Fermion-into-qubit code 

Description

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.

The first fermion-into-qubit code is the Jordan-Wigner transformation code, a trivial \([[n,n,1]]\) stabilizer code encoding Majorana operators into Pauli strings of weight \(O(n)\). This is necessary to ensure that Majorana operators satisfy the proper anti-commutation relations.

Subsequent encodings consisted of stabilizer codes with \(k < n\), ensuring anti-commutation through the long-range entanglement of the codestates. This makes it possible to reduce the Pauli weight of a Majorana operator by multiplying by a stabilizer. Stabilizer constraints are often associated with loops of a 2D lattice.

See [1; Table I][2; Table I] for comparisons of various fermion-into-qubit codes. In addition to the children of this entry, various custom encodings exist [3–7] that can be tailored to the quantum simulation problem of interest.

Encoding

Any two fermionic encodings in two spatial dimensions can be related to each other by a finite-depth generalized local unitary transformation [2].

Parent

  • Qubit stabilizer code — Fermion-into-qubit codes are qubit stabilizer codes that encode a logical fermionic Hilbert space into a physical space of \(n\) qubits.

Children

Cousins

  • Twist-defect surface code — Treating a twist-defect surface codespace as a logical fermion encoding yields a fermion-into-qubit code [8].
  • Gauss' law code — Gauge-group elements of a \(D\)-dimensional fermionic \(\mathbb{Z}_2\) gauge theory form a single-error-correcting linear binary code [9; Thm. 1].
  • Fermion code — Fermion (fermion-into-qubit) codes encode logical information into a physical space of fermionic modes (qubits). The Majorana operator algebra is isomorphic to the qubit Pauli-operator algebra via various fermion-into-qubit encodings. Various conditions on when a fermion code is exactly solvable via a fermion-into-qubit mapping have been formulated [10,11].
  • Kitaev honeycomb code — While the Kitaev honeycomb model is bosonic, a fermion-into-qubit mapping is useful for solving and understanding the model. Logical subspace of the Kitaev honeycomb code can be formulated as a joint eigenspace of certain Majorana operators [12; Sec. 4.1]. Logical Paulis are also be constructed out of Majorana operators.

References

[1]
C. Derby, J. Klassen, J. Bausch, and T. Cubitt, “Compact fermion to qubit mappings”, Physical Review B 104, (2021) arXiv:2003.06939 DOI
[2]
Y.-A. Chen and Y. Xu, “Equivalence between Fermion-to-Qubit Mappings in two Spatial Dimensions”, PRX Quantum 4, (2023) arXiv:2201.05153 DOI
[3]
R. W. Chien and J. D. Whitfield, “Custom fermionic codes for quantum simulation”, (2020) arXiv:2009.11860
[4]
A. Miller, Z. Zimborás, S. Knecht, S. Maniscalco, and G. García-Pérez, “Bonsai Algorithm: Grow Your Own Fermion-to-Qubit Mappings”, PRX Quantum 4, (2023) arXiv:2212.09731 DOI
[5]
T. Parella-Dilmé, K. Kottmann, L. Zambrano, L. Mortimer, J. S. Kottmann, and A. Acín, “Reducing Entanglement with Physically Inspired Fermion-To-Qubit Mappings”, PRX Quantum 5, (2024) arXiv:2311.07409 DOI
[6]
M. G. Algaba, P. V. Sriluckshmy, M. Leib, and F. Šimkovic IV, “Low-depth simulations of fermionic systems on square-grid quantum hardware”, Quantum 8, 1327 (2024) arXiv:2302.01862 DOI
[7]
Y. Liu, S. Che, J. Zhou, Y. Shi, and G. Li, “Fermihedral: On the Optimal Compilation for Fermion-to-Qubit Encoding”, arXiv (2024) DOI
[8]
A. J. Landahl and B. C. A. Morrison, “Logical fermions for fault-tolerant quantum simulation”, (2023) arXiv:2110.10280
[9]
L. Spagnoli, A. Roggero, and N. Wiebe, “Fault-tolerant simulation of Lattice Gauge Theories with gauge covariant codes”, (2024) arXiv:2405.19293
[10]
A. Chapman and S. T. Flammia, “Characterization of solvable spin models via graph invariants”, Quantum 4, 278 (2020) arXiv:2003.05465 DOI
[11]
S. J. Elman, A. Chapman, and S. T. Flammia, “Free Fermions Behind the Disguise”, Communications in Mathematical Physics 388, 969 (2021) arXiv:2012.07857 DOI
[12]
A. Roy and D. P. DiVincenzo, “Topological Quantum Computing”, (2017) arXiv:1701.05052
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Zoo Code ID: fermions_into_qubits

Cite as:
“Fermion-into-qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/fermions_into_qubits
BibTeX:
@incollection{eczoo_fermions_into_qubits, title={Fermion-into-qubit code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/fermions_into_qubits} }
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“Fermion-into-qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/fermions_into_qubits

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/fermion_into_qubit/fermions_into_qubits.yml.