Here is a list of all quantum codes with fault-tolerant gadgets.
Name Fault-tolerant gadget
Abelian topological code Fault-tolerant logical operations can be interpreted as anyon condensation events [1].Modular decoding, designed to overcome the backlog problem, is applicable to fault-tolerant protocols based on topological qubit stabilizer codes [2].
Bacon-Shor code Piecably fault-tolerant circuits can be employed to construct non-transversal gates effectively [3].
Bivariate bicycle code Fault-tolerant state initialization using Tanner graph techniques [4] and an ancillary surface code [5].
Cat code Universal set of error-corrected operations tolerating a single photon loss and an arbitrary ancilla fault [6].
Clifford-deformed surface code (CDSC) In order to leverage the benefits of CDSCs into practical universal computation, we have to implement syndrome measurement circuits and fault-tolerant logical gates in a bias-preserving way.
Cluster-state code Generalized foliation procedures exist for noise-bias preserving MBQC [7].
Color code Steane's ancilla-coupled measurement method [8]
Dual-rail quantum code Dual-rail qubits can be used to convert leakage and amplitude damping noise into erasure noise [9].
Expander code The flip decoding algorithm is fault tolerant against parity check errors [10]; see also course notes by M. Sudan.
Five-qubit perfect code Pieceable fault-tolerant CZ, CNOT, and CCZ gates [11].Syndrome measurement can be done with two ancillary flag qubits [12]. The depth of syndrome extraction circuits can be lowered by using past syndrome values [13].
Floquet color code Fault-tolerant measurement-based computation can be realized using the foliated Floquet color code [14].
Fusion-based quantum computing (FBQC) code Fusion networks are constructed in a fault tolerant way (as a stabilizer code), and they can be created in a way that naturally encodes topological fault tolerance.
GKP cluster-state code First encoding demonstrating the possibility of fault-tolerant measurement-based computation with CV cluster states. A fault-tolerance threshold can be achieved by concatenating existing fault-tolerant schemes for qubit-based cluster-state encodings with the GKP code [15].Hybrid cluster state consisting of GKP qubits at some modes and squeezed states at others has been proposed to work in a fault-tolerant scheme [16].
Gottesman-Kitaev-Preskill (GKP) code Logical Clifford operations are given by Gaussian unitaries, which map bounded-size errors to bounded-size errors [17].
Hastings-Haah Floquet code Floquet codes on tri-colorable lattices can be made fault-tolerant in the presence of dead qubits [18].
Heavy-hexagon code All logical gates can be fault-tolerantly implemented using lattice surgery and magic state injection.Stabilizer measurements are measured fault-tolerantly using one-flag circuits since some single-fault events can result in weight-two data qubit errors which are parallel to the code's logical operators. Hence, using information from the flag-qubit measurements is crucial to fault-tolerantly measure the code stabilizers.
Hierarchical code 2D geometrically local syndrome extraction circuits of depth \(O(\sqrt{n}/R)\) that utilize Clifford and SWAP gates of range \(R\) and that require order \(O(n)\) data and ancilla qubits. Such parameters (including a range of one) are possible while maintaining a threshold because of the concatenation step. This reduces the noise that would otherwise accumulate within a growing-depth syndrome extraction circuit. A key idea is that constant-depth syndrome extraction is not a necessary condition for fault-tolerance.
Homological product code Universal set of gates can be obtained by fault-tolerantly mapping between different encoded representations of a given logical state [19].
Honeycomb Floquet code One can run a fault-tolerant decoding algorithm by (1) bipartitioning the syndrome lattice into two graphs which are congruent to the Cayley graph of the free Abelian group with three generators (up to boundary conditions) and (2) performing a matching algorithm to deduce errors.
Kitaev surface code Transversal (non-Clifford) CCZ gate by bringing 2D surface codes together and using just-in-time decoding [20]. Gate can be simulated by taking 2D slices out of 3D surface codes [21].Homomorphic measurement protocols for arbitrary surface codes [22].Non-geometrically local connectivity can reduce overhead cost [23].Fault-tolerant post-selection framework yields magic states with low overhead [24].Framework of fault tolerance utilizing ZX calculus [25,26] that is applicable to MBQC, FBQC, and conventional computation versions of the surface code [27].Single-shot state preparation [28] and MWPM decoding [29].Syndrome extraction circuits consisting of CNOT gates and ancillary measurements [30], two-body measurements based on the Majorana mapping [31,32]. Circuits can be optimized to specific architectures [33] using spacetime circuit codes and ZX calculus [25,26].
Low-depth random Clifford-circuit qubit code Fault-tolerant state preparation [34].
Number-phase code Fault-tolerant computation schemes with number-phase codes have been proposed based on concatenation with Bacon-Shor subsystem codes [35].
Projective-plane surface code Fault-tolerant Hadamard gate [36].
Quantum Reed-Muller code Gate switching protocol for universal computation [37].
Quantum divisible code The \(T\) gate realized by concatenating members of the \([[2m − 1, 1 \leq k \leq 1 + \sum_{i=1}^{m-4}(m − i), 3]]\) quantum divisible code family with either the five-qubit \([[5,1,3]]\) or Steane \([[7,1,3]]\) code is fault-tolerant and does not require magic-state distillation. The gate is performed on the inner five-qubit/Steane code and does require encoding and decoding algorithms to pass between the inner and outer codes.
Quantum expander code Fault-tolerance with constant overhead can be achieved [38].
Quantum low-density parity-check (QLDPC) code Lattice surgery techniques with ancilla qubits [4,39].Fault-tolerance with constant overhead can be performed on certain QLDPC codes [40], e.g., quantum expander codes [38].GHz state distillation for Steane error correction [41].
Quantum polar code State preparation of a single logical qubit [42].
Quantum repetition code Toffoli magic-state preparation protocol [43].
Qubit CSS code Steane error correction, where fault-tolerance is ensured by preparing ancillary encoded states and extracting syndromes via \(CNOT\) gates [44].Homomorphic gadgets fault-tolerant measurement unify Steane and Shor error correction [22].Parallel syndrome extraction for distance-three codes can be done fault-tolerantly using one flag qubit [45].Distance-preserving flag fault-tolerant error correction using lookup tables for small codes [46].A fault-tolerant error-correction protocol using \(O(d\log d)\) syndrome measurements can be applied to any CSS code with distance \(d \geq \Omega(n^{\alpha})\) for any \(\alpha > 0\) [47].
Qubit code There are lower bounds on the overhead of fault-tolerant QEC in terms of the capacity of the noise channel [48]. A more stringent bound applies to geometrically local QEC due to the fact that locality constrains the growth of the entanglement that is needed for protection [49].
Qubit stabilizer code Characterizing fault-tolerant multi-qubit gates under the \(GF(4)\) representation may involve characterizing all global automorphisms of some number of copies of a code that preserve the symplectic inner product [50; pg. 9].Logical Bell measurements can be done transversally, and thus fault tolerantly, by performing bitwise Bell measurements for each pair of qubits (with each member of the pair taken from one of the two code blocks) and processing the result.With pieceable fault-tolerance, any nondegenerate stabilizer code with a complete set of fault-tolerant single-qubit Clifford gates has a universal set of non-transversal fault-tolerant gates [11].Shor error correction [51], in which fault tolerance against syndrome extraction errors is ensured by simply repeating syndrome measurements. A modification uses adaptive measurements [52].Generalization of Steane error correction stabilizer codes [3; Sec. 3.6].Fault-tolerant error correction scheme by Knill (a.k.a. telecorrection [53]), which is based on teleportation [54,55].GHz state distillation for Steane error correction [56].Syndrome extraction using flag qubits and classical codes [57].
Repetition code Triple modular redundancy (TMR) error-correction protocol [58] for fault-tolerant memory operations and classical gate operations; see section 2.6 and 2.7 Ref. [59] for a pedagogical explanation.
Rotated surface code A particular choice of CNOT gates during syndrome extraction is required to avoid hook errors and be fault-tolerant to syndrome qubit errors [30,60,61].
Square-lattice GKP code Clifford gates can be realized by performing linear-optical operations, sympletic transformations and displacements, all of which are Gaussian operations. Pauli gates can be performed using displacement operators. Clifford gates are fault tolerant in the sense that they map bounded-size errors to bounded-size errors [17].Error correction scheme is fault-tolerant to displacement noise as long as all input states have displacement errors less than \(\sqrt{\pi}/6\) [62].
Subsystem qubit stabilizer code Logical Clifford gates can be implemented fault-tolerantly for subsystem codes of distance at least three [63].
Subsystem surface code Gauge fixing and changing the order in which check operators are measured yields a fault-tolerant decoder [64].
Three-fermion (3F) model code Fault-tolerant MBQC protocol by encoding in, braiding, and fusing symmetry defects.
Triorthogonal code Universal fault-tolerant gates can be performed without magic-state distillation [63,65].
Two-component cat code Fault-tolerant error-correction procedure using small amplitude coherent states [66].Bias-preserving Hamiltonian-based CNOT gate [67] is part of a universal bias-preserving gate set that can be made fault tolerant using concatenation [67,68].Ancilla qubits encoded in two-component cat codes yield fault-tolerant syndrome extraction circuits [69].
Two-dimensional color code Clifford gates can be performed fault-tolerantly on a suitable 2D lattice [70].
Zero-pi qubit code One- and two-qubit phase gate errors can be suppressed [71].
\([[15, 7, 3]]\) Hamming-based CSS code Clifford gates can be performed fault-tolerantly using two ancillary flag qubits, and a CCZ gate can be performed using four ancilla qubits [72].
\([[15,1,3]]\) quantum Reed-Muller code Combining the Steane code and the 15-qubit Reed-Muller code through a fault-tolerant conversion can result in a universal transversal gate set that does not need magic state distillation [37,63,65,73].
\([[2^r-1, 2^r-2r-1, 3]]\) Hamming-based CSS code Syndrome measurement can be done with two ancillary flag qubits [12].Concatenations of Hamming-based CSS codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [74].
\([[2m,2m-2,2]]\) error-detecting code Two-qubit fault-tolerant state preparation, error detection and projective measurements [12] (see also [75]).CNOT and Hadamard gates using only two extra qubits and four-qubit fault-tolerant CCZ gate [72].
\([[4,2,2]]\) CSS code Preparation of certain states, both magic and non-magic, along with transversal gates can be performed fault-tolerantly, but requires post-selection because the code cannot correct errors [76]. Magic states can be injected into surface and color codes since the code is a small instance of both [77].
\([[7,1,3]]\) Steane code Fault-tolerant logical zero and magic state preparation [78].Pieceable fault-tolerant CCZ gate [11].Syndrome measurement can be done with ancillary flag qubits [12,79] or with no extra qubits [80]. The depth of syndrome extraction circuits can be lowered by using past syndrome values [13].
\([[8,3,2]]\) CSS code CCZ gate can be distilled in a fault-tolerant manner [81].

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