Here is a list of bosonic Fock-state codes.

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

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. |

Chuang-Leung-Yamamoto (CLY) code | Bosonic Fock-state code that encodes \(k\) qubits into \(n\) oscillators, with each oscillator restricted to having at most \(N\) excitations. Codewords are superpositions of oscillator Fock states which have exactly \(N\) total excitations, and are either uniform (i.e., balanced) superpositions or unbalanced superpositions. |

Dual-rail quantum code | Two-mode bosonic code encoding a logical qubit in Fock states with one excitation. The logical-zero state is represented by \(|10\rangle\), while the logical-one state is represented by \(|01\rangle\). This encoding is often realized in temporal or spatial modes, corresponding to a time-bin or frequency-bin encoding. Two different types of photon polarization can also be used. |

Fock-state bosonic code | Qudit-into-oscillator code whose protection against AD noise (i.e., photon loss) stems from the use of disjoint sets of Fock states for the construction of each code basis state. The simplest example is the dual-rail code, which has codewords consisting of single Fock states \(|10\rangle\) and \(|01\rangle\). This code can detect a single loss error since a loss operator in either mode maps one of the codewords to a different Fock state \(|00\rangle\). More involved codewords consist of several well-separated Fock states such that multiple loss events can be detected and corrected. |

Matrix-model code | Multimode-mode Fock-state bosonic approximate code derived from a matrix model, i.e., a non-Abelian bosonic gauge theory with a large gauge group. The model's degrees of freedom are matrix-valued bosons \(a\), each consisting of \(N^2\) harmonic oscillator modes and subject to an \(SU(N)\) gauge symmetry. |

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. |

One-hot quantum code | Encoding of a \(q\)-dimensional qudit into the single-excitation subspace of \(q\) modes. The \(j\)th logical state is the multi-mode Fock state with one photon in mode \(j\) and zero photons in the other modes. This code is useful for encoding and performing operations on qudits in multiple qubits [4–8]. |

Ouyang-Chao constant-excitation PI code | A constant-excitation PI Fock-state code whose construction is based on integer partitions. |

Pair-cat code | Two- or higher-mode extension of cat codes whose codewords are right eigenstates of powers of products of the modes' lowering operators. Many gadgets for cat codes have two-mode pair-cat analogues, with the advantage being that such gates can be done in parallel with a dissipative error-correction process. |

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

Two-mode binomial code | Two-mode constant-energy CLY code whose coefficients are square-roots of binomial coefficients. |

Very small logical qubit (VSLQ) code | The two logical codewords are \(|\pm\rangle \propto (|0\rangle\pm|2\rangle)(|0\rangle\pm|2\rangle)\), where the total Hilbert space is the tensor product of two transmon qudits (whose ground states \(|0\rangle\) and second excited states \(|2\rangle\) are used in the codewords). Since the code is intended to protect against losses, the qutrits can equivalently be thought of as oscillator Fock-state subspaces. |

Wasilewski-Banaszek code | Three-oscillator constant-excitation Fock-state code encoding a single logical qubit. |

\(\chi^{(2)}\) code | A \(3n\)-mode bosonic Fock-state code that requires only linear optics and the \(\chi^{(2)}\) optical nonlinear interaction for encoding, decoding, and logical gates. Codewords lie in Fock-state subspaces that are invariant under Hermitian combinations of the \(\chi^{(2)}\) nonlinearities \(abc^\dagger\) and \(i abc^\dagger\), where \(a\), \(b\), and \(c\) are lowering operators acting on one of the \(n\) triples of modes on which the codes are defined. Codewords are also \(+1\) eigenstates of stabilizer-like symmetry operators, and photon parities are error syndromes. |

## 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]
- R. D. Somma, “Quantum Computation, Complexity, and Many-Body Physics”, (2005) arXiv:quant-ph/0512209
- [5]
- M. R. Geller et al., “Universal quantum simulation with prethreshold superconducting qubits: Single-excitation subspace method”, (2015) arXiv:1505.04990
- [6]
- S. McArdle et al., “Digital quantum simulation of molecular vibrations”, Chemical Science 10, 5725 (2019) arXiv:1811.04069 DOI
- [7]
- N. P. D. Sawaya and J. Huh, “Quantum Algorithm for Calculating Molecular Vibronic Spectra”, The Journal of Physical Chemistry Letters 10, 3586 (2019) arXiv:1812.10495 DOI
- [8]
- N. P. D. Sawaya et al., “Resource-efficient digital quantum simulation of d-level systems for photonic, vibrational, and spin-s Hamiltonians”, npj Quantum Information 6, (2020) arXiv:1909.12847 DOI