Fermionic code 

Description

Finite-dimensional quantum error-correcting code encoding a logical 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].

Gates

Clifford operations on fermionic codes can often be formulated using Fermionic Linear Optics, a classically simulable model of computation [25].

Notes

See Ref. [6] for an introduction into Majorana-based qubits.

Parent

  • Qubit code — The Majorana operator algebra is isomorphic to the qubit Pauli-operator algebra via the Jordan-Wigner transformation [7]. However, Majorana codes and the noise they are designed for are based on a different notion of locality.

Children

Cousin

  • Bosonic code — Bosonic (fermionic) codes are associated with bosonic (fermionic) degrees of freedom.

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 and A. Kitaev, “Universal quantum computation with ideal Clifford gates and noisy ancillas”, Physical Review A 71, (2005) arXiv:quant-ph/0403025 DOI
[5]
S. Bravyi, “Lagrangian representation for fermionic linear optics”, (2004) arXiv:quant-ph/0404180
[6]
F. Hassler, “Majorana Qubits”, (2014) arXiv:1404.0897
[7]
A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires”, Physics-Uspekhi 44, 131 (2001) arXiv:cond-mat/0010440 DOI
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Zoo Code ID: fermions

Cite as:
“Fermionic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/fermions
BibTeX:
@incollection{eczoo_fermions, title={Fermionic 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:

“Fermionic 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/fermions/fermions.yml.