Here is a list of 1D stabilizer codes.
| Code | Description |
|---|---|
| 1D lattice stabilizer code | Lattice stabilizer code in one Euclidean dimension. |
| 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. |
| Kitaev chain code | An \([[n,1]]_{f}\) Majorana stabilizer code obtained from the ground-state subspace of the Kitaev Majorana chain in its fermionic topological phase [1]. Its codespace is stabilized by nearest-neighbor Majorana bilinears, while two unpaired edge Majoranas furnish one logical fermionic mode. Under parity-preserving noise, it behaves as the Majorana analogue of the repetition code [2]. |
| Kitaev current-mirror qubit code | Member of the family of \([[2n,(0,2),(2,n)]]_{\mathbb{Z}}\) homological rotor codes storing a logical qubit on a thin Möbius strip. The ideal code can be obtained from a Josephson-junction [3] system [4]. |
| Majorana box qubit | A Majorana stabilizer code which forms a fixed-parity subspace of the ground-state subspace of one or more Kitaev Majorana chain Hamiltonians. The \([[n,1,2]]_{f}\) Majorana box qubit forms the even-fermion-parity ground-state subspace of two parallel Kitaev Majorana chains in their fermionic topological phase. Its \([[2,1,2]]_{f}\) version is called the tetron Majorana code. An \([[3,2,2]]_{f}\) extension using three Kitaev chains and housing two logical qubits of the same parity is called the hexon Majorana code. Similarly, octon, decon, and dodecon are codes defined by the positive-parity subspace of \(4\), \(5\), and \(6\) fermionic modes, respectively [5]. |
| Quantum convolutional code | 1D translationally invariant qubit stabilizer code whose stabilizer group can be partitioned into constant-size subsets of constant support and of constant overlap between neighboring sets. Initially formulated as a quantum analogue of convolutional codes, which were designed to protect a continuous and never-ending stream of information. Precise formulations sometimes begin with a finite-dimensional lattice, with the intent to take the thermodynamic limit; logical dimension can be infinite as well. |
| Quantum irregular convolutional code (QIRCC) | Quantum convolutional code whose stabilizer group consists of different constant-size subsets. |
| 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 [6; Def. 30] of quantum convolutional codes. |
| 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 [7]. This code can be concatenated with various qubit codes such as surface codes and color codes. |
| Transverse-field Ising model (TFIM) code | A 1D translationally invariant stabilizer code whose encoding is a constant-depth circuit of nearest-neighbor gates on alternating even and odd bonds that consist of transverse-field Ising Hamiltonian interactions. The code allows for perfect state transfer of arbitrary distance using local operations and classical communications (LOCC). |
| \((5,1,2)\)-convolutional code | Family of quantum convolutional codes that are 1D lattice generalizations of the five-qubit perfect code, with the former’s lattice-translation symmetry being the extension of the latter’s cyclic permutation symmetry. |
| \([[5,1,3]]\) Five-qubit perfect code | Five-qubit cyclic stabilizer code that is the smallest qubit stabilizer code to correct a single-qubit error. |
References
- [1]
- A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires”, Physics-Uspekhi 44, 131 (2001) arXiv:cond-mat/0010440 DOI
- [2]
- A. Schuckert, E. Crane, A. V. Gorshkov, M. Hafezi, and M. J. Gullans, “Fault-tolerant fermionic quantum computing”, (2025) arXiv:2411.08955
- [3]
- S. M. Girvin, “Circuit QED: superconducting qubits coupled to microwave photons”, Quantum Machines: Measurement and Control of Engineered Quantum Systems 113 (2014) DOI
- [4]
- C. Vuillot, A. Ciani, and B. M. Terhal, “Homological Quantum Rotor Codes: Logical Qubits from Torsion”, Communications in Mathematical Physics 405, (2024) arXiv:2303.13723 DOI
- [5]
- D. Litinski and F. von Oppen, “Quantum computing with Majorana fermion codes”, Physical Review B 97, (2018) arXiv:1801.08143 DOI
- [6]
- D. Poulin, J.-P. Tillich, and H. Ollivier, “Quantum serial turbo-codes”, (2009) arXiv:0712.2888
- [7]
- T. Karzig et al., “Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes”, Physical Review B 95, (2017) arXiv:1610.05289 DOI