Fermion code[1]
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]. Majorana operators may either be considered individually or paired in various ways into creation and annihilation operators to yield fermionic modes. They form a Clifford algebra and can be interpreted as Ising anyons in certain contexts.
Admissible codewords include fermionic states, a subset of which is the Gaussian fermionic states [2–6]. Gaussian fermionic states are analogues of (non-displaced) Gaussian bosonic states; they are labeled by points in a Grassmannian and are sometimes called fermionic coherent states [7]. Fermionic analogues of ordinary (bosonic) coherent states are the fermionic coherent states labeled by Grassmann numbers [8,9]. A Wigner function formalism has been developed for fermionic states [10].
Protection
Majorana analogues of common qubit noise channels have been developed [11].Encoding
A fermionic code using fermion Fock states as codewords cannot protect against occupation-number errors (i.e., dephasing) and does not admit fermionic logical operators [12,13].Gates
Clifford operations on fermionic codes, shown [2] to be equivalent to match gates [14], can be formulated using Fermionic Linear Optics, a classically simulable model of computation [2–6,15]. The structure of the Majorana Clifford group has been studied [16].Non-Clifford gates can be done using gate teleportation, in which a gate can be obtained from a particular magic state (a.k.a. resource state) [15,17–20].General gates include qubit-like \(S\), \(T\), and \(CZ\) gates acting on either logical qubit or logical fermionic encodings. Fermionic gates include braiding gates which correspond to exchanging Majorana modes. Hybrid gates include \(CZ_{qf}\) gates between a logical qubit and a logical fermion. The braiding, \(CZ_{f}\), \(CZ_{qf}\), Hadamard, \(S\), and \(T\) gates are universal [12].Logical-fermion circuits constructed out of certain transversal gates do not admit a lower \(T\) gate count than logical-qubit circuits [12].Using fermion codes with logical fermion encodings and the fermionic fast Fourier transform [21] can yield exponential improvements in circuit depth over fermion-into-qubit encodings [12].Notes
See Ref. [22] for an introduction to Majorana-based qubits.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. Using fermion codes with logical fermion encodings and the fermionic fast Fourier transform [21] can yield exponential improvements in circuit depth over fermion-into-qubit encodings [12].
- Constant-excitation (CE) code— Fermion codewords lying in a fixed fermion-number subspace have to lie in the same subspace in order to protect against changes in fermion number [12].
- Majorana subsystem stabilizer code— Majorana subsystem stabilizer codes encode logical information into a physical Fock space of fermionic modes, with some degrees of freedom designated as gauge qubits.
Member of code lists
Primary Hierarchy
Parents
The Majorana operator algebra is isomorphic to the qubit Pauli-operator algebra via various fermion-into-qubit encodings.
Fermion code
Children
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]
- J. Gazeau, Coherent States in Quantum Physics (Wiley, 2009) DOI
- [8]
- F. A. Berezin, The Method of Second Quantization (Academic Press, New York, 1966).
- [9]
- M. Combescure and D. Robert, Coherent States and Applications in Mathematical Physics (Springer Netherlands, 2012) DOI
- [10]
- S. J. Park, C. Beny, and H. H. Lee, “Twisted Fourier analysis and pseudo-probability distributions”, (2020) arXiv:2004.13860
- [11]
- C. Knapp, M. Beverland, D. I. Pikulin, and T. Karzig, “Modeling noise and error correction for Majorana-based quantum computing”, Quantum 2, 88 (2018) arXiv:1806.01275 DOI
- [12]
- A. Schuckert, E. Crane, A. V. Gorshkov, M. Hafezi, and M. J. Gullans, “Fault-tolerant fermionic quantum computing”, (2025) arXiv:2411.08955
- [13]
- R. Ott, D. González-Cuadra, T. V. Zache, P. Zoller, A. M. Kaufman, and H. Pichler, “Error-corrected fermionic quantum processors with neutral atoms”, (2024) arXiv:2412.16081
- [14]
- L. G. Valiant, “Quantum computers that can be simulated classically in polynomial time”, Proceedings of the thirty-third annual ACM symposium on Theory of computing 114 (2001) DOI
- [15]
- R. Jozsa and A. Miyake, “Matchgates and classical simulation of quantum circuits”, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 464, 3089 (2008) arXiv:0804.4050 DOI
- [16]
- V. Bettaque and B. Swingle, “The Structure of the Majorana Clifford Group”, (2025) arXiv:2407.11319
- [17]
- Quantum Information and Computation 14, (2014) arXiv:1308.1463 DOI
- [18]
- D. J. Brod, “Efficient classical simulation of matchgate circuits with generalized inputs and measurements”, Physical Review A 93, (2016) arXiv:1602.03539 DOI
- [19]
- M. Hebenstreit, R. Jozsa, B. Kraus, S. Strelchuk, and M. Yoganathan, “All Pure Fermionic Non-Gaussian States Are Magic States for Matchgate Computations”, Physical Review Letters 123, (2019) arXiv:1905.08584 DOI
- [20]
- L. Coffman, G. Smith, and X. Gao, “Measuring Non-Gaussian Magic in Fermions: Convolution, Entropy, and the Violation of Wick’s Theorem and the Matchgate Identity”, (2025) arXiv:2501.06179
- [21]
- R. Babbush, N. Wiebe, J. McClean, J. McClain, H. Neven, and G. K.-L. Chan, “Low-Depth Quantum Simulation of Materials”, Physical Review X 8, (2018) arXiv:1706.00023 DOI
- [22]
- F. Hassler, “Majorana Qubits”, (2014) arXiv:1404.0897
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Michael Gullans (2025-01-12)
- Alexander Schuckert (2025-01-12)
- Victor V. Albert (2025-01-12)
- Victor V. Albert (2022-12-04)
- Victor V. Albert (2021-12-01)
Cite as:
“Fermion code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/fermions, arXiv:2606.11484