Here is a list of quantum codes defined on a single subsystem.

Code | Description |
---|---|

Binary dihedral PI code | Multi-qubit code designed to realize gates from the binary dihedral group transversally. Can also be interpreted as a single-spin code. The codespace projection is a projection onto an irrep of the binary dihedral group \( \mathsf{BD}_{2N} = \langle\omega I, X, P\rangle \) of order \(8N\), where \( \omega \) is a \( 2N \)th root of unity, and \( P = \text{diag} ( 1, \omega^2) \). |

Binomial code | Bosonic rotation codes designed to approximately protect against errors consisting of powers of raising and lowering operators up to some maximum power. Binomial codes can be thought of as spin-coherent states embedded into an oscillator [1]. |

Bosonic \(q\)-ary expansion | A one-to-one mapping between basis states on \(n\) prime-dimensional qudits (of dimension \(q=p\)) and the subspace of the first \(p^n\) single-mode Fock states. While this mapping offers a way to map qudits into a single mode, noise models for the two code families induce different notions of locality and thus qualitatively different physical interpretations [2]. |

Bosonic rotation code | A single-mode Fock-state bosonic code whose codespace is preserved by a phase-space rotation by a multiple of \(2\pi/N\) for some \(N\). The rotation symmetry ensures that encoded states have support only on every \(N^{\textrm{th}}\) Fock state. For example, single-mode Fock-state codes for \(N=2\) encoding a qubit admit basis states that are, respectively, supported on Fock state sets \(\{|0\rangle,|4\rangle,|8\rangle,\cdots\}\) and \(\{|2\rangle,|6\rangle,|10\rangle,\cdots\}\). |

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 \(\alpha\). |

Chebyshev code | Single-mode bosonic Fock-state code that can be used for error-corrected sensing of a signal Hamiltonian \({\hat n}^s\), where \({\hat n}\) is the occupation number operator. Codewords for the \(s\)th-order Chebyshev code are \begin{align} \begin{split} \ket{\overline 0} &=\sum_{k \text{~even}}^{[0,s]} \tilde{c}_k \Ket{\left\lfloor M\sin^2\left( k\pi/{2s}\right) \right\rfloor},\\ \ket{\overline 1} &= \sum_{k \text{~odd}}^{[0,s]} \tilde{c}_k \Ket{\left\lfloor M\sin^2 \left(k\pi/{2s}\right) \right\rfloor}, \end{split} \tag*{(1)}\end{align} where \(\tilde{c}_k>0\) can be obtained by solving a system of order \(O(s^2)\) linear equations, and where \(\lfloor x \rfloor\) is the floor function. The code approaches optimality for sensing the signal Hamiltonian as \(M\) increases. |

Clifford spin code | A single-spin code designed to realize a discrete group of gates using \(SU(2)\) rotations. Codewords are subspaces of a spin's Hilbert space that house irreducible representations (irreps) of a discrete subgroup of \(SU(2)\). |

Hexagonal GKP code | Single-mode GKP qudit-into-oscillator code based on the hexagonal lattice. Offers the best error correction against displacement noise in a single mode due to the optimal packing of the underlying lattice. |

Landau-level spin code | Approximate quantum code that encodes a qudit in the finite-dimensional Hilbert space of a single spin, i.e., a spherical Landau level. Codewords are approximately orthogonal Landau-level spin coherent states whose orientations are spaced maximally far apart along a great circle (equator) of the sphere. The larger the spin, the better the performance. |

Modular-qudit GKP code | Modular-qudit analogue of the GKP code. Encodes a qudit into a larger qudit and protects against Pauli shifts up to some maximum value. |

Molecular code | Encodes finite-dimensional Hilbert space into the Hilbert space of \(\ell^2\)-normalizable functions on the group \(SO_3\). Construction is based on nested subgroups \(H\subset K \subset SO_3\), where \(H,K\) are finite. The \(|K|/|H|\)-dimensional logical subspace is spanned by basis states that are equal superpositions of elements of cosets of \(H\) in \(K\). |

Monolithic quantum code | A code constructed in a single quantum system, i.e., a physical space that is not treated as a tensor product of \(n\) identical subsystems. Examples include codes in a single qudit, spin, oscillator, or molecule. |

Number-phase code | Bosonic rotation code consisting of superpositions of Pegg-Barnett phase states [3], \begin{align} |\phi\rangle \equiv \frac{1}{\sqrt{2\pi}}\sum_{n=0}^{\infty} \mathrm{e}^{\mathrm{i} n \phi} \ket{n}. \tag*{(2)}\end{align} Since phase states and thus the ideal codewords are not normalizable, approximate versions need to be constructed. The codes' key feature is that, in the ideal case, phase measurement has zero uncertainty, making it a good canditate for a syndrome measurement. |

Rotor GKP code | GKP code protecting against small angular position and momentum shifts of a planar rotor. |

Single-mode bosonic code | Encodes \(K\)-dimensional Hilbert space into a single bosonic mode. A trivial single-mode code encoding a qubit into the first two Fock states \(\{|0\rangle,|1\rangle\}\) is called the single-rail encoding [4,5]. |

Single-spin code | An encoding into a monolithic (i.e. non-tensor-product) Hilbert space that houses an irreducible representation of \(SU(2)\) or, more generally, another Lie group. In some cases, this space can be thought of as the permutation invariant subspace of a particular tensor-product space. |

Spin GKP code | An analogue of the single-mode GKP code designed for atomic ensembles. Was designed by using the Holstein-Primakoff mapping [6] (see also [7]) to pull back the phase-space structure of a bosonic system to the compact phase space of a quantum spin. A different construction emerges depending on which particular expression for GKP codewords is pulled back. |

Spin cat code | An analogue of the two-component cat code designed for a large spin, which is often realized in the PI subspace of atomic ensembles. |

Square-lattice GKP code | Single-mode GKP qudit-into-oscillator code based on the rectangular lattice. Its stabilizer generators are oscillator displacement operators \(\hat{S}_q(2\alpha)=e^{-2i\alpha \hat{p}}\) and \(\hat{S}_p(2\beta)=e^{2i\beta \hat{x}}\). To ensure \(\hat{S}_q(2\alpha)\) and \(\hat{S}_p(2\beta)\) generate a stabilizer group that is Abelian, there is a constraint that \(\alpha\beta=2q\pi\) where \(q\) is an integer denoting the logical dimension. |

Squeezed fock-state code | Approximate boosnic code that encodes a qubit into the same Fock state, but one which is squeezed in opposite directions. |

Two-component cat code | Code whose codespace is spanned by two coherent states \(\left|\pm\alpha\right\rangle\) for nonzero complex \(\alpha\). |

\(((5,3,2))_3\) qutrit code | Smallest qutrit block code realizing the \(\Sigma(360\phi)\) subgroup of \(SU(3)\) transversally. The next smallest code is \(((7,3,2))_3\). |

\(((7,2,3))\) Pollatsek-Ruskai code | Seven-qubit PI code that realizes gates from the binary icosahedral group transversally. Can also be interpreted as a spin-\(7/2\) single-spin code. The codespace projection is a projection onto an irrep of the binary icosahedral group \(2I\). |

\(SU(3)\) spin code | An extension of Clifford single-spin codes to the group \(SU(3)\), whose codespace is a projection onto a particular irrep of a subgroup of \(SU(3)\) of an underlying spin that houses some particular irrep of \(SU(3)\). |

## References

- [1]
- V. V. Albert et al., “Performance and structure of single-mode bosonic codes”, Physical Review A 97, (2018) arXiv:1708.05010 DOI
- [2]
- S. M. Girvin, “Introduction to quantum error correction and fault tolerance”, SciPost Physics Lecture Notes (2023) arXiv:2111.08894 DOI
- [3]
- S. M. Barnett and D. T. Pegg, “Phase in quantum optics”, Journal of Physics A: Mathematical and General 19, 3849 (1986) DOI
- [4]
- H.-W. Lee and J. Kim, “Quantum teleportation and Bell’s inequality using single-particle entanglement”, Physical Review A 63, (2000) arXiv:quant-ph/0007106 DOI
- [5]
- A. P. Lund and T. C. Ralph, “Nondeterministic gates for photonic single-rail quantum logic”, Physical Review A 66, (2002) arXiv:quant-ph/0205044 DOI
- [6]
- T. Holstein and H. Primakoff, “Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet”, Physical Review 58, 1098 (1940) DOI
- [7]
- C. D. Cushen and R. L. Hudson, “A quantum-mechanical central limit theorem”, Journal of Applied Probability 8, 454 (1971) DOI