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

Name | Realization(s) |
---|---|

Analog stabilizer code | One-sided device-independent QKD [1]. |

Bacon-Shor code | Trapped-ion qubits: state preparation, logical measurement, and syndrome extraction (deferred to the end) for nine-qubit Bacon-Shor code demonstrated on a 13-qubit device by M. Cetina and C. Monroe groups [2]. |

Binomial code | Microwave cavities coupled to superconducting circuits: state transfer between a binomial codeword to another system [3], error-correction protocol nearly reaching break-even [4], and a teleported CNOT gate [5]. 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 \(16\%\) [6] (see also [7]). See Ref. [8] for another experiment.Motional degree of freedom of a trapped ion: binomial state preparation for \(S=2\) realized by Tan group [9]. |

Cat code | Parity-syndrome measurement tested [10] and implemented for the four-component (\(S=1\)) cat code [11] in a microwave cavity coupled to a superconducting circuit. The latter work [11] 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 [12]. |

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 [13]. 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 [14], which is the same topological order as the \(G=D_4\) quantum double [15,16]. |

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 [17,18] for more details.Superconducting circuit devices: Gates have been demonstrated in the Schoelkopf group at Yale University [19]. Error detection has been demonstrated in 3D cavities in the Devoret group at Yale University [20] and Amazon Web Services [21] using transmon qubits, following earlier theoretical proposals [22,23]. Logical readout in 3D cavities has been demonstrated by Quantum Circuits Inc. [24].Photonic platforms: state preparation and measurement fidelity of \(99.98\%\) in the C telecom band by PsiQuantum [25]. |

Fibonacci string-net code | NMR: Implementation of braiding-based Hamamard gate on two qubits [26].Superconducting qubits: state preparation, fusion, and braiding [27]. |

Five-qubit perfect code | NMR: Implementation of perfect error correcting code on 5 spin subsystem of labeled crotonic acid for quantum network benchmarking [28]. Single-qubit logical gates [29]. Magic-state distillation using 7-qubit device [30].Superconducting qubits [31].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 [32]. Real-time magic-state distillation [33].Nitrogen-vacancy centers in diamond: fault-tolerant single-qubit Clifford operations using two ancillas [34]. 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 [35] |

Generalized Shor code | The \([[m^2,1,m]]\) codes for \(m\leq 7\) have been realized in trapped-ion quantum devices [36]. |

Group GKP code | Cryptographic applications stemming from the monogamy of entanglement of group GKP code and error words [1]. |

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 \(d=2\) [37,38] and \(d=3\) [39]. 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 [40]. |

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 [41]. |

Honeycomb Floquet code | Plaquette stabilizer measurement realized on the IBM Falcon superconducting-qubit device [35] |

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 [42].Superconducting circuits: initialization [43], braiding [44] and detection [44,45] 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 [46] and two-dimensional Rydberg atomic arrays [47]. |

Matching code | Braiding of defects has been demonstrated for a five-qubit version of code [48]. |

NTRU-GKP code | Public-key NTRU-based quantum communication protocol [49]. |

Pair-cat code | Microwave cavities coupled to superconducting circuits by the Wang group [50]. |

Quantum Tanner code | Used to obtain explicit lower bounds in the sum-of-squares game [51].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 [52] of the No low-energy trivial states (NLTS) conjecture [53]. |

Quantum divisible code | Triply-even codes can yield secure multi-party quantum computation [54]. |

Quantum parity code (QPC) | The \([[m^2,1,m]]\) codes for \(m\leq 7\) have been realized in trapped-ion quantum devices [36]. |

Quantum repetition code | NMR: 3-qubit phase-flip code [55,56], with up to two rounds of error correction in liquid-state NMR [57].Trapped ions: 3-qubit bit-flip code by Wineland group [58], and 3-qubit phase-flip algorithm implemented in 3 cycles on high fidelity gate operations [59]. Both phase- and bit-flip codes for 31 qubits and their stabilizer measurements have been realized by Quantinuum [60]. Multiple rounds of Steane error correction [61].Superconducting circuits: 3-qubit phase-flip and bit-flip code by Schoelkopf group [62]; 3-qubit bit-flip code [63]; 3-qubit phase-flip code up to 3 cycles of error correction [64]; IBM 15-qubit device [65]; IBM Rochester device using 43-qubit code [66]; Google system performing up to 8 error-correction cycles on 5 and 9 qubits [67]; Google Quantum AI Sycamore utilizing up to 11 physical qubits and running 50 correction rounds [68]; Google Quantum AI Sycamore utilizing up to 25 qubits for comparison of logical error scaling with a quantum code [69] (see also [70]).Continuous error correction protocols have been implemented on a 3-qubit superconducting qubit device [71].Semiconductor spin-qubit devices: 3-qubit devices at RIKEN [72] and Delft [73].Nitrogen-vacancy centers in diamond: 3-qubit phase-flip code [74,75] (see also Ref. [76]). |

Qubit CSS code | Fully homomorphic encryption [77].Entanglement purification protocols related to quantum key distribution (QKD) [78].Cryptographic applications stemming from the monogamy of entanglement of CSS code and error words [79]. |

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 [80,81], 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 [82]. State preparation also realized by Tan group [9].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 [41]. Subsequent paper by Devoret group [7] (see also [83]) 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 [6]). See Ref. [8] for another experiment.GKP states and homodyne measurements have been realized in propagating telecom light by the Furusawa group [84].Single-qubit \(Z\)-gate has been demonstrated [85] in the single-photon subspace of an infinite-mode space [86], 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. |

Surface-17 code | Implemented at ETH Zurich by the Wallraff group [87] and on the Zuchongzhi 2.1 superconducting quantum processor [88]. Both experimental error rates are above the pseudo-threshold for this code relative to a single qubit; see Physics viewpoint for a summary [89]. Magic state have been created on the latter processor [90]. |

Toric code | One cycle of syndrome readout on 19-qubit planar and 24-qubit toric codes realized in two-dimensional Rydberg atomic arrays [91]. |

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 [92]. |

Two-component cat code | Lindbladian-based [93,94] and Hamiltonian-based 'Kerr-cat' [95] encodings have been achieved in superconducting circuit devices by the Devoret group; Ref. [94] 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 [96]. A bit-flip time of up to 10s has been achieved for the two-legged cat code in the classical-bit regime [97–99]. A holonomic gate has been repurposed as a logical measurement [100]. 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 [101].T4C code realized in a superconducting circuit device by the Wang group [102]. |

Very small logical qubit (VSLQ) code | Star-code autonomous correction scheme realized using superconducting circuits [103]. |

XZZX surface code | Superconducting circuits: Distance-five 25-qubit code implemented on a superconducting quantum processor by Google Quantum AI [69]. This code outperformed the average of several instances of the smaller distance-three 9-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 [104]. Leakage errors have been handled in a separate work in a distance-three code [70].Rydberg atom arrays: Lukin group [105]. 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 [106]. |

\([[10,1,2]]\) 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 [107]. |

\([[12,2,4]]\) 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) [60] by Microsoft and Quantinuum [108]. |

\([[2m,2m-2,2]]\) error-detecting code | Trapped-ion devices: the \(m=5\) code has been realized on a 12-qubit device by Quantinuum [109]. |

\([[4,2,2]]\) CSS code | \([[4,1,2]]\) subcode implemented using four-qubit graph state of photons [110].Trapped-ion device by IonQ [111].Logical state preparation and flag-qubit error correction realized in superconducting-circuit devices by IBM [37,38,112,113].The subcode \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [114] and \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) [68], 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 [115].Logical gates for the \(\{|\overline{00}\rangle,|\overline{11}\rangle\}\) subcode, treated as a planar code, realized in superconducting circuits [116].The CZ magic state has been realized on an IBM heavy-hex superconducting circuit device [117]. |

\([[6,4,2]]\) error-detecting code | Trapped-ion devices: Bayesian quantum phase estimation on a device by Quantinuum [118]. |

\([[7,1,3]]\) Steane code | Trapped-ion devices: seven-qubit device in Blatt group [119]. Ten-qubit QCCD device by Quantinuum [120] 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 [121]). Fault-tolerant universal two-qubit gate set using T injection by Monz group [122]. 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 [32]; 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 [61]. Fault-tolerant universal gate set via code switching between the Steane code and the \([[10,1,2]]\) code [107]. 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 [108]. The quantum Fourier transform on three code blocks [123]. Fault-tolerant transversal and lattice-surgery state teleportation protocols as well as Knill error correction [124].Rydberg atom arrays: Lukin group [91]; transversal CNOT gate performed on distance \(3\), \(5\), and \(7\) codes, logical ten-qubit GHZ state initialized with break-even fidelity, fault-tolerant logical two-qubit GHZ state initialized [125]. |

\([[8,3,2]]\) CSS code | Trapped ions: one-qubit addition algorithm implemented fault-tolerantly on the Quantinuum H1-1 device [126].Superconducting circuits: fault-tolerant \(CZZ\) gate performed on IBM and IonQ devices [127].Rydberg atom arrays: Lukin group [105]. 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 [125]. |

\([[9,1,3]]\) 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 [36]. Variants of the code to handle coherent noise studied and realized by K. Brown and C. Monroe groups [128].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% [129]. All-photonic quantum repeater architecture tested on the same code [130]. |

\([[9,1,3]]_{\mathbb{R}}\) Lloyd-Slotine code | Optical network by the Furusawa group [131]. |

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