Here is a list of all quantum codes with fault-tolerant gadgets.

Name | Fault-tolerant gadget |
---|---|

3D surface code | Fault-tolerant Hadamard gate using teleportation and error correction [1]. |

Abelian TQD stabilizer code | Fault-tolerant circuits for all non-chiral abelian topological phases and the \(\mathbb{Z}_2^3\) code with a type-III cocycle [2]. |

Abelian topological code | Fault-tolerant logical operations can be interpreted as anyon condensation events [3].Modular decoding, designed to overcome the backlog problem, is applicable to fault-tolerant protocols based on topological qubit stabilizer codes [4]. |

Asymmetric quantum code | Fault-tolerant noise-bias-preserving computation scheme [5].Fault-tolerant circuits converting between asymmetric and symmetric subsystem codes [6,7]. |

Bacon-Shor code | Fault-tolerant teleportation-based computation scheme for asymmetric Bacon-Shor codes that is effective against highly biased noise [8].Pieceably fault-tolerant circuits can be employed to construct non-transversal gates effectively [9]. |

Bivariate bicycle (BB) code | Fault-tolerant state initialization using lattice surgery techniques [10,11] and an ancillary surface code [12]. |

Bosonic rotation code | Decoder based on measuring in the phase-state basis and using Knill error correction [13] is fault-tolerant under circuit-level noise [14]. |

Brown-Fawzi random Clifford-circuit code | Fault-tolerant state preparation [15]. |

Capped color code (CCC) | Fault-tolerant syndrome extraction and error correction for capped color codes in H form [16].Fault-tolerant T gate implementation [16]. |

Cat code | Universal set of error-corrected operations tolerating a single photon loss and an arbitrary ancilla fault [17]. |

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 | Photonic architecture [18].Generalized foliation procedures exist for noise-bias preserving MBQC [19]. |

Color code | The 6D color code is a self-correcting quantum memory and admits fault-tolerant universal gate set in 7D [20]. |

Concatenated Steane code | There exist fault-tolerant syndrome extraction protocols for the concatenated Steane code [21]. |

Concatenated qubit code | Fault-tolerant message passing between devices [22]. |

Dual-rail quantum code | Dual-rail qubits can be used to convert leakage and AD noise into erasure noise [23,24]. |

Expander code | The flip decoding algorithm is fault tolerant against parity check errors [25]; see also course notes by M. Sudan. |

Five-qubit perfect code | Pieceable fault-tolerant CZ, CNOT, and CCZ gates [26].Syndrome measurement can be done with two ancillary flag qubits [27]. The depth of syndrome extraction circuits can be lowered by using past syndrome values [28].Fault-tolerant logical one and logical minus state preparation in all-to-all and 2D grid connectivity [29]. |

Floquet color code | Fault-tolerant measurement-based computation can be realized using the foliated Floquet color code [30]. |

Freedman-Meyer-Luo code | The Freedman-Meter-Luo code has been generalized to a family with rate of order \(O(1/\sqrt{\log n})\) and minimum distance of order \(\Omega(\sqrt{\log n})\) which supports fault-tolerant non-Clifford gates [31]. |

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. There is a large family of fault-tolerant protocols [32]. |

Gottesman-Kitaev-Preskill (GKP) code | Logical Clifford operations are given by Gaussian unitaries, which map bounded-size errors to bounded-size errors [33]. For single-mode GKP codes, these operations correspond to non-trivial loops in the space of all single-mode GKP codes (the moduli space of elliptic curves, i.e., the three sphere with a trefoil knot removed) [34]. Such gates provide another example of monodromy under the particular notion of parallel transport introduced in Ref. [35]. |

Hastings-Haah Floquet code | Floquet codes on tri-colorable lattices can be made fault-tolerant in the presence of dead qubits [36]. |

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. |

Hermitian qubit 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 [37; pg. 9]. |

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 [38]. |

Honeycomb (6.6.6) color code | Fault-tolerant syndrome extraction circuits using flag qubits [39,40]. |

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. |

Hybrid cat code | Photonic architecture based on concatenation with RBH codes [41]. |

Kitaev honeycomb code | One can distill ancilla states to arbitrary precision for sufficiently small noise rates and assuming perfect Clifford operations [42]. |

Kitaev surface code | Transversal (non-Clifford) CCZ gate by bringing 2D surface codes together and using just-in-time decoding [43]. Gate can be simulated by taking 2D slices out of 3D surface codes [44].Flag fault-tolerant syndrome extraction [39].Homomorphic measurement protocols for arbitrary surface codes [45].Non-geometrically local connectivity can reduce overhead cost [46].Fault-tolerant post-selection framework yields magic states with low overhead [47].Framework of fault tolerance utilizing ZX calculus [48,49] that is applicable to MBQC, FBQC, and conventional computation versions of the surface code [50].Single-shot state preparation [51] and MWPM decoding [52].Syndrome extraction circuits consisting of CNOT gates and ancillary measurements [53]. Measurement schedules can be optimized using spacetime circuit codes to yield what is know as the 3CX surface code [54]. Schedules can also be optimized via ZX calculus [48,49]. Inspired by the honeycomb Floquet code, various weight-two measurement schemes have been designed [55–57]. |

Majorana color code | Ordinary and twist-based lattice surgery can be made fault tolerant [58] (see also [59]). |

Majorana surface code | Ordinary and twist-based lattice surgery can be made fault tolerant [58] (see also [59]). |

Number-phase code | Fault-tolerant computation schemes with number-phase codes have been proposed based on concatenation with Bacon-Shor subsystem codes [13]. |

Projective-plane surface code | Fault-tolerant Hadamard gate [60]. |

Quantum Golay code | Fault-tolerant depth-7 circuit consisting of 57 CNOT gates and preparing a logical-zero state [61]. |

Quantum LDPC (QLDPC) code | Lattice surgery techniques with ancilla qubits [10,11]. In one such technique, one first performs a logical measurement by code switching into a code whose stabilizer group includes the original stabilizers together with the logical Paulis that are to be measured. Then, one can reduce the weight of the output code using weight reduction.Fault-tolerance with constant overhead can be performed on certain QLDPC codes [62], e.g., quantum expander codes [63].GHZ state distillation for Steane error correction [64].Fault-tolerant logical measurements that generalize a previous construction [10] and that require an order \(O(d/\beta)\) ancilla qubits, where \(\beta\) is the Cheeger constant of the Tanner subgraph supporting the logical operator to be measured.Fault-tolerant constant-depth encoder and unencoder [65]. |

Quantum Reed-Muller code | Gate switching protocol for universal computation [66]. |

Quantum data-syndrome (QDS) code | Shor error correction [67,68], in which fault tolerance against syndrome extraction errors is ensured by simply repeating syndrome measurements \(\ell\) times, can be recast as a QDS code whose underlying matrix \(A\) is the identity matrix \(I_m\) repeated \(\ell\) times [69]. |

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 [70]. 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 [63]. |

Quantum polar code | State preparation of a single logical qubit [71]. |

Quantum repetition code | Toffoli magic-state preparation protocol [72]. |

Quasi-hyperbolic color code | There exists a family with rate of order \(O(1/\log n)\) and minimum distance of order \(\Omega(\log n)\) which supports fault-tolerant non-Clifford gates [31]. A construction based on the Torelli mapping yields a code with constant rate with similar gates [31]. |

Qubit CSS code | Steane error correction [73], where fault-tolerance is ensured by preparing ancillary encoded states and extracting syndromes via \(CNOT\) gates.Fault-tolerant error correction and logical measurements using flag qubits for distance-three cyclic CSS codes [74]. Parallel syndrome extraction for distance-three codes can be done fault-tolerantly using one flag qubit [75]. Distance-preserving flag fault-tolerant error correction can be done using lookup tables for small codes [76]. Any self-dual CSS code with bounded-weight stabilizer generators admits flag fault-tolerant syndrome extraction [39].Homomorphic gadgets fault-tolerant measurement unify Steane and Shor error correction [45].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 \(\alpha > 0\) [77].Fault-tolerant measurement-free scheme for low-distance CSS codes [78].Automated fault-tolerant encoding circuit synthesis [79]. |

Qubit code | There are lower bounds on the overhead of fault-tolerant QEC in terms of the capacity of the noise channel [80]. 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 [81].Arbitrary \(n\)-qubit circuits can be implemented fault-tolerantly in a 3D architecture using \(O(n^{3/2}\log^3 n)\) qubits, and in a 2D architecture using only \(O(n^2 \log^3 n)\) qubits [82]. |

Qubit stabilizer code | Gates in the Clifford hierarchy can be done using gate teleportation, in which a gate can be obtained from a particular magic state [83,84]. Such protocols can be made fault tolerant with the help of magic-state distillation [85].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 non-degenerate stabilizer code with a complete set of fault-tolerant single-qubit Clifford gates has a universal set of non-transversal fault-tolerant gates [26].Shor error correction [67,68], in which fault tolerance against syndrome extraction errors is ensured by simply repeating syndrome measurements. A modification uses adaptive measurements [86].Generalization of Steane error correction stabilizer codes [9; Sec. 3.6].Fault-tolerant error correction scheme by Knill (a.k.a. telecorrection [87]), which is based on teleportation [88,89]. A variant of it has been termed the Fibonacci scheme [90].Fault-tolerant error correction using flag qubits for codes satisfying certain conditions [39].GHZ state distillation for Steane error correction [91].Syndrome extraction using flag qubits and classical codes [92].Fault-tolerant constant-depth unencoder transforming logical states into physical states using single-qubit measurements [65].Post-selection based algorithm preparing magic state corresponding to arbitrary rotations [93]. |

Repetition code | Triple modular redundancy (TMR) error-correction protocol [94] for fault-tolerant memory operations and classical gate operations; see section 2.6 and 2.7 Ref. [95] 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 [53,96,97]. |

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 [33].Error correction scheme is fault-tolerant to displacement noise as long as all input states have displacement errors less than \(\sqrt{\pi}/6\) [98]. |

Square-octagon (4.8.8) color code | Color-code lattice surgery [99].Fault-tolerant syndrome extraction circuits [100]. |

Subsystem qubit stabilizer code | Logical Clifford gates can be implemented fault-tolerantly for subsystem codes of distance at least three [101]. |

Subsystem spacetime circuit code | Fault-tolerant measurement gadget that is a modification based on the DiVincenzo-Shor cat-state method [67,68]. |

Subsystem surface code | Gauge fixing and changing the order in which check operators are measured yields a fault-tolerant decoder [102]. |

Surface-17 code | Measurement-free fault-tolerant logical zero state preparation in nearest-neighbor qubit connectivity [103].Fault-tolerant logical zero and logical plus state preparation in all-to-all and 2D grid connectivity with flag qubits [29]. |

Tetrahedral color code | Fault-tolerant quantum computation designed for a 2D architecture [104]. |

Three-fermion (3F) Walker-Wang model code | Fault-tolerant MBQC protocol by encoding in, braiding, and fusing symmetry defects. |

Triangular surface code | The symmetry of triangle codes allows for fault-tolerant measurement and encoding in any Pauli basis [105].A non-fault-tolerant curcuit initializes the triangle code. To guarantee fault-tolerance, postselection is performed on trivial measurements of the syndrome and of the logical Pauli, depending on the basis of the logical states [105].Making syndrome extraction fault tolerant requires a specific ordering of syndrome measurements so as to avoid hook errors [105]. |

Triorthogonal code | Universal fault-tolerant gates can be performed without magic-state distillation [101,106]. |

Twist-defect surface code | Fault-tolerant measurement of defects [107].Twisted double covers of codes yield fault-tolerant Clifford gates performed via Dehn twists [108]. |

Twisted XZZX toric code | Fault-tolerant syndrome extraction circuits using flag qubits [109]. |

Two-component cat code | Fault-tolerant error-correction procedure using small amplitude coherent states [110].Bias-preserving Hamiltonian-based CNOT gate is part of a universal noise-bias-preserving gate set that can be made fault tolerant using concatenation [111,112].Ancilla qubits encoded in two-component cat codes yield fault-tolerant syndrome extraction circuits [113]. |

Zero-pi qubit code | One- and two-qubit phase gate errors can be suppressed [114]. |

\([[10,1,2]]\) CSS code | A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[10,1,2]]\) code [115]. |

\([[15, 7, 3]]\) quantum Hamming code | Clifford gates can be performed fault-tolerantly using two ancillary flag qubits, and a CCZ gate can be performed using four ancilla qubits [116]. |

\([[15,1,3]]\) quantum Reed-Muller code | A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[15,1,3]]\) code [66,101,106,117].Fault-tolerant logical zero and logical plus state preparation [29]. |

\([[16,6,4]]\) Tesseract color code | Post-selected fault-tolerant syndrome extraction [118,119]. |

\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code | Syndrome measurement can be done with two ancillary flag qubits [27].Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \([[6,2,2]]\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [120]. |

\([[2^r-1,1,3]]\) simplex code | Fault-tolerant syndrome extraction circuits using flag qubits [39]. |

\([[2m,2m-2,2]]\) error-detecting code | Logical SWAP gates can be performed fault tolerantly using an ancilla qubit [121; Sec. VII].Two-qubit fault-tolerant state preparation, error detection and projective measurements [27] (see also [122]).CNOT and Hadamard gates using only two extra qubits and four-qubit fault-tolerant CCZ gate [116].Fault-tolerant Clifford Trotter circuits using flag qubits [123].Weak fault tolerance: any single gate error can be detected by measuring stabilizers and utilizing extra ancillas [124]. |

\([[4,2,2]]\) Four-qubit 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 [125]. Magic states can be injected into surface and color codes since the code is a small instance of both [126].Concatenations of \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation schemes [88] (see also Ref. [127]).Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [120,128]. |

\([[5,1,2]]\) rotated surface code | Fault-tolerant implementation of the Clifford group based on transversal gates and SWAPs [119]. |

\([[6,2,2]]\) \(C_6\) code | Concatenations of \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation schemes [88] (see also Ref. [127]).Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [120,128]. |

\([[7,1,3]]\) Steane code | A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[15,1,3]]\) code [66,101,106,117].A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[10,1,2]]\) code [115].Fault-tolerant logical zero and magic state preparation [129]. Magic-state preparation converts unbiased noise into biased noise [130].Fault-tolerant logical zero and logical plus state preparation on all-to-all and 2D grid qubit connectivity [29].Pieceable fault-tolerant CCZ gate [26].Syndrome measurement can be done with ancillary flag qubits [27,105] or with no extra qubits [131]. The depth of syndrome extraction circuits can be lowered by using past syndrome values [28]. |

\([[8,3,2]]\) CSS code | CCZ gate can be distilled in a fault-tolerant manner [132]. |

\([[9,1,3]]\) Shor code | Fault-tolerant logical zero and logical plus state preparation [29]. |

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