Here is a list of concatenated quantum codes.

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Code Description
Analog repetition code An \([[n,1]]_{\mathbb{R}}\) analog stabilizer version of the quantum repetition code, encoding the position states of one mode into an odd number \(n\) of modes.
Auxiliary qubit mapping (AQM) code A concatenation of the JW transformation code with a qubit stabilizer code.
Binarized-and-concatenated (B&C) phantom code Member of a family of \(k=2\) CSS phantom codes obtained from a \(q=4\) Galois-qudit CSS code by binarizing each \(\mathbb{F}_4\) qudit into two qubits and then concatenating each qubit pair with the \([[4,2,2]]\) code [1].
Cat code Rotation-symmetric bosonic Fock-state code encoding a \(q\)-dimensional qudit into one oscillator which utilizes a constellation of \(q(S+1)\) coherent states distributed equidistantly around a circle in phase space of radius \(\alpha\).
Cat-repetition code A concatenated qubit-into-\(n\)-mode code obtained by encoding each qubit of a quantum repetition code into a two-component cat code in its cat-state basis.
Coherent-state repetition code A concatenated qubit-into-\(n\)-mode code (for odd \(n\)) whose inner code is a quantum repetition code and whose outer code is the two-component cat code in its coherent-state basis.
Concatenated GKP code A concatenated code whose outer code is a GKP code. In other words, a bosonic code that can be thought of as a concatenation of an arbitrary inner code and another bosonic outer code. Most examples encode physical qubits of an inner stabilizer code into the square-lattice GKP code.
Concatenated Steane code A member of the family of \([[7^m,1,3^m]]\) CSS codes, each of which is a recursive level-\(m\) concatenation of the Steane code. This family is one of the first to admit a concatenated threshold [2–6].
Concatenated bosonic code A concatenated code whose outer code is a bosonic code. In other words, a bosonic code that can be thought of as a concatenation of a possibly non-bosonic inner code and a bosonic outer code.
Concatenated cat code A concatenated code obtained by encoding the physical qubits of an inner qubit code into cat-code states. Most examples concatenate a qubit stabilizer code with the two-component cat code in its cat-state basis.
Concatenated quantum code A combination of two quantum codes, an inner code \(C_{\text{in}}\) and an outer code \(C_{\text{out}}\), where the physical subspace used for the inner code consists of the logical subspace of the outer code. In other words, one first encodes in the inner code, and then encodes each of its physical registers in the outer code. An inner \(C_{\text{in}} = ((n_1,K,d_1))_{q_1}\) and outer \(C_{\text{out}} = ((n_2,q_1,d_2))_{q_2}\) block quantum code yield an \(((n_1 n_2, K, d \geq d_1d_2))_{q_2}\) concatenated block quantum code [7].
Concatenated qubit code A concatenated code whose outer code is a qubit code. In other words, a qubit code that can be thought of as a concatenation of an inner qubit code and an outer qubit code. An inner \(C_{\text{in}} = ((n_1,K,d_1))\) and outer \(C_{\text{out}} = ((n_2,2,d_2))\) qubit code yield an \(((n_1 n_2, K, d \geq d_1d_2))\) concatenated qubit code.
GKP-surface code A concatenated code whose outer code is a GKP code and whose inner code is a surface code, including toric surface-code variants [8,9], rotated surface codes [10–13], and XZZX surface codes [14].
Group-based QPC An \([[m r,1,\min(m,r)]]_G\) generalization of the QPC.
Group-based quantum repetition code An \([[n,1]]_G\) generalization of the quantum repetition code.
Hierarchical code Member of a family of \([[n,k,d]]\) qubit stabilizer codes resulting from a concatenation of a constant-rate QLDPC code with a rotated surface code. Concatenation allows for syndrome extraction to be performed on a 2D geometry while maintaining a threshold at the expense of a logarithmically vanishing rate. The growing syndrome extraction circuit depth allows known bounds in the literature to be weakened [15,16].
Quantum parity code (QPC) A \([[m_1 m_2,1,\min(m_1,m_2)]]\) CSS code family obtained from concatenating an \(m_1\)-qubit bit-flip repetition code with an \(m_2\)-qubit phase-flip repetition code.
Quantum repetition code Encodes \(1\) qubit into \(n\) qubits according to \(|0\rangle\to|\phi_0\rangle^{\otimes n}\) and \(|1\rangle\to|\phi_1\rangle^{\otimes n}\). The code is called a bit-flip code when \(|\phi_i\rangle = |i\rangle\), and a phase-flip code when \(|\phi_0\rangle = |+\rangle\) and \(|\phi_1\rangle = |-\rangle\).
Quantum turbo code A quantum version of the turbo code, obtained from an interleaved serial quantum concatenation [17; Def. 30] of quantum convolutional codes. The interleaver induces long-range entanglement and can increase the minimum distance relative to the constituent convolutional codes [18].
Tetron code A \([[2,1,2]]_{f}\) Majorana box qubit encoding a logical qubit into four Majorana modes, equivalently into the fixed-total-parity sector of two physical fermionic modes. Four Majorana zero modes are the smallest aggregate that supports a qubit in a fixed fermion-parity sector [19]. This code can be concatenated with various qubit codes such as surface codes and color codes. Four-boundary Majorana surface-code patches are logical tetrons, i.e., higher-distance analogues of this physical tetron block [20].
Two-component cat code Code whose codespace is spanned by two coherent states \(\left|\pm\alpha\right\rangle\) for nonzero complex \(\alpha\).
XYZ\(^2\) hexagonal stabilizer code An instance of the matching code based on the Kitaev honeycomb model. It is described on a honeycomb tiling with \(XYZXYZ\) stabilizers on each hexagonal plaquette. Each vertical pair of qubits has an \(XX\), \(YY\), or \(ZZ\) link stabilizer depending on the orientation of the plaquette stabilizers.
Yoked surface code Member of a family of \([[n,k,d]]\) qubit CSS codes resulting from a concatenation of a QMDPC code with a rotated surface code. Concatenation does not impose additional connectivity constraints and can triple the number of logical qubits per physical qubit when compared to the original surface code. Concatenation with 1D (2D) QMDPC yields codes with twice (four times) the distance. Using the concatenation convention of the Zoo, the stabilizer generators of the inner QMDPC code are referred to as yokes in this context; the cited paper [21] uses the opposite inner/outer terminology.
\(D_4\) hyper-diamond GKP code Two-mode GKP qubit-into-oscillator code based on the \(D_4\) hyper-diamond lattice [22].
\([[12,2,4]]\) carbon code Twelve-qubit CSS code based on Knill’s \(C_4/C_6\) scheme [23]. Using the concatenation convention of the Zoo, the carbon code can be viewed as a block concatenation with inner code \([[4,2,2]]\) and outer code \(C_6\): three inner \([[4,2,2]]\) blocks encode six intermediate qubits, which are then encoded into two logical qubits by the outer \([[6,2,2]]\) code.
\([[14,3,3]]\) CSS phantom code CSS phantom code obtained by concatenating the \([[7,3,(d_X=3,d_Z=2)]]\) punctured hypercube code with the two-qubit phase-flip repetition code.
\([[4,1,2]]\) Leung-Nielsen-Chuang-Yamamoto (LNCY) code A four-qubit CSS stabilizer code that is the only qubit CSS code with such parameters.
\([[7,1,3]]\) Steane code A \([[7,1,3]]\) self-dual CSS code that is the smallest qubit CSS code to correct a single-qubit error [24]. The code is constructed using the classical binary \([7,4,3]\) Hamming code for protecting against both \(X\) and \(Z\) errors.
\([[9,1,3]]\) Shor code Nine-qubit CSS code that is the first quantum error-correcting code [25; ID 8802]. Among indecomposable \([[9,1,3]]\) CSS codes, the Shor code has the largest automorphism group [26].
\([[9,1,3]]_{\mathbb{R}}\) Lloyd-Slotine code An analog stabilizer version of Shor’s nine-qubit code, encoding one mode into nine and correcting arbitrary errors on any one mode.
\([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code Modular-qudit CSS code that generalizes the \([[9,1,3]]\) Shor code to \(q\)-level systems.

References

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