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 of \(n\) modes. |
| Kitaev chain code | An \([[n,1]]_{f}\) Majorana stabilizer code forming the ground-state of the Kitaev Majorana chain (a.k.a. Kitaev Majorana wire) in its fermionic topological phase, which is unitarily equivalent to the 1D quantum Ising model in the symmetry-breaking phase via the Jordan-Wigner transformation. The code is usually defined using the algebra of two anti-commuting Majorana operators called Majorana zero modes (MZMs) or Majorana edge modes (MEMs). It can be thought of as the Majorana stabilizer analogue of the quantum repetition code, and it encodes a logical fermion because its logical Majorana operator has odd weight [1]. |
| 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 [2] system [3]. |
| 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 [4]. |
| 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 [5; Def. 30] of quantum convolutional codes. |
| Tetron code | A \([[2,1,2]]_{f}\) Majorana box qubit, encoding two fixed-fermion states into the 4D ground-state space of two Kitaev chains, each of length two. The code encodes a logical qubit into four Majorana modes (i.e., two physical fermions), allowing it to be concatenated with various qubit codes such as surface codes and color codes. Embedding each physical qubit into two fermions via the tetron code is useful for exactly solving the Kitaev honeycomb model Hamiltonian [6] and other qubit Hamiltonians on certain graphs [7,8]. It has been used throughout condensed matter physics under the name of the Majorana representation [9,10] or parton construction [11], allowing for a mean-field treatment of many models that are otherwise not amenable. Majorana stabilizer groups can be converted into ordinary qubit stabilizer groups via the parton mapping, while their corresponding states are converted via the Gutzwiller projection [12]. |
| 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. |
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