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].
Cat code Bias-preserving CNOT gate [3] is part of a universal bias-preserving gate set that can be made fault tolerant using concatenation [4][3].
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 [5].Syndrome measurement [6].Steane's ancilla-coupled measurement method [6]
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.
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 [7].Error correction scheme is fault-tolerant to displacement noise as long as all input states have displacement errors less than \(\sqrt{\pi}/6\) [8].
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 [9].
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.
Multi-mode GKP code Logical Clifford operations are given by Gaussian unitaries, which map bounded-size errors to bounded-size errors [7].
Number-phase code Fault-tolerant computation schemes with number-phase codes have been proposed based on concatenation with Bacon-Shor subsystem codes [10].
Quantum Reed-Muller code Gate switching protocol for universal computation [11].
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 [12].
Quantum low-density parity-check (QLDPC) code Lattice surgery techniques with ancilla qubits [13].Fault-tolerance with constant overhead can be performed on certain QLDPC codes [14], e.g., quantum expander codes [12].
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 [15].Fault-tolerant error correction can be done using Shor error correction [16], which is based on repeated measurements, or Knill error correction, which is based on teleportation [17][18].
Steane \([[7,1,3]]\) code Pieceable fault-tolerant CCZ gate [15].Syndrome measurement can be done with ancillary flag qubits [19][20] or with no extra qubits [21].
\([[15,1,3]]\) 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 [11][22].
\([[2^r-1, 2^r-2r-1, 3]]\) Hamming-based CSS code Syndrome measurement can be done with two ancillary flag qubits [20].
\([[4,2,2]]\) 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 [23].
\([[5,1,3]]\) perfect code Pieceable fault-tolerant CZ and CCZ gates [15].Syndrome measurement can be done with two ancillary flag qubits [20].

References

[1]
Yoder, Theodore., DSpace@MIT Practical Fault-Tolerant Quantum Computation (2018)
[2]
A. M. Steane, “Active Stabilization, Quantum Computation, and Quantum State Synthesis”, Physical Review Letters 78, 2252 (1997). DOI; quant-ph/9611027
[3]
S. Puri et al., “Bias-preserving gates with stabilized cat qubits”, Science Advances 6, (2020). DOI; 1905.00450
[4]
J. Guillaud and M. Mirrahimi, “Repetition Cat Qubits for Fault-Tolerant Quantum Computation”, Physical Review X 9, (2019). DOI; 1904.09474
[5]
H. Bombin and M. A. Martin-Delgado, “Topological Quantum Distillation”, Physical Review Letters 97, (2006). DOI; quant-ph/0605138
[6]
Andrew J. Landahl and Ciaran Ryan-Anderson, “Quantum computing by color-code lattice surgery”. 1407.5103
[7]
D. Gottesman, A. Kitaev, and J. Preskill, “Encoding a qubit in an oscillator”, Physical Review A 64, (2001). DOI; quant-ph/0008040
[8]
S. Glancy and E. Knill, “Error analysis for encoding a qubit in an oscillator”, Physical Review A 73, (2006). DOI; quant-ph/0510107
[9]
T. Jochym-O'Connor, “Fault-tolerant gates via homological product codes”, Quantum 3, 120 (2019). DOI; 1807.09783
[10]
A. L. Grimsmo, J. Combes, and B. Q. Baragiola, “Quantum Computing with Rotation-Symmetric Bosonic Codes”, Physical Review X 10, (2020). DOI; 1901.08071
[11]
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). DOI; 1403.2734
[12]
Omar Fawzi, Antoine Grospellier, and Anthony Leverrier, “Constant overhead quantum fault-tolerance with quantum expander codes”. 1808.03821
[13]
Lawrence Z. Cohen et al., “Low-overhead fault-tolerant quantum computing using long-range connectivity”. 2110.10794
[14]
Daniel Gottesman, “Fault-Tolerant Quantum Computation with Constant Overhead”. 1310.2984
[15]
T. J. Yoder, R. Takagi, and I. L. Chuang, “Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes”, Physical Review X 6, (2016). DOI; 1603.03948
[16]
Peter W. Shor, “Fault-tolerant quantum computation”. quant-ph/9605011
[17]
E. Knill, “Quantum computing with realistically noisy devices”, Nature 434, 39 (2005). DOI; quant-ph/0410199
[18]
E. Knill, “Scalable Quantum Computation in the Presence of Large Detected-Error Rates”. quant-ph/0312190
[19]
T. J. Yoder and I. H. Kim, “The surface code with a twist”, Quantum 1, 2 (2017). DOI; 1612.04795
[20]
R. Chao and B. W. Reichardt, “Quantum Error Correction with Only Two Extra Qubits”, Physical Review Letters 121, (2018). DOI; 1705.02329
[21]
B. W. Reichardt, “Fault-tolerant quantum error correction for Steane’s seven-qubit color code with few or no extra qubits”, Quantum Science and Technology 6, 015007 (2020). DOI
[22]
D.-X. Quan et al., “Fault-tolerant conversion between adjacent Reed–Muller quantum codes based on gauge fixing”, Journal of Physics A: Mathematical and Theoretical 51, 115305 (2018). DOI; 1703.03860
[23]
Daniel Gottesman, “Quantum fault tolerance in small experiments”. 1610.03507