Here is a list of all quantum codes with fault-tolerant gadgets.
Name Fault-tolerant gadget
Bacon-Shor code Piecably fault-tolerant circuits can be employed to construct non-transversal gates effectively [1].
Calderbank-Shor-Steane (CSS) stabilizer code Steane error correction [2].Parallel syndrome extraction for distance-three codes can be done fault-tolerantly using one flag qubit [3].
Cat code Bias-preserving Hamiltonian-based CNOT gate [4] is part of a universal bias-preserving gate set that can be made fault tolerant using concatenation [5][4].Ancilla qubits encoded in cat codes yield fault-tolerant syndrome extraction circuits [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.
Color code Clifford gates can be performed fault-tolerantly on a suitable 2D lattice [7].Syndrome measurement [8].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].
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 concatenated 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 [13].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 [14].
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 [15].Error correction scheme is fault-tolerant to displacement noise as long as all input states have displacement errors less than \(\sqrt{\pi}/6\) [16].
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 [17].
Honeycomb 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 [18]. Gate can be simulated by taking 2D slices out of 3D surface codes [19].
Multi-mode GKP code Logical Clifford operations are given by Gaussian unitaries, which map bounded-size errors to bounded-size errors [15].
Number-phase code Fault-tolerant computation schemes with number-phase codes have been proposed based on concatenation with Bacon-Shor subsystem codes [20].
Quantum Reed-Muller code Gate switching protocol for universal computation [21].
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 [22].
Quantum low-density parity-check (QLDPC) code Lattice surgery techniques with ancilla qubits [23].Fault-tolerance with constant overhead can be performed on certain QLDPC codes [24], e.g., quantum expander codes [22].
Quantum polar code State preparation of a single logical qubit [25].
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 [26], which is based on repeated measurements. A modification uses adaptive measurements [27].Generalization of Steane error correction stabilizer codes [1; Sec. 3.6].Fault-tolerant error correction scheme by Knill (a.k.a. telecorrection [28]), which is based on teleportation [29][30].
Repetition code Triple modular redundancy (TMR) error-correction protocol [31]; see Ref. [32] 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 [33].
Steane \([[7,1,3]]\) code Pieceable fault-tolerant CCZ gate [11].Syndrome measurement can be done with ancillary flag qubits [34][12] or with no extra qubits [35].
\([[15,1,3]]\) 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 [21][36].
\([[2^r, 2^r-r-2, 3]]\) quantum Hamming code Concatenations of Hamming codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [37].
\([[2^r-1, 2^r-2r-1, 3]]\) Hamming-based CSS code Syndrome measurement can be done with two ancillary flag qubits [12].
\([[4,2,2]]\) 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 [38].

References

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