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].
Protection
Majorana analogues of common qubit noise channels have been developed [7].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 [8,9].Gates
Clifford operations on fermionic codes, shown [2] to be equivalent to match gates [10], can be formulated using Fermionic Linear Optics, a classically simulable model of computation [2–6,11]. The structure of the Majorana Clifford group has been studied [12].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) [11,13–16].General gates include 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 [8].Logical-fermion circuits constructed out of certain transversal gates do not admit a lower \(T\) gate count than logical-qubit circuits [8].Using fermion codes with logical fermion encodings and the fermionic fast Fourier transform [17] can yield exponential improvements in circuit depth over fermion-into-qubit encodings [8].Notes
See Ref. [18] for an introduction into 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 [17] can yield exponential improvements in circuit depth over fermion-into-qubit encodings [8].
- 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 [8].
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]
- 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
- [8]
- A. Schuckert, E. Crane, A. V. Gorshkov, M. Hafezi, and M. J. Gullans, “Fermion-qubit fault-tolerant quantum computing”, (2024) arXiv:2411.08955
- [9]
- 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
- [10]
- 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
- [11]
- 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
- [12]
- V. Bettaque and B. Swingle, “The Structure of the Majorana Clifford Group”, (2024) arXiv:2407.11319
- [13]
- Quantum Information and Computation 14, (2014) arXiv:1308.1463 DOI
- [14]
- D. J. Brod, “Efficient classical simulation of matchgate circuits with generalized inputs and measurements”, Physical Review A 93, (2016) arXiv:1602.03539 DOI
- [15]
- 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
- [16]
- 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
- [17]
- 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
- [18]
- F. Hassler, “Majorana Qubits”, (2014) arXiv:1404.0897
Page edit log
- Michael Gullans (2025-01-12) — most recent
- 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.), 2025. https://errorcorrectionzoo.org/c/fermions