Quantum repetition code[1] 


Encodes \(1\) qubit into \(n\) qubits according to \(|0\rangle\to|\phi_0\rangle^{\otimes n}\) and \(|1\rangle\to|\phi_1\rangle^{\otimes n}\). Also known as a bit-flip code when \(|\phi_i\rangle = |i\rangle\), and a phase-flip code when \(|\phi_0\rangle = |+\rangle\) and \(|\phi_1\rangle = |-\rangle\).

The \(\pm\)-basis codewords for the bit-flip code are GHz (a.k.a. cat) states \(|0\rangle^{\otimes n}\pm|1\rangle^{\otimes n}\). These are ground states of the one-dimensional classical Ising model Hamiltonian \(H=\sum_{i} Z_{i}Z_{i+1}\).

The \(\pm\)-basis codewords for the phase-flip code are expanded in the computational basis as \begin{align} \begin{split} |\overline{+}\rangle =\frac{1}{\sqrt{2^{n-1}}}\sum_{\sum_{i}v_{i}=0}|v_{1},\cdots,v_{n}\rangle~{\phantom{,}}\\ |\overline{-}\rangle =\frac{1}{\sqrt{2^{n-1}}}\sum_{\sum_{i}v_{i}=1}|v_{1},\cdots,v_{n}\rangle~, \end{split} \tag*{(1)}\end{align} showing that the phase-flip code stores information in the total parity of the qubits.


Bit-flip code detects bit-flip errors \(X\) on \(\left\lfloor (n-1)/2\right\rfloor\) qubits and does not detect any phase-flip errors \(Z\). Phase-flip code detects phase-flip errors \(Z\) on \(\left\lfloor (n-1)/2\right\rfloor\) qubits and does not detect any bit-flip errors \(X\).

Because they protect against only one type of noise, both codes can be thought of as a classical \([n,1,d]\) repetition code with classical distance \(d=\left\lfloor (n-1)/2\right\rfloor\) embedded in a quantum system. Nevertheless, the phase-flip code can offer some degree of protection in particular physical systems based on superconducting circuits [2,3].


Unitary circuit of depth logarithmic in \(n\) [4]. Any circuit has to have range \(n\) because Ghz states are locally indistinguishable [5].Adaptive constant-depth circuit with geometrically local gates and measurements throughout [6,7].Lindbladian-based dissipative encoding passively protecting against bit flips [8,9].


Toffoli magic-state preparation protocol [10].


Automaton-like decoders for the repetition code on a 2D lattice, otherwise known as the classical 2D Ising model, were developed by Toom [11,12]. An automaton by Gacs yields a decoder for a 1D lattice [13].Machine learning algorithm to implement continuous error-correction for the three-qubit quantum repetition code [14].

Fault Tolerance

Toffoli magic-state preparation protocol [10].

Code Capacity Threshold

Independent \(X\) noise: \(50\%\) with RG decoder for quantum repetition code arranged on a 1D or 2D lattice [15].


Phenomenological noise: \(11\%\) and \(17.2\%\) with RG decoder for quantum repetition code arranged on a 1D and 2D lattice, respectively [15].


NMR: 3-qubit phase-flip code [16,17], with up to two rounds of error correction in liquid-state NMR [18].Trapped ions: 3-qubit bit-flip code by Wineland group [19], and 3-qubit phase-flip algorithm implemented in 3 cycles on high fidelity gate operations [20]. Both phase- and bit-flip codes for 31 qubits and their stabilizer measurements have been realized by Quantinuum [21]. Multiple rounds of Steane error correction [22].Superconducting circuits: 3-qubit phase-flip and bit-flip code by Schoelkopf group [23]; 3-qubit bit-flip code [24]; 3-qubit phase-flip code up to 3 cycles of error correction [25]; IBM 15-qubit device [26]; IBM Rochester device using 43-qubit code [27]; Google system performing up to 8 error-correction cycles on 5 and 9 qubits [28]; Google Quantum AI Sycamore utilizing up to 11 physical qubits and running 50 correction rounds [29]; Google Quantum AI Sycamore utilizing up to 25 qubits for comparison of logical error scaling with a quantum code [30] (see also [31]).Continuous error correction protocols have been implemented on a 3-qubit superconducting qubit device [32].Semiconductor spin-qubit devices: 3-qubit devices at RIKEN [33] and Delft [34].Nitrogen-vacancy centers in diamond: 3-qubit phase-flip code [35,36] (see also Ref. [37]).


Repetition codes can be used to benchmark device performance [38].


  • Quantum parity code (QPC) — A \([[m_1 m_2,1,\min(m_1,m_2)]]\) QPC reduces to a repetition code when \(m_1\) or \(m_2\) is one.
  • Group-based quantum repetition code — Group-based quantum repetition codes reduce to quantum repetition codes for \(G = \mathbb{Z}_2\).
  • GNU PI code — GNU codewords for \(g=1\) reduce to the phase-flip repetition code.
  • Frustration-free Hamiltonian code — The codespace of the quantum repetition code is the ground-state space of a frustration-free classical Ising model with nearest-neighbor interactions.
  • Commuting-projector Hamiltonian code — The codespace of the quantum repetition code is the ground-state space of a frustration-free classical Ising model with nearest-neighbor interactions.


  • Fracton stabilizer code — The 1D quantum repetition code is an ingredient in product constructions that yield several fracton phases [39; Fig. 8].
  • Abelian topological code — The 1D quantum repetition code is an ingredient in product constructions that yield several topological phases [39; Fig. 8].
  • Repetition code — A quantum repetition code can be thought of as a classical \([n,1,d]\) repetition code with classical distance \(d=\left\lfloor (n-1)/2\right\rfloor\) embedded in a quantum system.
  • Two-component cat code — Two-legged cat and quantum repetition codes can be thought of as classical codes because they protect against only one type of noise. Two-legged cat codes (quantum repetition) codes suppress cavity dephasing (bit-flip) noise exponentially with \(|\alpha|^2\) (\(n\)). The stability offered by cat codes has been linked to several favorable properties of phases of matter associated with the repetition-code Hamiltonian [40,41].
  • Very small logical qubit (VSLQ) code — Parts of the VSLQ codewords resemble the two-qubit phase-flip repetition code, though the code cannot correct phase errors. Unlike the phase-flip code, the VSLQ code can correct for single photon loss because it uses the second excited state in the construction, which remains distinct from the vacuum even after photon loss.
  • \(D_4\) hyper-diamond GKP code — The \(D_4\) hyper-diamond GKP code can be seen as a concatenation of a rotated square-lattice GKP code with a repetition code [42]. This is related to the fact that the four-bit repetition code yields the \(D_4\) hyper-diamond lattice code via Construction A.
  • Self-correcting quantum code — The bit-flip repetition code associated with the 2D classical Ising model is a self-correcting classical memory [43; Sec. V.A].
  • XYZ ruby Floquet code — One third of the time during its measurement schedule, the ISG of the XYZ ruby Floquet code is that of the 6.6.6 color code concatenated with a three-qubit repetition code.
  • \([[2^D,D,2]]\) hypercube code — The hypercube code can be concatenated with a two-qubit quantum repetition code to yield a \([[2^{D+1},D,4]]\) error-detecting code family [44].
  • \([[9,1,3]]\) Shor code — The Shor code is a concatenation of a three-qubit bit-flip with a three-qubit phase-flip repetition code.
  • Transverse-field Ising model (TFIM) code — When written in the computational basis, the phase-flip and TFIM codewords are superpositions of qubit states of fixed total parity. The superposition is equal for the phase-flip code, whereas some states appear with a \(-1\) coefficient for TFIM code. However, the TFIM code can be encoded in constant depth.
  • X-cube model code — Generalized X-cube models [39] are constructed from a balanced product of the quantum repetion (1D Ising) code and the Newman-Moore code.


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“Quantum repetition code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_repetition
@incollection{eczoo_quantum_repetition, title={Quantum repetition code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quantum_repetition} }
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“Quantum repetition code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_repetition

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