Here is a list of quantum codes that have been realized in devices and/or that have practical real-world relevance.

[Jump to code graph excerpt]

Name Realization(s)
Analog repetition code Quantum teleportation [1], secret sharing [2], and superdense coding [3] protocols have been realized with analog GHZ states states for \(n=2,3\).
Analog stabilizer code One-sided device-independent QKD [4].
Analog-cluster-state code Analog cluster states on a number of modes ranging from tens to millions [5–7] have been synthesized in photonic degrees of freedom.Required primitives for Gaussian gates have been realized [8].
BPSK c-q code Linear-optical quantum receiver [9].Homodyne receiver [10].Kennedy receiver [10,11].Photon-number resolving detector [12].Communication over dephasing [13], time-varying phase-noise [14], and thermal-noise [15] channels.Adaptive decoder using displacements and photon detection [16].BPQM detector on a BPSK-modulated tree code [17].
Binomial code Microwave cavities coupled to superconducting circuits: state transfer between a binomial codeword to another system [18], error-correction protocol nearly reaching break-even [19], a teleported CNOT gate [20], and fault-tolerant logical operations utilizing three-level ancillas [21]. A realization of the "0-2-4" encoding is the first to go beyond break-even error-correction and yields a logical lifetime that exceeds the cavity lifetime by \(16\%\) [22] (see also [23]). See Ref. [24] for another experiment. The 0-2-4 binomial code has been used to store entangled states [25].Motional degree of freedom of a trapped ion: binomial state preparation for \(S=2\) realized by Tan group [26].
Cat code Parity-syndrome measurement tested [27] and implemented for the four-component (\(S=1\)) cat code [28] in a microwave cavity coupled to a superconducting circuit. The latter work [28] is the first to reach break-even error-correction, where the lifetime of a logical qubit is on par with the cavity lifetime, despite protection against dephasing not being implemented. A fault-tolerant version of parity measurement has also been realized [29].
Cat-repetition code Superconducting circuit devices: a repetition code out of two-component cat qubits has been realized for distances 3 and 5 [30].
Cluster-state code Quantum compututation with cluster states has been realized in the polarizations of photons [31,32].
Coherent FSK (CFSK) c-q code Time-resolving quantum receiver [33].Bondurant receiver [34].Bayesian inference [35].
Coherent-state c-q code Continuous-variable quantum key distribution (CV-QKD) [36–38].
Dihedral \(G=D_m\) quantum-double code Signatures of a phase equivalent to the \(G=D_4\) quantum double detected in a 27-qubit trapped-ion device by Quantinuum [39]. Preparation of ground states and braiding of anyons has also been performed. The phase was realized as a gauged \(G=\mathbb{Z}_3^2\) twisted quantum double [40], which is the same topological order as the \(G=D_4\) quantum double [41,42].
Dual-rail quantum code The dual-rail code is ubiquitous in linear-optical quantum devices and is behind the KLM protocol, one of the first proposals for fault-tolerant computation. See reviews [43–45] for more details.Superconducting circuit devices: Gates have been demonstrated in the Schoelkopf group at Yale University [46]. Error detection has been demonstrated in 3D cavities in the Devoret group at Yale University [47] and Amazon Web Services [48] using transmon qubits, following earlier theoretical proposals [49,50]. Logical readout in 3D cavities has been demonstrated by Quantum Circuits Inc. [51].Photonic platforms: state preparation and measurement fidelity of \(99.98\%\) in the C telecom band by PsiQuantum [52].
Error-corrected sensing code A single physical qubit entangled with an NV spin was used to measure an incoming signal in a way that bit-flip errors on the qubit were correctable [53].
Fibonacci string-net code NMR: Implementation of braiding-based Hamamard gate on two qubits [54].Superconducting qubits: state preparation, fusion, and braiding [55,56]. The latter work utilized DSNP and sampled the string-net wavefunction to estimate the underlying chromatic polynomial.
Five-qubit perfect code NMR: Implementation of perfect error correcting code on 5 spin subsystem of labeled crotonic acid for quantum network benchmarking [57]. Single-qubit logical gates [58]. Magic-state distillation using 7-qubit device [59].Superconducting qubits [60].Trapped-ion qubits: non-transversal CNOT gate between two logical qubits, including rounds of correction and fault-tolerant primitives such as flag qubits and pieceable fault tolerance, on a 12-qubit device by Quantinuum [61]. Real-time magic-state distillation [62].Nitrogen-vacancy centers in diamond: fault-tolerant single-qubit Clifford operations using two ancillas [63]. The fault-tolerant circuit yields better fidelity than the non-fault-tolerant circuit.
Floquet color code Plaquette stabilizer measurement realized on the IBM Falcon superconducting-qubit device [64]
Generalized Shor code The \([[m^2,1,m]]\) codes for \(m\leq 7\) have been realized in trapped-ion quantum devices [65].
Group GKP code Cryptographic applications stemming from the monogamy of entanglement of group GKP code and error words [4].
Heavy-hexagon code Superconducting qubits: Logical state preparation and flag-qubit error correction realized in superconducting-circuit devices (specifically, fixed-frequency transmon qubit architectures) by IBM for \(d=2\) [66,67] and \(d=3\) [68]. Simultaneous syndrome extraction and logical Bell-state preparation for both the embedded surface and Bacon-Shor codes of distance \(\leq 4\) on an IBM 133-qubit device [69]. Embedded rotated surface code magic-state injection implemented on IBM fez device [70].
Hexagonal GKP code Microwave cavity coupled to superconducting circuits: reduced form of GKP error correction, where displacement error syndromes are measured to one bit of precision using an ancillary transmon [71].
Honeycomb (6.6.6) color code Superconducting qubits: transversal Clifford gates, randomized logical benchmarking, and magic-state injection demonstrated on distance-three and five triangular color codes on the Willow device by Google Quantum AI [72]. Logical state teleportation using lattice surgery performed between two distance-three color codes.
Honeycomb Floquet code Plaquette stabilizer measurement realized on the IBM Falcon superconducting-qubit device [64]
Kitaev chain code Photonic systems: braiding of defects has been simulated in a device that has a different notion of locality than a bona-fide fermionic system [73].Superconducting circuits: initialization [74], braiding [75] and detection [75,76] of defects has been simulated in devices that have a different notion of locality than a bona-fide fermionic system.
Kitaev surface code Signatures of corresponding topological phase of matter detected in superconducting circuits [77] and two-dimensional Rydberg atomic arrays [78]. Measurement schedules associated with the 3CX surface code realized in superconducting qubits on the Willow device by Google Quantum AI [79].
Long-range enhanced surface code (LRESC) Preparation of GHZ state of four logical qubits with beyond break-even fidelity in a \([[25,4,3]]\) LRESC [80].
Loop toric code Trapped ions: single-shot QEC realized using the \([[33,1,4]]\) loop toric code on the Quantinuum H2 device [81].
Matching code Braiding of defects has been demonstrated for a five-qubit version of code [82].
Modular-qudit cluster-state code Quantum compututation with cluster states has been realized using photons in the time and frequency domains [83].
Modular-qudit surface code State preparation, anyon creation, anyon fusion, and transfer of entanglement between anyons and defect in a 24 qubit trapped ion device by Quantinuum [84].
Monitored random-circuit code Measurement induced quantum phases have been realized in a trapped-ion processor [85].
NTRU-GKP code Public-key NTRU-based quantum communication protocol [86].
Niset-Andersen-Cerf code Realized in Ref. [87] in an optical system with 3 beam-splitters. The fidelity peaked around \(0.6\) for deterministic approach, and around \(0.77\) for the probabilistic approach (with a 25% chance of error).
Number-phase code Motional degree of freedom of a trapped ion: state initialization [88].
On-off keyed (OOK) c-q code Proof-of-principle experiments using Dolinar [89] and TES receivers [90].
PPM c-q code Conditional pulse nulling (CPN) receiver [91].
PSK c-q code Unambiguous state discrimination using displacement-based receiver for 4-PSK [92].Multi-stage quantum receivers [93–96].Bayesian inference [97].Time resolving quantum receiver opertaing in the telecom C band [98].Displacements and photon detection [99].Adaptive decoder using linear-optical elements and photon detection [16].
Pair-cat code Microwave cavities coupled to superconducting circuits by the Wang group [100].
Quantum Tanner code Used to obtain explicit lower bounds in the sum-of-squares game [101].States that, on average, achieve small violations of check operators for quantum Tanner codes require a circuit of non-constant depth to make. They are used in the proof [102] of the NLTS conjecture [103].
Quantum divisible code Triply even codes can yield secure multi-party quantum computation [104].
Quantum parity code (QPC) The \([[m^2,1,m]]\) codes for \(m\leq 7\) have been realized in trapped-ion quantum devices [65].QPCs have been discussed independently in the context of superconducting circuits [105; Eq. (1)][106; Eqs. (8-10)], and aspects of such designs have been realized in experiments [107].
Quantum repetition code NMR: 2-qubit phase-flip code [108]; 3-qubit phase-flip code [109,110], with up to two rounds of error correction in liquid-state NMR [111]. Such codes were used to characterize noise [112].Linear optics: 2-qubit phase-flip code [113].Trapped ions: 3-qubit bit-flip code by Wineland group [114], and 3-qubit phase-flip algorithm implemented in 3 cycles on high fidelity gate operations [115]. Both phase- and bit-flip codes for 31 qubits and their stabilizer measurements have been realized by Quantinuum [116]. Multiple rounds of Steane error correction [117].Superconducting circuits: 3-qubit phase-flip and bit-flip code by Schoelkopf group [118,119]; 3-qubit bit-flip code [120]; 3-qubit phase-flip code up to 3 cycles of error correction [121]; IBM 15-qubit device [122]; IBM Rochester device using 43-qubit code [123]; Google system performing up to 8 error-correction cycles on 5 and 9 qubits [124]; Google Quantum AI Sycamore utilizing up to 11 physical qubits and running 50 correction rounds [125]; Google Quantum AI Sycamore utilizing up to 25 qubits for comparison of logical error scaling with a quantum code [126] (see also [127]). Google Quantum AI follow-up experiment on codes up to (classical) distance 29, demonstrating exponential suppression to an error floor of \(10^{-10}\) [128]. Ising-model Nishimori phase transition realized for GHZ states on 54 qubits on a 127 qubit IBM device [129]. GHZ state on 75 qubits made on an IBM device [130]. Implementation of planar decoder for codes with distances between 3 and 11 on 72-qubit superconducting device [131]. Lattice surgery on the surface-17 code has been realized by splitting the code into two repetition codes by the Wallraff group [132]. Continuous-time QEC protocols have been implemented on a 3-qubit superconducting qubit device [133].Semiconductor spin-qubit devices: 3-qubit devices at RIKEN [134] and Delft [135].Nitrogen-vacancy centers in diamond: 3-qubit phase-flip code [136,137] (see also Ref. [138]).Repetition codes are used in quantum annealing protocols [139–141].
Qubit CSS code Fully homomorphic encryption [142].Cryptographic applications stemming from the monogamy of entanglement of CSS code and error words [143].
Qubit c-q code Quantum enhancement was demonstrated using a polarization-based non-error-correcting c-q encodings [144].
Smolin-Smith-Wehner (SSW) code The \(((5,5,2))\) SSW code has been realized in an NMR device [145].
Square-lattice GKP code Motional degree of freedom of a trapped ion: square-lattice GKP encoding realized with the help of post-selection by Home group [146,147], followed by realization of reduced form of GKP error correction, where displacement error syndromes are measured to one bit of precision using an ion electronic state [148]. State preparation also realized by Tan group [26]. Universal gate set, including a two-qubit entangling gate, realized by Tan group [149]. State initialization and application to measuring displacements [88].Microwave cavity coupled to superconducting circuits: reduced form of square-lattice GKP error correction, where displacement error syndromes are measured to one bit of precision using an ancillary transmon [71]. Subsequent paper by Devoret group [23] uses reinforcement learning for error-correction cycle design and is the first to go beyond break-even error-correction, with the lifetime of a logical qubit exceeding the cavity lifetime by about a factor of two (see also [22]). See Ref. [24] for another experiment. A feed-forward-free, i.e., fully autonomous protocol has also been implemented by Nord Quantique [150]. Qudit encodings with \(q=3,4\) have been realized, with logical error rates also beyond break even [151].GKP states and homodyne measurements have been realized in propagating telecom light by the Furusawa group [152].Single-qubit \(Z\)-gate has been demonstrated [153] in the single-photon subspace of an infinite-mode space [154], in which time and frequency become bosonic conjugate variables of a single effective bosonic mode. In this context, GKP position-state wavefunctions are called Dirac combs or frequency combs.
Square-octagon (4.8.8) color code Rydberg atomic devices: logical magic-state distillation using distance-three and five 4.8.8 color codes, observing an improvement in logical fidelity on a device by Quera [155].
Squeezed cat code Squeezed cat code has been realized in a superconducting circuit device by the Gao group [156].
Subsystem QECC A two-qubit unitarily correctable subsystem code recovery has been realized in an optical system [157].
Toric code One cycle of syndrome readout on 19-qubit planar and 24-qubit toric codes realized in two-dimensional Rydberg atomic arrays [158].
Twist-defect surface code Ground state of the toric code has been implemented with and without twists, and the non-Abelian braiding behavior of the twists, which realize Ising anyons, has been demonstrated [159]. Logical Clifford gates for \([[8,2,2]]\) and \([[10,2,3]]\) twist-defect surface codes realized in a trapped ion device by Quantinuum [160].
Two-component cat code Lindbladian-based [161,162] and Hamiltonian-based 'Kerr-cat' [163] encodings have been achieved in superconducting circuit devices by the Devoret group; Ref. [162] also demonstrated a displacement-based gate. The Lindbladian-based scheme has further achieved a suppression of bit-flip errors that is exponential in the average photon number up to a bit-flip time of 1ms [164]. A bit-flip time of up to 10s has been achieved for the two-legged cat code in the classical-bit regime [165–167]. A holonomic gate has been repurposed as a logical measurement [168]. The 'Kerr-cat' encoding and a \(\pi/2\) gate have been realized with the help of a band-block filter, yielding a bit-flip lifetime of 1 ms in the 10-photon regime [169] (see also Ref. [170]). Lindblad-based encoding achieved in a 2D cavity by AWS [171].T4C code realized in a superconducting circuit device by the Wang group [172].Squeezed cat code has been realized in superconducting circuit device by the Gao group [156].
Very small logical qubit (VSLQ) code Star-code autonomous correction scheme realized using superconducting circuits [173].
XZZX surface code Superconducting circuits: Distance-five 25-qubit code implemented on a superconducting quantum processor by Google Quantum AI [126]. This code outperformed the average of several instances of the smaller distance-three nine-qubit \(XZZX\) variant of the surface-17 code realized on the same device, both in terms of logical error probability over 25 cycles and in terms of logical error per cycle. This increase in error-correcting capabilities while using more physical qubits supports the notion of an error threshold. Braiding of defects has been demonstrated for the distance-five code [174]. Leakage errors have been handled in a separate work in a distance-three code [127]. Google Quantum AI follow-up experiment realizing distance-5 and distance-7 codes with 100 rounds of correction using the Libra and transformer-based decoders. The logical error rate is suppressed by a factor of \(\approx 2\), demonstrating beyond-break-even error correction with a block quantum code [128]. Rydberg atom arrays: Lukin group [175]. Transversal CNOT gates performed on distance \(3\), \(5\), and \(7\) codes.
Zero-pi qubit code A related superconducting circuit has been realized by the Houck group [176].
\([[10,1,2]]\) CSS code Trapped-ion devices: fault-tolerant universal gate set via code switching between the Steane code and the \([[10,1,2]]\) code on a device from the Monz group [177].
\([[12,2,4]]\) carbon code Trapped-ion devices: Three rounds of error correction and post-selected fault-tolerant logical Bell-state preparation with logical error rates at least 5 times lower than physical rate on a quantum charge-coupled device (QCCD) [116] by Microsoft and Quantinuum [178].
\([[16,6,4]]\) Tesseract color code Trapped-ion devices: logical graph and GHZ states of up to 12 logical qubits constructed using three copies of the \([[16,4,2,4]]\) tesseract subsystem code, along with five rounds of post-selected fault-tolerant error correction in a device by Quantinuum [179].
\([[2m,2m-2,2]]\) error-detecting code Trapped-ion devices: the \(m=5\) code has been realized on a 12-qubit device by Quantinuum [180].
\([[4,2,2]]\) Four-qubit code See also [181; Tab. I] for more details on each experimental realization.\([[4,1,2]]\) subcodes implemented in linear optical networks [182,183].Trapped-ion device by IonQ [184].Logical state preparation and flag-qubit error correction realized in superconducting-circuit devices by IBM [66,67,185,186].The subcode \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [187] and \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) [125], treated as a planar surface code, has been realized in superconducting-circuit devices.Logical gates between two copies of the subcode \(\{|\overline{10}\rangle,|\overline{11}\rangle\}\), interpreted as lattice surgery between planar surface codes, realized in superconducting circuits [188].Logical gates for the \(\{|\overline{00}\rangle,|\overline{11}\rangle\}\) subcode, treated as a planar code, realized in superconducting circuits [189].The CZ magic state has been realized on an IBM heavy-hex superconducting circuit device [190].Logical Clifford gates for a twist-defect surface code that is single-qubit Clifford equivalent to a \([[4,1,2]]\) realized in a trapped ion device by Quantinuum [160].CPC gadgets for the \([[4,2,2]]\) code have been implemented on the IBM 5Q superconducting device [191].An FPGA implementation of the collision clustering decoder [192] realized on a Rigetti superconducting device [193].Rydberg atomic devices: error detection, erasure correction, and post-selected fault-tolerant circuits demonstrated on 24 logical qubits on a 256-qubit device by Atom Computing, with each qubit encoded in the \([[4,2,2]]\) code [194]. The device also ran the Bernstein-Vazirani algorithm on up to 28 logical qubits encoded in a \([[4,1,2]]\) subcode [194]. Post-selected fault-tolerant realization of a benchmarking protocol [195], preparation of the ground state of the single-impurity Anderson impurity model, and post-selected fault-tolerant logical Bell-state preparation demonstrated on one copy of the \([[4,2,2]]\) code on a device by Infleqtion [181].
\([[6,4,2]]\) error-detecting code Trapped-ion devices: Bayesian quantum phase estimation on a device by Quantinuum [196].
\([[7,1,3]]\) Steane code Trapped-ion devices: seven-qubit device in Blatt group [197]. Ten-qubit QCCD device by Quantinuum [198] realizing repeated syndrome extraction, real-time look-up-table decoding (yielding lower logical SPAM error rate than physical SPAM), and non-fault-tolerant magic-state distillation (see APS Physics Synopsis [199]). Fault-tolerant universal two-qubit gate set using T injection by Monz group [200]. Logical CNOT gate and Bell-pair creation between two logical qubits (yielding a logical fidelity higher than physical), including rounds of correction and fault-tolerant primitives such as flag qubits and pieceable fault tolerance, on a 20-qubit device by Quantinuum [61]; logical fidelity interval of the combined preparation-CNOT-measurement procedure was higher than that of the unencoded physical qubits. Multiple rounds of Steane error correction [117]. Fault-tolerant universal gate set via code switching between the Steane code and the \([[10,1,2]]\) code [177]. Post-selected fault-tolerant logical Bell-state preparation with logical error rates at least 10 times lower than physical rate on a device by Quantinuum [178]. The quantum Fourier transform on three code blocks [201]. Fault-tolerant transversal and lattice-surgery state teleportation protocols as well as Knill error correction [202]. Rains shadow enumerators have been measured [203].Rydberg atom arrays: Lukin group [158]; ten logical qubits, transversal CNOT gate performed, logical ten-qubit GHZ state initialized with break-even fidelity, and fault-tolerant logical two-qubit GHZ state initialized [175].
\([[8,3,2]]\) CSS code Trapped ions: one-qubit addition algorithm implemented fault-tolerantly on the Quantinuum H1-1 device [204].Superconducting circuits: fault-tolerant \(CZZ\) gate performed on IBM and IonQ devices [205].Rydberg atom arrays: Lukin group [175]. 48 logical qubits, 228 logical two-qubit gates, 48 logical CCZ gates, and error detection peformed in 16 blocks. Circuit outcomes were sampled and cross-entropy (XEB) was calculated to verify quantumness. Logical entanglement entropy was measured [175].
\([[9,1,3,3]]\) Nine-qubit Bacon-Shor code Trapped-ion qubits: state preparation, logical measurement, and syndrome extraction (deferred to the end) demonstrated on a 13-qubit device by M. Cetina and C. Monroe groups [206].Rydberg atomic devices: repeated error correction demonstrated on a device by Atom Computing [194].
\([[9,1,3]]\) Shor code Trapped-ion qubits: state preparation with 98.8(1)% and 98.5(1)% fidelity for state \(|\overline{0}\rangle\) and \(|\overline{1}\rangle\), respectively, by N. Linke group [65]. Variants of the code to handle coherent noise studied and realized by K. Brown and C. Monroe groups [207].Optical systems: quantum teleportation of information implemented by J.-W. Pan group on maximally entangled pair of one physical and one logical qubit with fidelity rate of up to 78.6% [208]. All-photonic quantum repeater architecture tested on the same code [209].
\([[9,1,3]]\) Surface-17 code Implemented at ETH Zurich by the Wallraff group [210] and on the Zuchongzhi 2.1 superconducting quantum processor [211]. Both experimental error rates are above the pseudo-threshold for this code relative to a single qubit; see Physics viewpoint for a summary [212]. Magic state have been created on the latter processor [213]. Lattice surgery on the surface-17 code has been realized by splitting the code into two repetition codes by the Wallraff group [132].
\([[9,1,3]]_{\mathbb{R}}\) Lloyd-Slotine code Optical network by the Furusawa group [214].

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