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

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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] (see also Ref. [5]).
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 [6; Table I]. It has been extended for models with several modes per site [7].
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, S. Bravyi, A. Mezzacapo, and J. D. Whitfield, “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]
A. Y. Vlasov, “Clifford Algebras, Spin Groups and Qubit Trees”, Quanta 11, 97 (2022) arXiv:1904.09912 DOI
[6]
C. Derby, J. Klassen, J. Bausch, and T. Cubitt, “Compact fermion to qubit mappings”, Physical Review B 104, (2021) arXiv:2003.06939 DOI
[7]
L. Clinton, T. Cubitt, B. Flynn, F. M. Gambetta, J. Klassen, A. Montanaro, S. Piddock, R. A. Santos, and E. Sheridan, “Towards near-term quantum simulation of materials”, Nature Communications 15, (2024) arXiv:2205.15256 DOI
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