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]. |
Analog-cluster-state code | Analog cluster states on a number of modes ranging from tens to millions [2–4] have been synthesized in photonic degrees of freedom.Required primitives for Gaussian gates have been realized [5]. |
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 [6]. |
Binomial code | Microwave cavities coupled to superconducting circuits: state transfer between a binomial codeword to another system [7], error-correction protocol nearly reaching break-even [8], and a teleported CNOT gate [9]. 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 \(16\%\) [10] (see also [11]). See Ref. [12] for another experiment.Motional degree of freedom of a trapped ion: binomial state preparation for \(S=2\) realized by Tan group [13]. |
Cat code | Parity-syndrome measurement tested [14] and implemented for the four-component (\(S=1\)) cat code [15] in a microwave cavity coupled to a superconducting circuit. The latter work [15] 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 [16]. |
Cluster-state code | Quantum compututation with cluster states has been realized in the polarizations of photons [17,18]. |
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 [19]. 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 [20], which is the same topological order as the \(G=D_4\) quantum double [21,22]. |
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 [23–25] for more details.Superconducting circuit devices: Gates have been demonstrated in the Schoelkopf group at Yale University [26]. Error detection has been demonstrated in 3D cavities in the Devoret group at Yale University [27] and Amazon Web Services [28] using transmon qubits, following earlier theoretical proposals [29,30]. Logical readout in 3D cavities has been demonstrated by Quantum Circuits Inc. [31].Photonic platforms: state preparation and measurement fidelity of \(99.98\%\) in the C telecom band by PsiQuantum [32]. |
Fibonacci string-net code | NMR: Implementation of braiding-based Hamamard gate on two qubits [33].Superconducting qubits: state preparation, fusion, and braiding [34,35]. 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 [36]. Single-qubit logical gates [37]. Magic-state distillation using 7-qubit device [38].Superconducting qubits [39].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 [40]. Real-time magic-state distillation [41].Nitrogen-vacancy centers in diamond: fault-tolerant single-qubit Clifford operations using two ancillas [42]. 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 [43] |
Generalized Shor code | The \([[m^2,1,m]]\) codes for \(m\leq 7\) have been realized in trapped-ion quantum devices [44]. |
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 \(d=2\) [45,46] and \(d=3\) [47]. 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 [48]. |
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 [49]. |
Honeycomb Floquet code | Plaquette stabilizer measurement realized on the IBM Falcon superconducting-qubit device [43] |
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 [50].Superconducting circuits: initialization [51], braiding [52] and detection [52,53] 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 [54] and two-dimensional Rydberg atomic arrays [55]. |
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 [56]. |
Matching code | Braiding of defects has been demonstrated for a five-qubit version of code [57]. |
Modular-qudit cluster-state code | Quantum compututation with cluster states has been realized using photons in the time and frequency domains [58]. |
NTRU-GKP code | Public-key NTRU-based quantum communication protocol [59]. |
Pair-cat code | Microwave cavities coupled to superconducting circuits by the Wang group [60]. |
Quantum Tanner code | Used to obtain explicit lower bounds in the sum-of-squares game [61].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 [62] of the No low-energy trivial states (NLTS) conjecture [63]. |
Quantum divisible code | Triply-even codes can yield secure multi-party quantum computation [64]. |
Quantum parity code (QPC) | The \([[m^2,1,m]]\) codes for \(m\leq 7\) have been realized in trapped-ion quantum devices [44].QPCs have been discussed independently in the context of superconducting circuits [65; Eq. (1)][66; Eqs. (8-10)], and aspects of such designs have been realized in experiments [67]. |
Quantum repetition code | NMR: 3-qubit phase-flip code [68,69], with up to two rounds of error correction in liquid-state NMR [70].Trapped ions: 3-qubit bit-flip code by Wineland group [71], and 3-qubit phase-flip algorithm implemented in 3 cycles on high fidelity gate operations [72]. Both phase- and bit-flip codes for 31 qubits and their stabilizer measurements have been realized by Quantinuum [73]. Multiple rounds of Steane error correction [74].Superconducting circuits: 3-qubit phase-flip and bit-flip code by Schoelkopf group [75]; 3-qubit bit-flip code [76]; 3-qubit phase-flip code up to 3 cycles of error correction [77]; IBM 15-qubit device [78]; IBM Rochester device using 43-qubit code [79]; Google system performing up to 8 error-correction cycles on 5 and 9 qubits [80]; Google Quantum AI Sycamore utilizing up to 11 physical qubits and running 50 correction rounds [81]; Google Quantum AI Sycamore utilizing up to 25 qubits for comparison of logical error scaling with a quantum code [82] (see also [83]).Continuous error correction protocols have been implemented on a 3-qubit superconducting qubit device [84].Semiconductor spin-qubit devices: 3-qubit devices at RIKEN [85] and Delft [86].Nitrogen-vacancy centers in diamond: 3-qubit phase-flip code [87,88] (see also Ref. [89]). |
Qubit CSS code | Fully homomorphic encryption [90].Entanglement purification protocols related to quantum key distribution (QKD) [91].Cryptographic applications stemming from the monogamy of entanglement of CSS code and error words [92]. |
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 [93,94], 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 [95]. State preparation also realized by Tan group [13].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 [49]. Subsequent paper by Devoret group [11] (see also [96]) 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 [10]). See Ref. [12] for another experiment.GKP states and homodyne measurements have been realized in propagating telecom light by the Furusawa group [97].Single-qubit \(Z\)-gate has been demonstrated [98] in the single-photon subspace of an infinite-mode space [99], 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 [100] and on the Zuchongzhi 2.1 superconducting quantum processor [101]. Both experimental error rates are above the pseudo-threshold for this code relative to a single qubit; see Physics viewpoint for a summary [102]. Magic state have been created on the latter processor [103]. |
Toric code | One cycle of syndrome readout on 19-qubit planar and 24-qubit toric codes realized in two-dimensional Rydberg atomic arrays [104]. |
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 [105]. Logical Clifford gates for \([[8,2,2]]\) and \([[10,2,3]]\) twist-defect surface codes realized in a trapped ion device by Quantinuum [106]. |
Two-component cat code | Lindbladian-based [107,108] and Hamiltonian-based 'Kerr-cat' [109] encodings have been achieved in superconducting circuit devices by the Devoret group; Ref. [108] 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 [110]. A bit-flip time of up to 10s has been achieved for the two-legged cat code in the classical-bit regime [111–113]. A holonomic gate has been repurposed as a logical measurement [114]. 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 [115].T4C code realized in a superconducting circuit device by the Wang group [116]. |
Very small logical qubit (VSLQ) code | Star-code autonomous correction scheme realized using superconducting circuits [117]. |
XZZX surface code | Superconducting circuits: Distance-five 25-qubit code implemented on a superconducting quantum processor by Google Quantum AI [82]. 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 [118]. Leakage errors have been handled in a separate work in a distance-three code [83].Rydberg atom arrays: Lukin group [119]. 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 [120]. |
\([[10,1,2]]\) 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 [121]. |
\([[12,2,4]]\) 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) [73] by Microsoft and Quantinuum [122]. |
\([[2m,2m-2,2]]\) error-detecting code | Trapped-ion devices: the \(m=5\) code has been realized on a 12-qubit device by Quantinuum [123]. |
\([[4,2,2]]\) Four-qubit code | \([[4,1,2]]\) subcode implemented using four-qubit graph state of photons [124].Trapped-ion device by IonQ [125].Logical state preparation and flag-qubit error correction realized in superconducting-circuit devices by IBM [45,46,126,127].The subcode \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [128] and \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) [81], 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 [129].Logical gates for the \(\{|\overline{00}\rangle,|\overline{11}\rangle\}\) subcode, treated as a planar code, realized in superconducting circuits [130].The CZ magic state has been realized on an IBM heavy-hex superconducting circuit device [131].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 [106].CPC gadgets for the \([[4,2,2]]\) code have been implemented on the IBM 5Q superconducting device [132]. |
\([[6,4,2]]\) error-detecting code | Trapped-ion devices: Bayesian quantum phase estimation on a device by Quantinuum [133]. |
\([[7,1,3]]\) Steane code | Trapped-ion devices: seven-qubit device in Blatt group [134]. Ten-qubit QCCD device by Quantinuum [135] 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 [136]). Fault-tolerant universal two-qubit gate set using T injection by Monz group [137]. 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 [40]; 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 [74]. Fault-tolerant universal gate set via code switching between the Steane code and the \([[10,1,2]]\) code [121]. 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 [122]. The quantum Fourier transform on three code blocks [138]. Fault-tolerant transversal and lattice-surgery state teleportation protocols as well as Knill error correction [139].Rydberg atom arrays: Lukin group [104]; 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 [140]. |
\([[8,3,2]]\) CSS code | Trapped ions: one-qubit addition algorithm implemented fault-tolerantly on the Quantinuum H1-1 device [141].Superconducting circuits: fault-tolerant \(CZZ\) gate performed on IBM and IonQ devices [142].Rydberg atom arrays: Lukin group [119]. 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 [140]. |
\([[9,1,3]]\) 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 [44]. Variants of the code to handle coherent noise studied and realized by K. Brown and C. Monroe groups [143].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% [144]. All-photonic quantum repeater architecture tested on the same code [145]. |
\([[9,1,3]]_{\mathbb{R}}\) Lloyd-Slotine code | Optical network by the Furusawa group [146]. |
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