40; Appx. B].
\([[6,4,2]]\) error-detecting code CNOT and Hadamard gates [69; Appx. B].A \(CZ\) gate implemented by transversal \(S\) and \(S^{\dagger}\) [67]; see also [55].
\([[7,1,3]]\) Steane code All single-qubit Clifford gates, which realize the \(2O\) binary octahedral subgroup of \(SU(2)\) [37,47].
\([[7,1,3]]\) twist-defect surface code Single-qubit octahedral Clifford gates [51,67].
\([[8, 3, 3]]\) code Automorphism groups of the underlying classical codes can yield transversal operations when combined with qubit permutations, but such operations do not generally preserve the code [39; Sec. IV.D].No gates outside of the Pauli group were found in Ref. [67].
\([[8,2,2]]\) hyperbolic color code Applying transversal \(S\) and \(S^{\dagger}\), \(\sqrt{X}\), and Hadamard gates yields various logical gates [67].
\([[8,3,2]]\) CSS code CZ gates between any two logical qubits [67] and (weakly) transversal CCZ gate [6,67,70].
\([[k+4,k,2]]\) 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]
H. Bombin, “Clifford gates by code deformation”, New Journal of Physics 13, 043005 (2011) arXiv:1006.5260 DOI
[4]
A. G. Fowler, “Two-dimensional color-code quantum computation”, Physical Review A 83, (2011) arXiv:0806.4827 DOI
[5]
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”, (2024) arXiv:2311.05674
[6]
A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
[7]
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
[8]
P. Aliferis and A. W. Cross, “Subsystem Fault Tolerance with the Bacon-Shor Code”, Physical Review Letters 98, (2007) arXiv:quant-ph/0610063 DOI
[9]
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
[10]
T. J. Yoder, “Universal fault-tolerant quantum computation with Bacon-Shor codes”, (2017) arXiv:1705.01686
[11]
A. Kubica and M. E. Beverland, “Universal transversal gates with color codes: A simplified approach”, Physical Review A 91, (2015) arXiv:1410.0069 DOI
[12]
S. Bravyi and A. Cross, “Doubled Color Codes”, (2015) arXiv:1509.03239
[13]
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
[14]
D. Gottesman, “Theory of fault-tolerant quantum computation”, Physical Review A 57, 127 (1998) arXiv:quant-ph/9702029 DOI
[15]
D. Gottesman, “Stabilizer Codes and Quantum Error Correction”, (1997) arXiv:quant-ph/9705052
[16]
E. M. Rains, “Quantum codes of minimum distance two”, (1997) arXiv:quant-ph/9704043
[17]
J. Hu et al., “Mitigating Coherent Noise by Balancing Weight-2 Z-Stabilizers”, IEEE Transactions on Information Theory 68, 1795 (2022) arXiv:2011.00197 DOI
[18]
D. Gottesman and L. L. Zhang, “Fibre bundle framework for unitary quantum fault tolerance”, (2017) arXiv:1309.7062
[19]
J. Haah, “Towers of generalized divisible quantum codes”, Physical Review A 97, (2018) arXiv:1709.08658 DOI
[20]
P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 DOI
[21]
E. M. Rains, “Nonbinary quantum codes”, (1997) arXiv:quant-ph/9703048
[22]
K. Dolev et al., “Gauging the bulk: generalized gauging maps and holographic codes”, Journal of High Energy Physics 2022, (2022) arXiv:2108.11402 DOI
[23]
S. Cree et al., “Fault-Tolerant Logical Gates in Holographic Stabilizer Codes Are Severely Restricted”, PRX Quantum 2, (2021) arXiv:2103.13404 DOI
[24]
P. Webster and S. D. Bartlett, “Locality-preserving logical operators in topological stabilizer codes”, Physical Review A 97, (2018) arXiv:1709.00020 DOI
[25]
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
[26]
J. E. Moussa, “Transversal Clifford gates on folded surface codes”, Physical Review A 94, (2016) arXiv:1603.02286 DOI
[27]
F. H. E. Watson et al., “Qudit color codes and gauge color codes in all spatial dimensions”, Physical Review A 92, (2015) arXiv:1503.08800 DOI
[28]
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
[29]
N. Rengaswamy et al., “On Optimality of CSS Codes for Transversal T”, IEEE Journal on Selected Areas in Information Theory 1, 499 (2020) arXiv:1910.09333 DOI
[30]
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
[31]
E. T. Campbell and M. Howard, “Unifying Gate Synthesis and Magic State Distillation”, Physical Review Letters 118, (2017) arXiv:1606.01906 DOI
[32]
J. Haah and M. B. Hastings, “Codes and Protocols for DistillingT, controlled-S, and Toffoli Gates”, Quantum 2, 71 (2018) arXiv:1709.02832 DOI
[33]
M. B. Hastings and J. Haah, “Distillation with Sublogarithmic Overhead”, Physical Review Letters 120, (2018) arXiv:1709.03543 DOI
[34]
C. Vuillot and N. P. Breuckmann, “Quantum Pin Codes”, IEEE Transactions on Information Theory 68, 5955 (2022) arXiv:1906.11394 DOI
[35]
J. Hu, Q. Liang, and R. Calderbank, “Designing the Quantum Channels Induced by Diagonal Gates”, Quantum 6, 802 (2022) arXiv:2109.13481 DOI
[36]
J. Hu, Q. Liang, and R. Calderbank, “Divisible Codes for Quantum Computation”, (2022) arXiv:2204.13176
[37]
P. W. Shor, “Fault-tolerant quantum computation”, (1997) arXiv:quant-ph/9605011
[38]
N. P. Breuckmann and S. Burton, “Fold-Transversal Clifford Gates for Quantum Codes”, (2022) arXiv:2202.06647
[39]
M. Grassl and M. Roetteler, “Leveraging automorphisms of quantum codes for fault-tolerant quantum computation”, 2013 IEEE International Symposium on Information Theory (2013) arXiv:1302.1035 DOI
[40]
S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012) arXiv:1209.2426 DOI
[41]
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
[42]
Webster, Mark. The XP Stabilizer Formalism. Dissertation, University of Sydney, 2023.
[43]
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
[44]
M. M. Wilde, “Logical operators of quantum codes”, Physical Review A 79, (2009) arXiv:0903.5256 DOI
[45]
N. Rengaswamy et al., “Synthesis of Logical Clifford Operators via Symplectic Geometry”, 2018 IEEE International Symposium on Information Theory (ISIT) (2018) arXiv:1803.06987 DOI
[46]
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
[47]
B. Zeng, A. Cross, and I. L. Chuang, “Transversality versus Universality for Additive Quantum Codes”, (2007) arXiv:0706.1382
[48]
J. T. Anderson and T. Jochym-O’Connor, “Classification of transversal gates in qubit stabilizer codes”, (2014) arXiv:1409.8320
[49]
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
[50]
M. Newman and Y. Shi, “Limitations on Transversal Computation through Quantum Homomorphic Encryption”, (2017) arXiv:1704.07798
[51]
T. J. Yoder and I. H. Kim, “The surface code with a twist”, Quantum 1, 2 (2017) arXiv:1612.04795 DOI
[52]
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
[53]
E. Kubischta and I. Teixeira, “Quantum Codes and Irreducible Products of Characters”, (2024) arXiv:2403.08999
[54]
E. Kubischta and I. Teixeira, “Family of Quantum Codes with Exotic Transversal Gates”, Physical Review Letters 131, (2023) arXiv:2305.07023 DOI
[55]
M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
[56]
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
[57]
M. P. da Silva et al., “Demonstration of logical qubits and repeated error correction with better-than-physical error rates”, (2024) arXiv:2404.02280
[58]
R. Chao and B. W. Reichardt, “Fault-tolerant quantum computation with few qubits”, npj Quantum Information 4, (2018) arXiv:1705.05365 DOI
[59]
S. Koutsioumpas, D. Banfield, and A. Kay, “The Smallest Code with Transversal T”, (2022) arXiv:2210.14066
[60]
E. Knill, R. Laflamme, and W. Zurek, “Threshold Accuracy for Quantum Computation”, (1996) arXiv:quant-ph/9610011
[61]
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
[62]
E. T. Campbell, B. M. Terhal, and C. Vuillot, “Roads towards fault-tolerant universal quantum computation”, Nature 549, 172 (2017) arXiv:1612.07330 DOI
[63]
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
[64]
B. Zeng et al., “Local unitary versus local Clifford equivalence of stabilizer and graph states”, Physical Review A 75, (2007) arXiv:quant-ph/0611214 DOI
[65]
S. X. Cui, D. Gottesman, and A. Krishna, “Diagonal gates in the Clifford hierarchy”, Physical Review A 95, (2017) arXiv:1608.06596 DOI
[66]
D. Gottesman, “Quantum fault tolerance in small experiments”, (2016) arXiv:1610.03507
[67]
H. Chen et al., “Automated discovery of logical gates for quantum error correction (with Supplementary (153 pages))”, Quantum Information and Computation 22, 947 (2022) arXiv:1912.10063 DOI
[68]
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
[69]
H. Goto, “Many-hypercube codes: High-rate quantum error-correcting codes for high-performance fault-tolerant quantum computation”, (2024) arXiv:2403.16054
[70]
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.