| Abelian topological code |
Fault-tolerant logical operations can be interpreted as anyon condensation events [1]. |
| Bacon-Shor code |
Piecably fault-tolerant circuits can be employed to construct non-transversal gates effectively [2]. |
| Calderbank-Shor-Steane (CSS) stabilizer code |
Steane error correction [3].Homomorphic gadgets fault-tolerant measurement unify Steane and Shor error correction [4].Parallel syndrome extraction for distance-three codes can be done fault-tolerantly using one flag qubit [5]. |
| 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. |
| Color code |
Clifford gates can be performed fault-tolerantly on a suitable 2D lattice [6].Steane's ancilla-coupled measurement method [7]Gauge fixing can be used to switch between 2D and 3D color codes, thereby yielding fault-tolerant with constant time overhead using only local quantum operations [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]. |
| Floquet color code |
Fault-tolerant measurement-based computation can be realized using the foliated Floquet color code [13]. |
| 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 [14].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 [15]. |
| Gottesman-Kitaev-Preskill (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 [16].Error correction scheme is fault-tolerant to displacement noise as long as all input states have displacement errors less than \(\sqrt{\pi}/6\) [17]. |
| 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. |
| Homological product code |
Universal set of gates can be obtained by fault-tolerantly mapping between different encoded representations of a given logical state [18]. |
| 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 [19]. Gate can be simulated by taking 2D slices out of 3D surface codes [20].Homomorphic measurement protocols for arbitrary surface codes [4].Non-geometrically local connectivity can reduce overhead cost [21].Fault-tolerant post-selection framework yields magic states with low overhead [22]. |
| Multi-mode GKP code |
Logical Clifford operations are given by Gaussian unitaries, which map bounded-size errors to bounded-size errors [16]. |
| Number-phase code |
Fault-tolerant computation schemes with number-phase codes have been proposed based on concatenation with Bacon-Shor subsystem codes [23]. |
| Quantum Reed-Muller code |
Gate switching protocol for universal computation [24]. |
| 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 [25]. |
| Quantum low-density parity-check (QLDPC) code |
Lattice surgery techniques with ancilla qubits [26].Fault-tolerance with constant overhead can be performed on certain QLDPC codes [27], e.g., quantum expander codes [25].GHz state distillation for Steane error correction [28]. |
| Quantum polar code |
State preparation of a single logical qubit [29]. |
| Quantum repetition code |
Toffoli magic-state preparation protocol [30]. |
| Qubit stabilizer code |
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].Fault-tolerant error correction scheme by Shor [31], which is based on repeated measurements. A modification uses adaptive measurements [32].Generalization of Steane error correction stabilizer codes [2; Sec. 3.6].Fault-tolerant error correction scheme by Knill (a.k.a. telecorrection [33]), which is based on teleportation [34][35].GHz state distillation for Steane error correction [36].Syndrome extraction using flag qubits and classical codes [37]. |
| Repetition code |
Triple modular redundancy (TMR) error-correction protocol [38]; see Ref. [39] for a pedagogical explanation. |
| Rotated surface code |
A particular choice of CNOT gates during syndrome extraction is required to be fault-tolerant to syndrome qubit errors [40]. |
| Stabilizer code over \(GF(4)\) |
Characterizing fault-tolerant multi-qubit gates may involve characterizing all global automorphisms of some number of copies of a code that preserve the symplectic inner product [41; pg. 9]. |
| Subsystem qubit stabilizer code |
Logical Clifford gates can be implemented fault-tolerantly for subsystem codes of distance at least three [42]. |
| Subsystem surface code |
Gauge fixing and changing the order in which check operators are measured yields a fault-tolerant decoder [43]. |
| Triorthogonal code |
Universal fault-tolerant gates can be performed without magic-state distillation [44][42]. |
| Two-component cat code |
Fault-tolerant error-correction procedure using small amplitude coherent states [45].Bias-preserving Hamiltonian-based CNOT gate [46] is part of a universal bias-preserving gate set that can be made fault tolerant using concatenation [47][46].Ancilla qubits encoded in two-component cat codes yield fault-tolerant syndrome extraction circuits [48]. |
| \([[15, 7, 3]]\) 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 [49]. |
| \([[15,1,3]]\) 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 [44][24][50][42]. |
| \([[2^r, 2^r-r-2, 3]]\) quantum Hamming code |
Concatenations of Hamming codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [51]. |
| \([[2^r-1, 2^r-2r-1, 3]]\) Hamming-based CSS code |
Syndrome measurement can be done with two ancillary flag qubits [12]. |
| \([[2m,2m-2,2]]\) error-detecting code |
Two-qubit fault-tolerant state preparation, error detection and projective measurements [12] (see also [52]).CNOT and Hadamard gates using only two extra qubits and four-qubit fault-tolerant CCZ gate [49]. |
| \([[4,2,2]]\) CSS code |
Preparation of certain states along with transversal gates can be performed fault-tolerantly, but requires post-selection because the code cannot correct errors [53]. |
| \([[7,1,3]]\) Steane code |
Pieceable fault-tolerant CCZ gate [11].Syndrome measurement can be done with ancillary flag qubits [54][12] or with no extra qubits [55]. |
| \([[8,3,2]]\) code |
CCZ gate can be distilled in a fault-tolerant manner [56]. |