The Error Correction Zoo

Victor V. Albert; Philippe Faist

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 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].
Capped color code (CCC)	Fault-tolerant syndrome extraction and error correction for capped color codes in H form [15].Fault-tolerant T gate implementation [15].
Cat code	Universal set of error-corrected operations tolerating a single photon loss and an arbitrary ancilla fault [16].Linear-optical noise suppression and mitigation scheme [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	There is a simple proof of a threshold for MBQC [18].Photonic architecture [19].Generalized foliation procedures exist for noise-bias preserving MBQC [20].Fault complexes yield fault-tolerance properties of cluster states that come from foliations of codes [21].
Color code	The 6D color code is a self-correcting quantum memory and admits fault-tolerant universal gate set in 7D [22].
Concatenated Steane code	Fault-tolerant computation can be done on nearest-neighbor arrays [23].There exist fault-tolerant syndrome extraction protocols for the concatenated Steane code [24].The combination of the concatenated Steane code and QLDPC codes with non-vanishing rate yield fault-tolerant quantum computation with constant space and polylogarithmic time overheads, even when classical computation time is taken into account [25].
Concatenated qubit code	Fault-tolerant message passing between devices [26].Blocklet concatenation uses concatenation and transversal gates in a way that is tailored to FBQC platforms [27].
Dihedral \(G=D_m\) quantum-double code	Universal topological quantum computation is possible for certain groups such as \(G=D_3=S_3\) [28,29].\(U\)-model gate set [30], which can protect from circuit-level noise with the help of an anyon interferometer for the case of \(G=S_3\) [31].
Distance-balanced code	Single-ancilla syndrome extraction circuits that, for the most part, preserve the effective distance of weight-reduced qLDPC codes [32]. The distance balancing technique of Ref. [33] preserves effective distance [32].
Dual-rail quantum code	Dual-rail qubits can be used to convert leakage and AD noise into erasure noise [34,35].
Floquet color code	Fault-tolerant measurement-based computation can be realized using the foliated Floquet color code [36].
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 [37].
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 [38].
Gottesman-Kitaev-Preskill (GKP) code	Logical Clifford operations are given by Gaussian unitaries, which map bounded-size errors to bounded-size errors [39]. 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) [40]. Such gates provide another example of monodromy under the particular notion of parallel transport introduced in Ref. [41].The 4D square-lattice GKP code admits the isthmus property, which allows certain ancilla errors to be detectable [42].
Hastings-Haah Floquet code	Periodic Floquet codes on tri-colorable lattices can be made fault-tolerant in the presence of dead qubits [43,44].
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 [45; pg. 9].
Hexagonal \(CZ\) code	The hexagonal \(CZ\) code can be obtained from two surface codes by gauging [46–55] their logical \(CZ\) gate [56]. Gates on the two surface codes in the third level of the Clifford hierarchy, such as \(CZ\) gates, can be realized fault tolerantly by performing this procedure and reversing it [56].
Hierarchical code	2D geometrically local syndrome extraction circuits of depth of order \(O(\sqrt{n}/R)\) 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 [57].
Honeycomb (6.6.6) color code	Fault-tolerant syndrome extraction circuits using flag qubits [58,59].
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 [60].
Hypergraph product (HGP) code	Single-ancilla syndrome extraction circuits do not admit hook errors [32].There is a fault-tolerant universal computation scheme for hypergraph-product codes concatenated with the \([[4,2,2]]\) code [61].
Kitaev honeycomb code	One can distill ancilla states to arbitrary precision for sufficiently small noise rates and assuming perfect Clifford operations [62].
Kitaev surface code	Transversal (non-Clifford) \(CCZ\) gate by bringing 2D surface codes together and using just-in-time decoding [63,64]. Gate can be simulated by taking 2D slices out of 3D surface codes [65].Flag fault-tolerant syndrome extraction [58].Homomorphic measurement protocols for arbitrary surface codes [66].Non-geometrically local connectivity can reduce overhead cost [67].Magic-state distillation protocols [68–71] leading up to magic-state cultivation [72].Framework of fault tolerance utilizing ZX calculus [73,74] that is applicable to MBQC, FBQC, and conventional computation versions of the surface code [75].Single-shot state preparation [76] and MWPM decoding [77].Syndrome extraction circuits consisting of CNOT gates and ancillary measurements [68]. Measurement schedules can be optimized using spacetime circuit codes to yield what is known as the 3CX surface code [78]. Schedules can also be optimized via ZX calculus [73,74]. Inspired by the honeycomb Floquet code, various weight-two measurement schemes have been designed [79–81], with the scheme in Ref. [80] being a special case of DWR.LUCI framework for syndrome extraction circuits [82,83].
Log-depth geometrically local Clifford-circuit code	Fault-tolerant state preparation [84].
Majorana box qubit	Fault-tolerant computation scheme [85].
Majorana color code	Ordinary and twist-based lattice surgery can be made fault tolerant [86] (see also [87]).
Majorana surface code	Ordinary and twist-based lattice surgery can be made fault tolerant [86] (see also [87]).
Number-phase code	Fault-tolerant computation schemes with number-phase codes have been proposed based on concatenation with Bacon-Shor subsystem codes [13].
Perfect-tensor code	Fault-tolerant \(d\)-uniform state preparation [88].
Projective-plane surface code	Fault-tolerant Hadamard gate via constant-depth Clifford circuit [89].
Quantum LDPC (QLDPC) code	Lattice surgery techniques with ancilla qubits [10,11,90]. 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 [91], e.g., quantum expander codes [92].GHZ state distillation for Steane error correction [93].Fault-tolerant logical measurements [55] 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. This can be used for a generalization of lattice surgery for CSS QLDPC codes [94].Fault-tolerant constant-depth encoder and unencoder [95].Fault-tolerant logical measurements based on an extractor system and allowing for universal computation [96].
Quantum Reed-Muller (RM) code	Gate switching protocol for universal computation [97].Fault-tolerant universal computation can be achieved via code switching between the \([[127,1,15]]\) self-dual doubly even punctured quantum RM code and the \([[127,1,7]]\) triply even punctured quantum RM code [98].
Quantum data-syndrome (QDS) code	Shor error correction [99,100], 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 [101].
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 [102]. 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 [92].
Quantum polar code	State preparation of a single logical qubit [103].
Quantum repetition code	Fault-tolerant syndrome detection [104].Toffoli magic-state preparation protocol [105].
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 [37]. A construction based on the Torelli mapping yields a code with constant rate with similar gates [37].
Qubit CSS code	Steane error correction [106], 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 [107]. Parallel syndrome extraction for distance-three codes can be done fault-tolerantly using one flag qubit [108]. Distance-preserving flag fault-tolerant error correction can be done using lookup tables for small codes [109]. Any self-dual CSS code with bounded-weight stabilizer generators admits flag fault-tolerant syndrome extraction [58].Homomorphic gadgets fault-tolerant measurement unify Steane and Shor error correction [66].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 \(\alpha > 0\) [110].Fault-tolerant measurement-free scheme for low-distance CSS codes [111].Automated fault-tolerant encoding circuit synthesis [112].Fault-tolerant homological measurement of logical Pauli operators [113].Fault-tolerant CNOT gate using generalized lattice surgery [114].
Qubit code	There are lower bounds on the overhead of fault-tolerant QEC in terms of the capacity of the noise channel [115]. 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 [116].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 [117].Fault-tolerant gates can be done for any code supporting a transversal implementation of Pauli gates using generalized gate teleportation [118].
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 [119,120]. Such protocols can be made fault tolerant with the help of magic-state distillation [121]. See review on magic-state distillation [122].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 [123].Shor error correction [99,100] (see also Steane's ancilla factory [124]), in which fault tolerance against syndrome extraction errors is ensured by simply repeating syndrome measurements. A modification uses adaptive measurements [125].Generalization of Steane error correction stabilizer codes [9; Sec. 3.6].Fault-tolerant error correction scheme by Knill (a.k.a. telecorrection [126]), which is based on teleportation [127,128]. A variant of it has been termed the Fibonacci scheme [129].Fault-tolerant error correction using flag qubits for codes satisfying certain conditions [58].GHZ state distillation for Steane error correction [130].Syndrome extraction using flag qubits and classical codes [131].Fault-tolerant constant-depth unencoder transforming logical states into physical states using single-qubit measurements [95].Post-selection based algorithm preparing magic state corresponding to arbitrary rotations [132].Code switching can be done using only transversal gates for qubit stabilizer codes [133].Flag-Proxy Networks (FPNs) [134].A logical Pauli can be gauged out to yield a fault-tolerant measurement that requires a qubit overhead linear in the Pauli's support [55].Automated fault-tolerant circuit synthesis using boolean satisfiability [135].Algorithm for fault-tolerant magic-state initialization [136].Fault tolerance of qubit stabilizer codes has been formalized [137].
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 [68,138,139].
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 [39].Error correction scheme is fault-tolerant to displacement noise as long as all input states have displacement errors less than \(\sqrt{\pi}/6\) [140].
Square-octagon (4.8.8) color code	Color-code lattice surgery [141].Fault-tolerant syndrome extraction circuits [142].
Subsystem Hypergraph Product Simplex (SHYPS) code	Logical Clifford operation on \(b\) blocks can be implemented fault-tolerantly in depth roughly \(4b r^2\) [143].
Subsystem qubit stabilizer code	Logical Clifford gates can be implemented fault-tolerantly for subsystem codes of distance at least three [144].
Subsystem spacetime circuit code	Fault-tolerant measurement gadget that is a modification based on the DiVincenzo-Shor cat-state method [99,100].
Subsystem surface code	Gauge fixing and changing the order in which check operators are measured yields a fault-tolerant decoder [145].
Tetrahedral color code	Fault-tolerant quantum computation designed for a 2D architecture [146].
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 [147].A non-fault-tolerant curcuit initializes the triangle code. To guarantee fault-tolerance, post-selection is performed on trivial measurements of the syndrome and of the logical Pauli, depending on the basis of the logical states [147].Making syndrome extraction fault tolerant requires a specific ordering of syndrome measurements so as to avoid hook errors [147].
Triorthogonal code	Universal fault-tolerant gates can be performed without magic-state distillation [144,148].
Twist-defect surface code	Fault-tolerant measurement of defects [149].Twisted double covers of codes yield fault-tolerant Clifford gates performed via Dehn twists [150].
Twisted XZZX toric code	Fault-tolerant syndrome extraction circuits using flag qubits [151].
Two-component cat code	Fault-tolerant error-correction procedure using small amplitude coherent states [152].Bias-preserving Hamiltonian-based CNOT gate is part of a universal noise-bias-preserving gate set that can be made fault tolerant using concatenation [153,154].
Zero-pi qubit code	One- and two-qubit phase gate errors can be suppressed [155].
\([[10,1,2]]\) CSS code	A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[10,1,2]]\) code [156].
\([[144,12,12]]\) gross code	Fault-tolerant modular quantum computing framework [157].
\([[15, 7, 3]]\) 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 [158].
\([[15,1,3]]\) quantum RM code	A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[15,1,3]]\) code [97,144,148,159].Fault-tolerant logical zero and logical plus state preparation using reinforcement learning [160].
\([[16,6,4]]\) Tesseract color code	Post-selected fault-tolerant syndrome extraction [161,162].
\([[17,1,5]]\) 4.8.8 color code	Fault-tolerant logical zero state preparation using reinforcement learning [160].
\([[23, 1, 7]]\) Quantum Golay code	Fault-tolerant depth-7 circuit consisting of 57 CNOT gates and preparing a logical-zero state [163].
\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code	Syndrome measurement can be done with two ancillary flag qubits [164].Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [165,166]. This scheme can be augmented to fit on a 1D qubit lattice with nearest-neighbor connectivity [167].
\([[2^r-1,1,3]]\) simplex code	Fault-tolerant syndrome extraction circuits using flag qubits [58].
\([[2m,2m-2,2]]\) error-detecting code	Logical SWAP gates can be performed fault tolerantly using an ancilla qubit [168; Sec. VII].Two-qubit fault-tolerant state preparation, error detection and projective measurements [164] (see also [169]).CNOT and Hadamard gates using only two extra qubits and four-qubit fault-tolerant \(CCZ\) gate [158].Fault-tolerant Clifford Trotter circuits that are linear in \(k\) using flag qubits via a solve-and-stitch algorithm and application of a logical identity circuit [170].Weak fault tolerance: any single gate error can be detected by measuring stabilizers and utilizing extra ancillas [171].
\([[4,2,2]]\) 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 [172]. Magic states can be injected into surface and color codes since the code is a small instance of both [173].Concatenations of \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation schemes [127] admitting a post-selected threshold [174,175] (see also Ref. [176]).Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [165,166]. This scheme can be augmented to fit on a 1D qubit lattice with nearest-neighbor connectivity [167].Fault-tolerant implementation of the Deutsch-Josza algorithm [177].
\([[5,1,2]]\) rotated surface code	Fault-tolerant implementation of the Clifford group based on transversal gates and SWAPs [178].
\([[6,2,2]]\) \(C_6\) code	Concatenations of \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation schemes [127] admitting a post-selected threshold [174,175] (see also Ref. [176]).Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [165,166]. This scheme can be augmented to fit on a 1D qubit lattice with nearest-neighbor connectivity [167].Fault-tolerant magic-state preparation [179].One of the code's logical qubits can be relaxed to a gauge qubit to yield a \([[6,1,1,2]]\) subsystem qubit stabilizer code with a particular set of transversal gates. This code admits a fault-tolerant circuit relevant to magic-state preparation [180].
\([[7,1,3]]\) Steane code	A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[15,1,3]]\) code [97,144,148,159].A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[10,1,2]]\) code [156].Fault-tolerant logical zero and magic state preparation [181]. Magic-state preparation converts unbiased noise into biased noise [182].Fault-tolerant logical zero and logical plus state preparation on all-to-all and 2D grid qubit connectivity [160].Pieceable fault-tolerant \(CCZ\) gate [123].Syndrome measurement can be done with ancillary flag qubits [147,164] or with no extra qubits [183]. The depth of syndrome extraction circuits can be lowered by using past syndrome values [184].Computation of ground-state energy of the hydrogen molecule [185].Fault-tolerant measurement-free error-correction cycle [111].
\([[8,3,2]]\) Smallest interesting color code	\(CCZ\) gate can be distilled in a fault-tolerant manner [186].Fault-tolerant and teleportation-free logical Hadamard [187].
\([[9,1,3]]\) Shor code	Fault-tolerant logical zero and logical plus state preparation using reinforcement learning [160].Fault-tolerant measurement-free logical-zero state preparation [188].
\([[9,1,3]]\) Surface-17 code	Measurement-free fault-tolerant logical zero state preparation in nearest-neighbor qubit connectivity [188].Fault-tolerant logical zero and logical plus state preparation in all-to-all and 2D grid connectivity with flag qubits [160].

References

[1]: M. Vasmer and D. E. Browne, “Three-dimensional surface codes: Transversal gates and fault-tolerant architectures”, Physical Review A 100, (2019) arXiv:1801.04255 DOI
[2]: A. Bauer, “Low-overhead non-Clifford fault-tolerant circuits for all non-chiral abelian topological phases”, Quantum 9, 1673 (2025) arXiv:2403.12119 DOI
[3]: M. S. Kesselring, J. C. Magdalena de la Fuente, F. Thomsen, J. Eisert, S. D. Bartlett, and B. J. Brown, “Anyon Condensation and the Color Code”, PRX Quantum 5, (2024) arXiv:2212.00042 DOI
[4]: H. Bombín, C. Dawson, Y.-H. Liu, N. Nickerson, F. Pastawski, and S. Roberts, “Modular decoding: parallelizable real-time decoding for quantum computers”, (2023) arXiv:2303.04846
[5]: P. Aliferis, F. Brito, D. P. DiVincenzo, J. Preskill, M. Steffen, and B. M. Terhal, “Fault-tolerant computing with biased-noise superconducting qubits: a case study”, New Journal of Physics 11, 013061 (2009) arXiv:0806.0383 DOI
[6]: A. M. Stephens, Z. W. E. Evans, S. J. Devitt, and L. C. L. Hollenberg, “Asymmetric quantum error correction via code conversion”, Physical Review A 77, (2008) arXiv:0708.3969 DOI
[7]: Z. W. E. Evans, A. M. Stephens, J. H. Cole, and L. C. L. Hollenberg, “Error correction optimisation in the presence of X/Z asymmetry”, (2007) arXiv:0709.3875
[8]: P. Brooks and J. Preskill, “Fault-tolerant quantum computation with asymmetric Bacon-Shor codes”, Physical Review A 87, (2013) arXiv:1211.1400 DOI
[9]: Yoder, Theodore., DSpace@MIT Practical Fault-Tolerant Quantum Computation (2018)
[10]: L. Z. Cohen, I. H. Kim, S. D. Bartlett, and B. J. Brown, “Low-overhead fault-tolerant quantum computing using long-range connectivity”, Science Advances 8, (2022) arXiv:2110.10794 DOI
[11]: Q. Xu, J. P. B. Ataides, C. A. Pattison, N. Raveendran, D. Bluvstein, J. Wurtz, B. Vasic, M. D. Lukin, L. Jiang, and H. Zhou, “Constant-Overhead Fault-Tolerant Quantum Computation with Reconfigurable Atom Arrays”, (2023) arXiv:2308.08648
[12]: S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, “High-threshold and low-overhead fault-tolerant quantum memory”, Nature 627, 778 (2024) arXiv:2308.07915 DOI
[13]: A. L. Grimsmo, J. Combes, and B. Q. Baragiola, “Quantum Computing with Rotation-Symmetric Bosonic Codes”, Physical Review X 10, (2020) arXiv:1901.08071 DOI
[14]: L. D. H. My, S. Qin, and H. K. Ng, “Circuit-level fault tolerance of cat codes”, (2024) arXiv:2406.04157
[15]: T. Tansuwannont and D. Leung, “Achieving Fault Tolerance on Capped Color Codes with Few Ancillas”, PRX Quantum 3, (2022) arXiv:2106.02649 DOI
[16]: Q. Xu, P. Zeng, D. Xu, and L. Jiang, “Fault-Tolerant Operation of Bosonic Qubits with Discrete-Variable Ancillae”, (2023) arXiv:2310.20578
[17]: Y. S. Teo, S. U. Shringarpure, S. Cho, and H. Jeong, “Linear-optical protocols for mitigating and suppressing noise in bosonic systems”, Quantum Science and Technology 10, 035003 (2025) arXiv:2411.11313 DOI
[18]: P. Aliferis and D. W. Leung, “Simple proof of fault tolerance in the graph-state model”, Physical Review A 73, (2006) arXiv:quant-ph/0503130 DOI
[19]: S. J. Devitt, A. G. Fowler, A. M. Stephens, A. D. Greentree, L. C. L. Hollenberg, W. J. Munro, and K. Nemoto, “Architectural design for a topological cluster state quantum computer”, New Journal of Physics 11, 083032 (2009) arXiv:0808.1782 DOI
[20]: J. Claes, J. E. Bourassa, and S. Puri, “Tailored cluster states with high threshold under biased noise”, npj Quantum Information 9, (2023) arXiv:2201.10566 DOI
[21]: T. Hillmann, G. Dauphinais, I. Tzitrin, and M. Vasmer, “Single-shot and measurement-based quantum error correction via fault complexes”, (2024) arXiv:2410.12963
[22]: H. Bombin, R. W. Chhajlany, M. Horodecki, and M. A. Martin-Delgado, “Self-Correcting Quantum Computers”, (2012) arXiv:0907.5228
[23]: A. M. Stephens, A. G. Fowler, and L. C. L. Hollenberg, “Universal fault tolerant quantum computation on bilinear nearest neighbor arrays”, (2008) arXiv:quant-ph/0702201
[24]: B. Pato, T. Tansuwannont, and K. R. Brown, “Concatenated Steane code with single-flag syndrome checks”, Physical Review A 110, (2024) arXiv:2403.09978 DOI
[25]: S. Tamiya, M. Koashi, and H. Yamasaki, “Polylog-time- and constant-space-overhead fault-tolerant quantum computation with quantum low-density parity-check codes”, (2024) arXiv:2411.03683
[26]: M. Christandl, O. Fawzi, and A. Goswami, “Fault-tolerant quantum input/output”, (2024) arXiv:2408.05260
[27]: D. Litinski, “Blocklet concatenation: Low-overhead fault-tolerant protocols for fusion-based quantum computation”, (2025) arXiv:2506.13619
[28]: C. Mochon, “Anyon computers with smaller groups”, Physical Review A 69, (2004) arXiv:quant-ph/0306063 DOI
[29]: G. K. Brennen, M. Aguado, and J. I. Cirac, “Simulations of quantum double models”, New Journal of Physics 11, 053009 (2009) arXiv:0901.1345 DOI
[30]: S. X. Cui, S.-M. Hong, and Z. Wang, “Universal quantum computation with weakly integral anyons”, Quantum Information Processing 14, 2687 (2015) arXiv:1401.7096 DOI
[31]: L. Chen, Y. Ren, R. Fan, and A. Jaffe, “A Universal Circuit Set Using the \(S_3\) Quantum Double”, (2025) arXiv:2411.09697
[32]: S. J. S. Tan and L. Stambler, “Effective Distance of Higher Dimensional HGPs and Weight-Reduced Quantum LDPC Codes”, (2025) arXiv:2409.02193
[33]: S. Evra, T. Kaufman, and G. Zémor, “Decodable quantum LDPC codes beyond the \(\sqrt{n}\) distance barrier using high dimensional expanders”, (2020) arXiv:2004.07935
[34]: A. S. Fletcher, P. W. Shor, and M. Z. Win, “Channel-Adapted Quantum Error Correction for the Amplitude Damping Channel”, (2007) arXiv:0710.1052
[35]: A. Kubica, A. Haim, Y. Vaknin, H. Levine, F. Brandão, and A. Retzker, “Erasure Qubits: Overcoming the T1 Limit in Superconducting Circuits”, Physical Review X 13, (2023) arXiv:2208.05461 DOI
[36]: S. Paesani and B. J. Brown, “High-Threshold Quantum Computing by Fusing One-Dimensional Cluster States”, Physical Review Letters 131, (2023) arXiv:2212.06775 DOI
[37]: G. Zhu, S. Sikander, E. Portnoy, A. W. Cross, and B. J. Brown, “Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries”, (2024) arXiv:2310.16982
[38]: H. Bombin, C. Dawson, T. Farrelly, Y. Liu, N. Nickerson, M. Pant, F. Pastawski, and S. Roberts, “Fault-tolerant complexes”, (2023) arXiv:2308.07844
[39]: D. Gottesman, A. Kitaev, and J. Preskill, “Encoding a qubit in an oscillator”, Physical Review A 64, (2001) arXiv:quant-ph/0008040 DOI
[40]: J. Conrad, A. G. Burchards, and S. T. Flammia, “Lattices, Gates, and Curves: GKP codes as a Rosetta stone”, (2024) arXiv:2407.03270
[41]: D. Gottesman and L. L. Zhang, “Fibre bundle framework for unitary quantum fault tolerance”, (2017) arXiv:1309.7062
[42]: B. Royer, S. Singh, and S. M. Girvin, “Encoding Qubits in Multimode Grid States”, PRX Quantum 3, (2022) arXiv:2201.12337 DOI
[43]: D. Aasen, J. Haah, P. Bonderson, Z. Wang, and M. Hastings, “Fault-Tolerant Hastings-Haah Codes in the Presence of Dead Qubits”, (2023) arXiv:2307.03715
[44]: C. McLauchlan, G. P. Gehér, and A. E. Moylett, “Accommodating Fabrication Defects on Floquet Codes with Minimal Hardware Requirements”, Quantum 8, 1562 (2024) arXiv:2405.15854 DOI
[45]: E. M. Rains, “Nonbinary quantum codes”, (1997) arXiv:quant-ph/9703048
[46]: M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
[47]: J. Haegeman, K. Van Acoleyen, N. Schuch, J. I. Cirac, and F. Verstraete, “Gauging Quantum States: From Global to Local Symmetries in Many-Body Systems”, Physical Review X 5, (2015) arXiv:1407.1025 DOI
[48]: S. Vijay, J. Haah, and L. Fu, “Fracton topological order, generalized lattice gauge theory, and duality”, Physical Review B 94, (2016) arXiv:1603.04442 DOI
[49]: D. J. Williamson, “Fractal symmetries: Ungauging the cubic code”, Physical Review B 94, (2016) arXiv:1603.05182 DOI
[50]: L. Bhardwaj, D. Gaiotto, and A. Kapustin, “State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter”, Journal of High Energy Physics 2017, (2017) arXiv:1605.01640 DOI
[51]: A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
[52]: W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
[53]: K. Dolev, V. Calvera, S. S. Cree, and D. J. Williamson, “Gauging the bulk: generalized gauging maps and holographic codes”, Journal of High Energy Physics 2022, (2022) arXiv:2108.11402 DOI
[54]: T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes I. Gauging and dualities”, (2023) arXiv:2310.16032
[55]: D. J. Williamson and T. J. Yoder, “Low-overhead fault-tolerant quantum computation by gauging logical operators”, (2024) arXiv:2410.02213
[56]: M. Davydova, A. Bauer, J. C. M. de la Fuente, M. Webster, D. J. Williamson, and B. J. Brown, “Universal fault tolerant quantum computation in 2D without getting tied in knots”, (2025) arXiv:2503.15751
[57]: T. Jochym-O’Connor, “Fault-tolerant gates via homological product codes”, Quantum 3, 120 (2019) arXiv:1807.09783 DOI
[58]: C. Chamberland and M. E. Beverland, “Flag fault-tolerant error correction with arbitrary distance codes”, Quantum 2, 53 (2018) arXiv:1708.02246 DOI
[59]: C. Chamberland, A. Kubica, T. J. Yoder, and G. Zhu, “Triangular color codes on trivalent graphs with flag qubits”, New Journal of Physics 22, 023019 (2020) arXiv:1911.00355 DOI
[60]: J. Lee, N. Kang, S.-H. Lee, H. Jeong, L. Jiang, and S.-W. Lee, “Fault-Tolerant Quantum Computation by Hybrid Qubits with Bosonic Cat Code and Single Photons”, PRX Quantum 5, (2024) arXiv:2401.00450 DOI
[61]: N. Berthusen, S. J. S. Tan, E. Huang, and D. Gottesman, “Adaptive Syndrome Extraction”, (2025) arXiv:2502.14835
[62]: S. Bravyi, “Universal quantum computation with theν=5∕2fractional quantum Hall state”, Physical Review A 73, (2006) arXiv:quant-ph/0511178 DOI
[63]: B. J. Brown, “A fault-tolerant non-Clifford gate for the surface code in two dimensions”, Science Advances 6, (2020) arXiv:1903.11634 DOI
[64]: T. R. Scruby, K. Nemoto, and Z. Cai, “Fault-tolerant Quantum Computation without Distillation on a 2D Device”, (2025) arXiv:2412.12529
[65]: T. R. Scruby, D. E. Browne, P. Webster, and M. Vasmer, “Numerical Implementation of Just-In-Time Decoding in Novel Lattice Slices Through the Three-Dimensional Surface Code”, Quantum 6, 721 (2022) arXiv:2012.08536 DOI
[66]: S. Huang, T. Jochym-O’Connor, and T. J. Yoder, “Homomorphic Logical Measurements”, (2022) arXiv:2211.03625
[67]: D. Litinski and N. Nickerson, “Active volume: An architecture for efficient fault-tolerant quantum computers with limited non-local connections”, (2022) arXiv:2211.15465
[68]: A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: Towards practical large-scale quantum computation”, Physical Review A 86, (2012) arXiv:1208.0928 DOI
[69]: A. G. Fowler and S. J. Devitt, “A bridge to lower overhead quantum computation”, (2013) arXiv:1209.0510
[70]: H. Bombín, M. Pant, S. Roberts, and K. I. Seetharam, “Fault-Tolerant Postselection for Low-Overhead Magic State Preparation”, PRX Quantum 5, (2024) arXiv:2212.00813 DOI
[71]: T. Itogawa, Y. Takada, Y. Hirano, and K. Fujii, “Efficient Magic State Distillation by Zero-Level Distillation”, PRX Quantum 6, (2025) arXiv:2403.03991 DOI
[72]: C. Gidney, N. Shutty, and C. Jones, “Magic state cultivation: growing T states as cheap as CNOT gates”, (2024) arXiv:2409.17595
[73]: B. Coecke and R. Duncan, “Interacting Quantum Observables”, Lecture Notes in Computer Science 298 DOI
[74]: B. Coecke and R. Duncan, “Interacting quantum observables: categorical algebra and diagrammatics”, New Journal of Physics 13, 043016 (2011) arXiv:0906.4725 DOI
[75]: H. Bombin, D. Litinski, N. Nickerson, F. Pastawski, and S. Roberts, “Unifying flavors of fault tolerance with the ZX calculus”, Quantum 8, 1379 (2024) arXiv:2303.08829 DOI
[76]: S. Bravyi, D. Gosset, R. König, and M. Tomamichel, “Quantum advantage with noisy shallow circuits”, Nature Physics 16, 1040 (2020) arXiv:1904.01502 DOI
[77]: S. H. Choe and R. Koenig, “Long-range data transmission in a fault-tolerant quantum bus architecture”, (2022) arXiv:2209.09774
[78]: M. McEwen, D. Bacon, and C. Gidney, “Relaxing Hardware Requirements for Surface Code Circuits using Time-dynamics”, Quantum 7, 1172 (2023) arXiv:2302.02192 DOI
[79]: R. Chao, M. E. Beverland, N. Delfosse, and J. Haah, “Optimization of the surface code design for Majorana-based qubits”, Quantum 4, 352 (2020) arXiv:2007.00307 DOI
[80]: C. Gidney, “A Pair Measurement Surface Code on Pentagons”, Quantum 7, 1156 (2023) arXiv:2206.12780 DOI
[81]: L. Grans-Samuelsson, R. V. Mishmash, D. Aasen, C. Knapp, B. Bauer, B. Lackey, M. P. da Silva, and P. Bonderson, “Improved Pairwise Measurement-Based Surface Code”, Quantum 8, 1429 (2024) arXiv:2310.12981 DOI
[82]: D. M. Debroy, M. McEwen, C. Gidney, N. Shutty, and A. Zalcman, “LUCI in the Surface Code with Dropouts”, (2024) arXiv:2410.14891
[83]: D. M. Debroy, “Diamond Circuits for Surface Codes”, (2025) arXiv:2502.10355
[84]: J. Nelson, G. Bentsen, S. T. Flammia, and M. J. Gullans, “Fault-Tolerant Quantum Memory using Low-Depth Random Circuit Codes”, (2023) arXiv:2311.17985
[85]: D. Aasen et al., “Roadmap to fault tolerant quantum computation using topological qubit arrays”, (2025) arXiv:2502.12252
[86]: D. Litinski and F. von Oppen, “Quantum computing with Majorana fermion codes”, Physical Review B 97, (2018) arXiv:1801.08143 DOI
[87]: C. McLauchlan and B. Béri, “A new twist on the Majorana surface code: Bosonic and fermionic defects for fault-tolerant quantum computation”, Quantum 8, 1400 (2024) arXiv:2211.11777 DOI
[88]: S. Majidy, D. Hangleiter, and M. J. Gullans, “Scalable and fault-tolerant preparation of encoded k-uniform states”, (2025) arXiv:2503.14506
[89]: R. Kobayashi and G. Zhu, “Cross-Cap Defects and Fault-Tolerant Logical Gates in the Surface Code and the Honeycomb Floquet Code”, PRX Quantum 5, (2024) arXiv:2310.06917 DOI
[90]: A. Cross, Z. He, P. Rall, and T. Yoder, “Improved QLDPC Surgery: Logical Measurements and Bridging Codes”, (2024) arXiv:2407.18393
[91]: D. Gottesman, “Fault-Tolerant Quantum Computation with Constant Overhead”, (2014) arXiv:1310.2984
[92]: O. Fawzi, A. Grospellier, and A. Leverrier, “Constant Overhead Quantum Fault-Tolerance with Quantum Expander Codes”, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) 743 (2018) arXiv:1808.03821 DOI
[93]: N. Rengaswamy, N. Raveendran, A. Raina, and B. Vasić, “Entanglement Purification with Quantum LDPC Codes and Iterative Decoding”, Quantum 8, 1233 (2024) arXiv:2210.14143 DOI
[94]: A. Cowtan, Z. He, D. J. Williamson, and T. J. Yoder, “Parallel Logical Measurements via Quantum Code Surgery”, (2025) arXiv:2503.05003
[95]: Y. Shi, A. Patil, and S. Guha, “Stabilizer Entanglement Distillation and Efficient Fault-Tolerant Encoders”, PRX Quantum 6, (2025) arXiv:2408.06299 DOI
[96]: Z. He, A. Cowtan, D. J. Williamson, and T. J. Yoder, “Extractors: QLDPC Architectures for Efficient Pauli-Based Computation”, (2025) arXiv:2503.10390
[97]: 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) arXiv:1403.2734 DOI
[98]: A. Gong and J. M. Renes, “Computation with quantum Reed-Muller codes and their mapping onto 2D atom arrays”, (2024) arXiv:2410.23263
[99]: P. W. Shor, “Fault-tolerant quantum computation”, (1997) arXiv:quant-ph/9605011
[100]: D. P. DiVincenzo and P. W. Shor, “Fault-Tolerant Error Correction with Efficient Quantum Codes”, Physical Review Letters 77, 3260 (1996) arXiv:quant-ph/9605031 DOI
[101]: A. Ashikhmin, C.-Y. Lai, and T. A. Brun, “Quantum Data-Syndrome Codes”, IEEE Journal on Selected Areas in Communications 38, 449 (2020) arXiv:1907.01393 DOI
[102]: J. Hu, Q. Liang, and R. Calderbank, “Divisible Codes for Quantum Computation”, (2022) arXiv:2204.13176
[103]: A. Goswami, M. Mhalla, and V. Savin, “Fault-tolerant preparation of quantum polar codes encoding one logical qubit”, Physical Review A 108, (2023) arXiv:2209.06673 DOI
[104]: Y. C. Cheng and R. J. Silbey, “Microscopic quantum dynamics study on the noise threshold of fault-tolerant quantum error correction”, Physical Review A 72, (2005) arXiv:quant-ph/0412168 DOI
[105]: C. Chamberland et al., “Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes”, PRX Quantum 3, (2022) arXiv:2012.04108 DOI
[106]: A. M. Steane, “Active Stabilization, Quantum Computation, and Quantum State Synthesis”, Physical Review Letters 78, 2252 (1997) arXiv:quant-ph/9611027 DOI
[107]: T. Tansuwannont, C. Chamberland, and D. Leung, “Flag fault-tolerant error correction, measurement, and quantum computation for cyclic Calderbank-Shor-Steane codes”, Physical Review A 101, (2020) arXiv:1803.09758 DOI
[108]: P.-H. Liou and C.-Y. Lai, “Parallel syndrome extraction with shared flag qubits for Calderbank-Shor-Steane codes of distance three”, Physical Review A 107, (2023) arXiv:2208.00581 DOI
[109]: B. Pato, T. Tansuwannont, S. Huang, and K. R. Brown, “Optimization Tools for Distance-Preserving Flag Fault-Tolerant Error Correction”, PRX Quantum 5, (2024) arXiv:2306.12862 DOI
[110]: N. Delfosse, B. W. Reichardt, and K. M. Svore, “Beyond Single-Shot Fault-Tolerant Quantum Error Correction”, IEEE Transactions on Information Theory 68, 287 (2022) arXiv:2002.05180 DOI
[111]: S. Heußen, D. F. Locher, and M. Müller, “Measurement-Free Fault-Tolerant Quantum Error Correction in Near-Term Devices”, PRX Quantum 5, (2024) arXiv:2307.13296 DOI
[112]: T. Peham, L. Schmid, L. Berent, M. Müller, and R. Wille, “Automated Synthesis of Fault-Tolerant State Preparation Circuits for Quantum Error-Correction Codes”, PRX Quantum 6, (2025) arXiv:2408.11894 DOI
[113]: B. Ide, M. G. Gowda, P. J. Nadkarni, and G. Dauphinais, “Fault-tolerant logical measurements via homological measurement”, (2024) arXiv:2410.02753
[114]: C. Poirson, J. Roffe, and R. I. Booth, “Engineering CSS surgery: compiling any CNOT in any code”, (2025) arXiv:2505.01370
[115]: O. Fawzi, A. Müller-Hermes, and A. Shayeghi, “A Lower Bound on the Space Overhead of Fault-Tolerant Quantum Computation”, (2022) arXiv:2202.00119 DOI
[116]: N. Baspin, O. Fawzi, and A. Shayeghi, “A lower bound on the overhead of quantum error correction in low dimensions”, (2023) arXiv:2302.04317
[117]: S. H. Choe and R. Koenig, “How to fault-tolerantly realize any quantum circuit with local operations”, (2024) arXiv:2402.13863
[118]: E. Kubischta and I. Teixeira, “Flexible Fault Tolerant Gate Gadgets”, (2024) arXiv:2409.11616
[119]: D. Gottesman and I. L. Chuang, “Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations”, Nature 402, 390 (1999) arXiv:quant-ph/9908010 DOI
[120]: X. Zhou, D. W. Leung, and I. L. Chuang, “Methodology for quantum logic gate construction”, Physical Review A 62, (2000) arXiv:quant-ph/0002039 DOI
[121]: S. Bravyi and A. Kitaev, “Universal quantum computation with ideal Clifford gates and noisy ancillas”, Physical Review A 71, (2005) arXiv:quant-ph/0403025 DOI
[122]: C. J. Trout and K. R. Brown, “Magic state distillation and gate compilation in quantum algorithms for quantum chemistry”, International Journal of Quantum Chemistry 115, 1296 (2015) DOI
[123]: T. J. Yoder, R. Takagi, and I. L. Chuang, “Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes”, Physical Review X 6, (2016) arXiv:1603.03948 DOI
[124]: A. M. Steane, “Space, Time, Parallelism and Noise Requirements for Reliable Quantum Computing”, Fortschritte der Physik 46, 443 (1998) arXiv:quant-ph/9708021 DOI
[125]: T. Tansuwannont, B. Pato, and K. R. Brown, “Adaptive syndrome measurements for Shor-style error correction”, Quantum 7, 1075 (2023) arXiv:2208.05601 DOI
[126]: C. M. Dawson, H. L. Haselgrove, and M. A. Nielsen, “Noise thresholds for optical cluster-state quantum computation”, Physical Review A 73, (2006) arXiv:quant-ph/0601066 DOI
[127]: E. Knill, “Quantum computing with realistically noisy devices”, Nature 434, 39 (2005) arXiv:quant-ph/0410199 DOI
[128]: E. Knill, “Scalable Quantum Computation in the Presence of Large Detected-Error Rates”, (2004) arXiv:quant-ph/0312190
[129]: P. Aliferis and J. Preskill, “Fibonacci scheme for fault-tolerant quantum computation”, Physical Review A 79, (2009) arXiv:0809.5063 DOI
[130]: N. Rengaswamy, A. Raina, N. Raveendran, and B. Vasić, “Distilling GHZ States using Stabilizer Codes”, (2022) arXiv:2109.06248
[131]: B. Anker and M. Marvian, “Flag Gadgets based on Classical Codes”, (2024) arXiv:2212.10738
[132]: H. Choi, F. T. Chong, D. Englund, and Y. Ding, “Fault Tolerant Non-Clifford State Preparation for Arbitrary Rotations”, (2023) arXiv:2303.17380
[133]: S. Heußen and J. Hilder, “Efficient fault-tolerant code switching via one-way transversal CNOT gates”, (2024) arXiv:2409.13465
[134]: S. Vittal, A. Javadi-Abhari, A. W. Cross, L. S. Bishop, and M. Qureshi, “Flag Proxy Networks: Tackling the Architectural, Scheduling, and Decoding Obstacles of Quantum LDPC codes”, (2024) arXiv:2409.14283
[135]: L. Schmid, T. Peham, L. Berent, M. Müller, and R. Wille, “Deterministic Fault-Tolerant State Preparation for Near-Term Quantum Error Correction: Automatic Synthesis Using Boolean Satisfiability”, (2025) arXiv:2501.05527
[136]: N. Fazio, M. Webster, and Z. Cai, “Low-overhead Magic State Circuits with Transversal CNOTs”, (2025) arXiv:2501.10291
[137]: K. Chen, Y. Liu, W. Fang, J. Paykin, X.-C. Wu, A. Schmitz, S. Zdancewic, and G. Li, “Verifying Fault-Tolerance of Quantum Error Correction Codes”, (2025) arXiv:2501.14380
[138]: E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
[139]: Y. Tomita and K. M. Svore, “Low-distance surface codes under realistic quantum noise”, Physical Review A 90, (2014) arXiv:1404.3747 DOI
[140]: S. Glancy and E. Knill, “Error analysis for encoding a qubit in an oscillator”, Physical Review A 73, (2006) arXiv:quant-ph/0510107 DOI
[141]: A. J. Landahl and C. Ryan-Anderson, “Quantum computing by color-code lattice surgery”, (2014) arXiv:1407.5103
[142]: A. J. Landahl, J. T. Anderson, and P. R. Rice, “Fault-tolerant quantum computing with color codes”, (2011) arXiv:1108.5738
[143]: A. J. Malcolm et al., “Computing Efficiently in QLDPC Codes”, (2025) arXiv:2502.07150
[144]: D. Banfield, H. Leitch, and A. Kay, “Implementing Clifford Gates on Stabilizer Codes via Measurement”, (2025) arXiv:2210.14074
[145]: O. Higgott and N. P. Breuckmann, “Subsystem Codes with High Thresholds by Gauge Fixing and Reduced Qubit Overhead”, Physical Review X 11, (2021) arXiv:2010.09626 DOI
[146]: H. Bombin, “2D quantum computation with 3D topological codes”, (2018) arXiv:1810.09571
[147]: T. J. Yoder and I. H. Kim, “The surface code with a twist”, Quantum 1, 2 (2017) arXiv:1612.04795 DOI
[148]: A. Paetznick and B. W. Reichardt, “Universal Fault-Tolerant Quantum Computation with Only Transversal Gates and Error Correction”, Physical Review Letters 111, (2013) arXiv:1304.3709 DOI
[149]: M. B. Hastings and A. Geller, “Reduced Space-Time and Time Costs Using Dislocation Codes and Arbitrary Ancillas”, (2015) arXiv:1408.3379
[150]: S. Burton, E. Durso-Sabina, and N. C. Brown, “Genons, Double Covers and Fault-tolerant Clifford Gates”, (2024) arXiv:2406.09951
[151]: Q. Xu, N. Mannucci, A. Seif, A. Kubica, S. T. Flammia, and L. Jiang, “Tailored XZZX codes for biased noise”, (2022) arXiv:2203.16486
[152]: A. P. Lund, T. C. Ralph, and H. L. Haselgrove, “Fault-Tolerant Linear Optical Quantum Computing with Small-Amplitude Coherent States”, Physical Review Letters 100, (2008) arXiv:0707.0327 DOI
[153]: J. Guillaud and M. Mirrahimi, “Repetition Cat Qubits for Fault-Tolerant Quantum Computation”, Physical Review X 9, (2019) arXiv:1904.09474 DOI
[154]: S. Puri et al., “Bias-preserving gates with stabilized cat qubits”, Science Advances 6, (2020) arXiv:1905.00450 DOI
[155]: P. Brooks, A. Kitaev, and J. Preskill, “Protected gates for superconducting qubits”, Physical Review A 87, (2013) arXiv:1302.4122 DOI
[156]: I. Pogorelov, F. Butt, L. Postler, C. D. Marciniak, P. Schindler, M. Müller, and T. Monz, “Experimental fault-tolerant code switching”, (2024) arXiv:2403.13732
[157]: T. J. Yoder, E. Schoute, P. Rall, E. Pritchett, J. M. Gambetta, A. W. Cross, M. Carroll, and M. E. Beverland, “Tour de gross: A modular quantum computer based on bivariate bicycle codes”, (2025) arXiv:2506.03094
[158]: R. Chao and B. W. Reichardt, “Fault-tolerant quantum computation with few qubits”, npj Quantum Information 4, (2018) arXiv:1705.05365 DOI
[159]: D.-X. Quan, L.-L. Zhu, C.-X. Pei, and B. C. Sanders, “Fault-tolerant conversion between adjacent Reed–Muller quantum codes based on gauge fixing”, Journal of Physics A: Mathematical and Theoretical 51, 115305 (2018) arXiv:1703.03860 DOI
[160]: R. Zen, J. Olle, L. Colmenarez, M. Puviani, M. Müller, and F. Marquardt, “Quantum Circuit Discovery for Fault-Tolerant Logical State Preparation with Reinforcement Learning”, (2024) arXiv:2402.17761
[161]: N. Delfosse and B. W. Reichardt, “Short Shor-style syndrome sequences”, (2020) arXiv:2008.05051
[162]: P. Prabhu and B. W. Reichardt, “Distance-four quantum codes with combined postselection and error correction”, Physical Review A 110, (2024) arXiv:2112.03785 DOI
[163]: A. Paetznick and B. W. Reichardt, “Fault-tolerant ancilla preparation and noise threshold lower bounds for the 23-qubit Golay code”, (2013) arXiv:1106.2190
[164]: R. Chao and B. W. Reichardt, “Quantum Error Correction with Only Two Extra Qubits”, Physical Review Letters 121, (2018) arXiv:1705.02329 DOI
[165]: H. Yamasaki and M. Koashi, “Time-Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation”, Nature Physics 20, 247 (2024) arXiv:2207.08826 DOI
[166]: S. Yoshida, S. Tamiya, and H. Yamasaki, “Concatenate codes, save qubits”, npj Quantum Information 11, (2025) arXiv:2402.09606 DOI
[167]: C. Gidney and T. Bergamaschi, “A Constant Rate Quantum Computer on a Line”, (2025) arXiv:2502.16132
[168]: D. Gottesman, “Theory of fault-tolerant quantum computation”, Physical Review A 57, 127 (1998) arXiv:quant-ph/9702029 DOI
[169]: C. N. Self, M. Benedetti, and D. Amaro, “Protecting expressive circuits with a quantum error detection code”, Nature Physics 20, 219 (2024) arXiv:2211.06703 DOI
[170]: Z. Chen and N. Rengaswamy, “Tailoring Fault-Tolerance to Quantum Algorithms”, (2024) arXiv:2404.11953
[171]: C. Gerhard and T. A. Brun, “Weakly Fault-Tolerant Computation in a Quantum Error-Detecting Code”, (2024) arXiv:2408.14828
[172]: D. Gottesman, “Quantum fault tolerance in small experiments”, (2016) arXiv:1610.03507
[173]: R. S. Gupta et al., “Encoding a magic state with beyond break-even fidelity”, Nature 625, 259 (2024) arXiv:2305.13581 DOI
[174]: B. Reichardt, “Postselection threshold against biased noise”, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS’06) 420 (2006) arXiv:quant-ph/0608018 DOI
[175]: P. Aliferis, D. Gottesman, and J. Preskill, “Accuracy threshold for postselected quantum computation”, (2007) arXiv:quant-ph/0703264
[176]: J. Cho, “Fault-tolerant linear optics quantum computation by error-detecting quantum state transfer”, Physical Review A 76, (2007) arXiv:quant-ph/0612073 DOI
[177]: D. Singh and S. Prakash, “Fault-Tolerant Implementation of the Deutsch-Jozsa Algorithm”, (2024) arXiv:2412.04791
[178]: M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
[179]: S. Dasu, S. Burton, K. Mayer, D. Amaro, J. A. Gerber, K. Gilmore, D. Gresh, D. DelVento, A. C. Potter, and D. Hayes, “Breaking even with magic: demonstration of a high-fidelity logical non-Clifford gate”, (2025) arXiv:2506.14688
[180]: A. Cross and D. Vandeth, “Small Binary Stabilizer Subsystem Codes”, (2025) arXiv:2501.17447
[181]: H. Goto, “Minimizing resource overheads for fault-tolerant preparation of encoded states of the Steane code”, Scientific Reports 6, (2016) DOI
[182]: N. Fazio, R. Harper, and S. D. Bartlett, “Logical Noise Bias in Magic State Injection”, Quantum 9, 1779 (2025) arXiv:2401.10982 DOI
[183]: 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
[184]: D. Bhatnagar, M. Steinberg, D. Elkouss, C. G. Almudever, and S. Feld, “Low-Depth Flag-Style Syndrome Extraction for Small Quantum Error-Correction Codes”, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 63 (2023) arXiv:2305.00784 DOI
[185]: K. Yamamoto et al., “Quantum Error-Corrected Computation of Molecular Energies”, (2025) arXiv:2505.09133
[186]: J. Haah and M. B. Hastings, “Measurement sequences for magic state distillation”, Quantum 5, 383 (2021) arXiv:2007.07929 DOI
[187]: E. J. Kuehnke, K. Levi, J. Roffe, J. Eisert, and D. Miller, “Hardware-tailored logical Clifford circuits for stabilizer codes”, (2025) arXiv:2505.20261
[188]: H. Goto, Y. Ho, and T. Kanao, “Measurement-free fault-tolerant logical-zero-state encoding of the distance-three nine-qubit surface code in a one-dimensional qubit array”, Physical Review Research 5, (2023) arXiv:2303.17211 DOI