# 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 unitarily 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

## Realizations

## Notes

## Parent

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

## Cousin

- 3D fermionic surface code — The 3D fermionic surface code is the result of applying the 3D bosonization mapping to a trivial fermonic theory [8]. Twist defects in the 3D fermionic surface code take the form of Kitaev chains after the mapping [8,9].

## References

- [1]
- A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires”, Physics-Uspekhi 44, 131 (2001) arXiv:cond-mat/0010440 DOI
- [2]
- S. Bravyi and R. König, “Disorder-Assisted Error Correction in Majorana Chains”, Communications in Mathematical Physics 316, 641 (2012) arXiv:1108.3845 DOI
- [3]
- J.-S. Xu et al., “Simulating the exchange of Majorana zero modes with a photonic system”, Nature Communications 7, (2016) arXiv:1411.7751 DOI
- [4]
- 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
- [5]
- 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
- [6]
- X. Mi et al., “Noise-resilient edge modes on a chain of superconducting qubits”, Science 378, 785 (2022) arXiv:2204.11372 DOI
- [7]
- A. Kitaev and C. Laumann, “Topological phases and quantum computation”, (2009) arXiv:0904.2771
- [8]
- M. Barkeshli et al., “Codimension-2 defects and higher symmetries in (3+1)D topological phases”, SciPost Physics 14, (2023) arXiv:2208.07367 DOI
- [9]
- P. Webster and S. D. Bartlett, “Fault-tolerant quantum gates with defects in topological stabilizer codes”, Physical Review A 102, (2020) arXiv:1906.01045 DOI

## Page edit log

- Victor V. Albert (2023-03-07) — most recent

## Cite as:

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