Here is a list of codes encoding fermionic degrees of freedom into qubit systems.

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

2D bosonization code | A mapping between a 2D lattice of qubits and a 2D lattice quadratic Hamiltonian of Majorana modes. This family also includes a super-compact fermionic encoding with a qubit-to-fermion ratio of \(1.25\) [1; Table I]. |

3D bosonization code | A mapping that maps a 3D lattice quadratic Hamiltonian of Majorana modes into a lattice of qubits which realize a \(\mathbb{Z}_2\) gauge theory with a particular Gauss law. |

Auxiliary qubit mapping (AQM) code | A concatenation of the JW transformation code with a qubit stabilizer code. |

Ball-Verstraete-Cirac (BVC) code | A 2D fermion-into-qubit encoding that builds upon the JW transformation encoding by eliminating the weight-\(O(n)\) \(X\)-type string at the expense introducing additional qubits. See [1; Sec. IV.B] for details. |

Bosonization code | A mapping that maps a \(D\)-dimensional lattice quadratic Hamiltonian of Majorana modes into a lattice of qubits. The resulting qubit code can realize various topological phases, depending on the initial Majorana-mode Hamiltonian and its symmetries. |

Bravyi-Kitaev superfast (BKSF) code | An single error-detecting fermion-into-qubit encoding defined on 2D qubit lattice whose stabilizers are associated with loops in the lattice. The code can be generalized to a single error-correcting code (i.e., with distance three) on graphs of degree \(\geq 6\) [2]. |

Bravyi-Kitaev transformation (BKT) code | A fermion-into-qubit encoding that maps Majorana operators into Pauli strings of weight \(\lceil \log (n+1) \rceil\). The code can be reformulated in terms of Fenwick trees [3], and the Pauli-string weight can be further optimized to yield the segmented Bravyi-Kitaev (SBK) transformation code [4]. |

Derby-Klassen (DK) code | A fermion-into-qubit code defined on regular tilings with maximum degree 4 whose stabilizers are associated with loops in the tiling. The code outperforms several other encodings in terms of encoding rate [5; Table I]. It has been extended for models with several modes per site [6]. |

Fermion-into-qubit code | Qubit stabilizer code encoding a logical fermionic Hilbert space into a physical space of \(n\) qubits. Such codes are primarily intended for simulating fermionic systems on quantum computers, and some of them have error-detecting, correcting, and transmuting properties. |

Jordan-Wigner transformation code | A mapping between qubit Pauli strings and Majorana operators that can be thought of as a trivial \([[n,n,1]]\) code. The mapping is best described as converting a chain of \(n\) qubits into a chain of \(2n\) Majorana modes (i.e., \(n\) fermionic modes). It maps Majorana operators into Pauli strings of weight \(O(n)\). |

Majorana loop stabilizer code (MLSC) | An single error-correcting fermion-into-qubit encoding defined on a 2D qubit lattice whose stabilizers are associated with loops in the lattice. |

Ternary-tree fermion-into-qubit code | A fermion-into-qubit encoding defined on ternary trees that maps Majorana operators into Pauli strings of weight \(\lceil \log_3 (2n+1) \rceil\). |

## References

- [1]
- Y.-A. Chen and Y. Xu, “Equivalence between Fermion-to-Qubit Mappings in two Spatial Dimensions”, PRX Quantum 4, (2023) arXiv:2201.05153 DOI
- [2]
- K. Setia et al., “Superfast encodings for fermionic quantum simulation”, Physical Review Research 1, (2019) arXiv:1810.05274 DOI
- [3]
- P. M. Fenwick, “A new data structure for cumulative frequency tables”, Software: Practice and Experience 24, 327 (1994) DOI
- [4]
- V. Havlíček, M. Troyer, and J. D. Whitfield, “Operator locality in the quantum simulation of fermionic models”, Physical Review A 95, (2017) arXiv:1701.07072 DOI
- [5]
- C. Derby et al., “Compact fermion to qubit mappings”, Physical Review B 104, (2021) arXiv:2003.06939 DOI
- [6]
- L. Clinton et al., “Towards near-term quantum simulation of materials”, Nature Communications 15, (2024) arXiv:2205.15256 DOI