Fermion code 

Description

Finite-dimensional quantum error-correcting code encoding a logical (qudit or fermionic) Hilbert space into a physical Fock space of fermionic modes. Codes are typically described using Majorana operators, which are linear combinations of fermionic creation and annihilation operators [1].

Admissible codewords are called fermionic states, a subset of which is the Gaussian fermionic states [2–6].

Gates

Clifford operations on fermionic codes can often be formulated using Fermionic Linear Optics, a classically simulable model of computation [2–6]. The structure of the Majorana Clifford group has been studied [7].

Notes

See Ref. [8] for an introduction into Majorana-based qubits.

Parent

  • Qubit code — The Majorana operator algebra is isomorphic to the qubit Pauli-operator algebra via various fermion-into-qubit encodings.

Children

Cousins

  • Bosonic code — Bosonic (fermionic) codes are associated with bosonic (fermionic) degrees of freedom.
  • Fermion-into-qubit 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 [9,10].

References

[1]
S. B. Bravyi and A. Yu. Kitaev, “Fermionic Quantum Computation”, Annals of Physics 298, 210 (2002) arXiv:quant-ph/0003137 DOI
[2]
E. Knill, “Fermionic Linear Optics and Matchgates”, (2001) arXiv:quant-ph/0108033
[3]
B. M. Terhal and D. P. DiVincenzo, “Classical simulation of noninteracting-fermion quantum circuits”, Physical Review A 65, (2002) arXiv:quant-ph/0108010 DOI
[4]
S. Bravyi, “Lagrangian representation for fermionic linear optics”, (2004) arXiv:quant-ph/0404180
[5]
L. Hackl and E. Bianchi, “Bosonic and fermionic Gaussian states from Kähler structures”, SciPost Physics Core 4, (2021) arXiv:2010.15518 DOI
[6]
T. Guaita, L. Hackl, and T. Quella, “Representation theory of Gaussian unitary transformations for bosonic and fermionic systems”, (2024) arXiv:2409.11628
[7]
V. Bettaque and B. Swingle, “The Structure of the Majorana Clifford Group”, (2024) arXiv:2407.11319
[8]
F. Hassler, “Majorana Qubits”, (2014) arXiv:1404.0897
[9]
A. Chapman and S. T. Flammia, “Characterization of solvable spin models via graph invariants”, Quantum 4, 278 (2020) arXiv:2003.05465 DOI
[10]
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
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Zoo Code ID: fermions

Cite as:
“Fermion code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/fermions
BibTeX:
@incollection{eczoo_fermions, title={Fermion code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/fermions} }
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Cite as:

“Fermion code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/fermions

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/majorana/fermions.yml.