Kitaev chain code[1] 

Description

An \([[n,1,1]]_{f}\) Majorana stabilizer code forming the ground-state of the Kitaev Majorana chain (a.k.a. Kitaev Majorana wire) in its fermionic topological phase, which is equivalent to the 1D quantum Ising model in the symmetry-breaking phase via the Jordan-Wigner transformation. The code is usually defined using the algebra of two anti-commuting Majorana operators called Majorana zero modes (MZMs) or Majorana edge modes (MEMs).

Codewords have different values of the fermionic parity. As a result, this code is considered unphysical because, in the fermionic context, fermion parity conservation prevents one from realizing coherent superpositions between them. However, controlling the MZMs associated with this code is an important step to realizing its parent the tetron code, whose codewords have the same parity.

Protection

In the fermionic context, code states are insensitive to any local Majorana operator product that respects fermion parity symmetry. However, the distance is one because the code does not protect against single Majorana operators, which do not commute with the parity symmetry.

Realizations

Photonic systems: braiding of defects has been simulated in a device that has a different notion of locality than a bona-fide fermionic system [2].Superconducting circuits: initialization [3], braiding [4] and detection [4,5] of defects has been simulated in devices that have a different notion of locality than a bona-fide fermionic system.

Notes

See notes [6] for a description of this code.

Parent

  • Tetron Majorana code — Kitaev chain codewords can be obtained by restricting to only one Kitaev chain out of the two chains that define the tetron Majorana code.

References

[1]
A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires”, Physics-Uspekhi 44, 131 (2001) arXiv:cond-mat/0010440 DOI
[2]
J.-S. Xu et al., “Simulating the exchange of Majorana zero modes with a photonic system”, Nature Communications 7, (2016) arXiv:1411.7751 DOI
[3]
K. J. Sung et al., “Simulating Majorana zero modes on a noisy quantum processor”, Quantum Science and Technology 8, 025010 (2023) arXiv:2206.00563 DOI
[4]
N. Harle, O. Shtanko, and R. Movassagh, “Observing and braiding topological Majorana modes on programmable quantum simulators”, Nature Communications 14, (2023) arXiv:2203.15083 DOI
[5]
X. Mi et al., “Noise-resilient edge modes on a chain of superconducting qubits”, Science 378, 785 (2022) arXiv:2204.11372 DOI
[6]
A. Kitaev and C. Laumann, “Topological phases and quantum computation”, (2009) arXiv:0904.2771
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Zoo Code ID: kitaev_chain

Cite as:
“Kitaev chain code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/kitaev_chain
BibTeX:
@incollection{eczoo_kitaev_chain,
  title={Kitaev chain code},
  booktitle={The Error Correction Zoo},
  year={2023},
  editor={Albert, Victor V. and Faist, Philippe},
  url={https://errorcorrectionzoo.org/c/kitaev_chain}
}
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Permanent link:
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Cite as:

“Kitaev chain code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/kitaev_chain

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/fermions/kitaev_chain.yml.