Quantum Error Correction Code implementations

Quantum error correction is expected to be essential for achieving a quantum advantage over classical methods. Each quantum error correction code (QECC) is described by three parameters \([[ n, k, d ]]\), where \(k\) logical qubits are encoded into \(n\) physical qubits. The distance of the code is specified by the parameter \(d\), which determines the code’s ability to detect and correct errors. Classical error correction codes are defined by the same parameters but only with single brackets \([ n, k, d ]\). In quantum computing, repetition codes are considered classical, as they can detect either bit-flip or phase-flip errors but not arbitrary combinations of both. Each QECC is characterized by an error threshold \(p_{th}\), beyond which the quantum error correction method becomes ineffective — meaning that the error correction process introduces more errors than it mitigates. For a successful implementation, all physical quantum operations must exhibit error rates below this threshold. When the physical error rate is much smaller than the threshold, the error rate of the logical qubit \(\epsilon_{log}\) is expected to decrease with the code distance exponentially:

\[\epsilon_\mathrm{log} \propto \left( \frac{p}{p_\mathrm{th}} \right)^{(d+1)/2}\]

One popular method for quantum error correction is stabilizer codes, which constitute the majority of the quantum codes presented in the following table. In this framework, the logical state is encoded across multiple physical qubits with a specific initialization procedure. The logical qubits are then repeatedly stabilized with several syndrome measurements that permit the extraction of errors (and eventually their correction). This stabilization process occurs at regular intervals throughout the computation. The stabilization process is considered active when syndrome extraction and correction are carried out dynamically during the execution of the quantum circuit. In contrast, the approach is referred to as passive when error detection and correction are deferred after the final measurement (it is sometimes called post-selection). At scale, a stabilizer QECC must be active, which is obviously harder to implement as it requires low syndrome decoding latency with efficient dynamic corrections (sometimes, the correction can be transferred at the end of the circuit).

In general, a large set of logical operations can be efficiently implemented using transversal operations, wherein applying a logical gate to a logical qubit corresponds to the independent application of that gate to each constituent physical qubit. However, a universal logical gate set requires at least one non-transversal gate, which is difficult to realize in practice and involves the preparation of specific states called magic states. Magic state distillation is a process that enhances the fidelity of a magic state.

The following table summarizes experimental implementations of QECC conducted on real quantum hardware. Part of the presented data is adapted from the comparative study by R. Larose [1].

Date & Ref Affiliation Hardware Code Code \(n, k, d\) # Ancillas
Per log		 #logical qubits Max #cycles
Feedback loop		 Transversal
Logic gate		 Non-transversal
Logic gates		 Magic state
Preparation		 Magic state
Distillation
1998 [2] MIT, LANL,
Hardvard		 NMR Repetition \([3, 1, 3]\) 1 no no no no
2001 [3] LANL NMR Perfect [4] \([[5, 1, 3]]\) 1 no no no no
2004 [5] NIST Trapped-ions Repetition \([3, 1, 3]\) 1 no no no no
2011 [6] Innsbruck Univ
et al.		 Trapped-ions Repetition \([3, 1, 3]\) 1 3 no no no no
2011 [7] Waterloo Univ NMR Repetition \([3, 1, 3]\) 1 2 no no no no
2012 [8] Dortmund Univ
Waterloo Univ		 NMR Perfect [4] \([[5, 1, 3]]\) 1 yes yes no no
2012 [9] Yale Univ
Delft Univ		 Superconducting Repetition \([3, 1, 3]\) 1 no no no no
2014 [10] Britsol Univ
et al.		 Photonic Graph state \([[4, 1, 2]]\) 1 1 no no no no
2014 [11] Google
et al.		 Superconducting Repetition \([5, 1, 5]\) 4 1 8 no no no no
2014 [12] Innsbruck Univ
et al.		 Trapped-ions Steane [13] \([[7, 1, 3]]\) 1 yes no no no
2014 [14] Stuttgart Univ
et al.		 NV center Repetition \([3, 1, 3]\) 1 no no no no
2015 [15] QuTech Superconducting Repetition \([3, 1, 3]\) 2 1 no no no no
2015 [16] IBM Superconducting Repetition \([2, 0, 2]\) 2 1 no no no no
2015 [17] QuTech
et al.		 NV center Repetition \([3, 1, 3]\) 1 1 no no no no
2017 [18] IBM Superconducting Detection code [19] \([[4, 2, 2]]\) 1 1 no no no no
2017 [20] Maryland Univ
IonQ
et al.		 Trapped-ions Detection code [19] \([[4, 2, 2]]\) 1 1 yes no no no
2018 [21] Basel Univ Superconducting Repetition \([8, 1, 8]\) 7 1 1 yes no no no
2019 [22] Hefei Univ
et al.		 Superconducting Perfect [4] \([[5, 1, 3]]\) 0 1 no no no no
2019 [23] Sydney Univ Superconducting Detection code [19] \([[4, 2, 2]]\) 0 1 yes yes no no
2020 [24] IBM Superconducting Repetition \([22, 1, 22]\) 21 1 no no no no
2020 [25] ETH Zurich Superconducting Surface \([[4, 1, 2]]\) 3 1 10 no no no no
2020 [26] Maryland Univ
et al.		 Trapped-ions Bacon-Shor \([[9, 1, 3]]\) 4 1 1 yes no yes no
2020 [27] Hefei Univ
et al.		 Photonic Bacon-Shor \([[9, 1, 3]]\) 2 1 1 yes yes yes no
2021 [28] Google Superconducting Repetition \([11, 1, 11]\) 10 1 50 no no no no
2021 [28] Google Superconducting Surface \([[4, 1, 2]]\) 3 1 15 no no no no
2021 [29] Quantinuum Trapped-ions Steane [13] \([[7, 1, 3]]\) 3 1 6 yes yes yes no
2022 [30] Quantinuum Trapped-ions Perfect [4] \([[5, 1, 3]]\) 2 2 no yes no no
2022 [30] Quantinuum Trapped-ions Steane [13] \([[7, 1, 3]]\) 1 2 yes no no no
2022 [31] Shanghai Univ
et al.		 Superconducting Surface \([[9, 1, 3]]\) 8 1 11 no no no no
2022 [32] ETH Zurich
et al.		 Superconducting Surface \([[9, 1, 3]]\) 8 1 16 no no no no
2022 [33] QuTech
et al.		 NV center Perfect [4] \([[5, 1, 3]]\) 2 1 1 yes no no no
2023 [34] Google Superconducting Repetition \([25, 1, 25]\) 24 1 no no no no
2023 [34] Google Superconducting Surface \([20, 1, 5]]\) 24 1 25 no no no no
2023 [35] Toronto Univ
et al.		 Superconducting
ibmq_mumbaï		 CSS \([[8, 3, 2]]\) 0 3 yes no no no
2023 [35] Toronto Univ
et al.		 Trapped-ions
ionq-11q		 CSS \([[8, 3, 2]]\) 0 3 yes no yes no
2023 [36] QuTech
Quantinuum		 Trapped-ions CSS \([[8, 3, 2]]\) 2 3 yes yes no no
2024 [37] Microsoft
Quantinuum		 Trapped-ions Steane [13] \([[7, 1, 3]]\) 3 2 1 yes no no no
2024 [37] Microsoft
Quantinuum		 Trapped-ions Concatenated [37] \([[12, 2, 4]]\)
\([[4, 2, 2]]\) & \([[6, 2, 2]]\)		 3 2 3 yes no no no
2024 [38] Harvard
QuERA
Et al.		 Neutral atoms Surface \([[49, 1, 7]]\) 20 2 yes no no no
2024 [38] Harvard
QuERA
Et al.		 Neutral atoms Code block \([[8, 3, 2]]\) 0 48 yes no no no
2024 [39] Google
et al.		 Superconducting Surface \([[49, 1, 7]]\) 48 1 250 no no no no
2024 [39] Google
et al.		 Superconducting Repetition \([29, 1, 29]\) 28 1 no no no no
2024 [39] Google
et al.		 Superconducting Surface \([20, 1, 5]]\) 24 1 5000000 no no no no
2024 [40] Quantinuum Trapped-ions Concatenated \([[10, 1, 4]]\)
\([[2, 1, 1]]\) & \([[5, 1, 3]]\)		 1 10 no no no no

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