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). It can be thought of as the Majorana stabilizer analogue of the quantum repetition code, and it encodes a logical fermion because its logical Majorana operator has odd weight [2].
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. One can combine two such code blocks to form a Majorana box qubit, which is the fixed-parity subspace of the combined codespace. Odd numbers of code blocks also contain fixed-parity logical subspaces in their codespace.
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. Disorder may help with protection [3].Gates
Braiding gates, fermionic \(S\) gates, braid-based fermionic \(T\) gates, and fermion and hybrid qubit-fermion \(CZ\) gates [2].Decoding
Local automaton decoder based on self-dual cellular automaton [4].Syndrome extraction can be performed by interfacing with a qubit ancilla and a hybrid qubit-fermion gate [2].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 [5].Superconducting circuits: initialization [6], braiding [7] and detection [7,8] of defects has been simulated in devices that have a different notion of locality than a bona-fide fermionic system.Notes
See notes [9] for a description of this code.Cousins
- Quantum repetition code— The Kitaev chain code can be thought of as the Majorana stabilizer analogue of the quantum repetition code [2].
- 3D fermionic surface code— The 3D fermionic surface code is the result of applying the 3D bosonization mapping to a trivial fermonic theory [10]. Twist defects in the 3D fermionic surface code take the form of Kitaev chains after the mapping [10,11].
Primary Hierarchy
References
- [1]
- A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires”, Physics-Uspekhi 44, 131 (2001) arXiv:cond-mat/0010440 DOI
- [2]
- A. Schuckert, E. Crane, A. V. Gorshkov, M. Hafezi, and M. J. Gullans, “Fermion-qubit fault-tolerant quantum computing”, (2024) arXiv:2411.08955
- [3]
- S. Bravyi and R. König, “Disorder-Assisted Error Correction in Majorana Chains”, Communications in Mathematical Physics 316, 641 (2012) arXiv:1108.3845 DOI
- [4]
- N. Lang and H. P. Büchler, “Strictly local one-dimensional topological quantum error correction with symmetry-constrained cellular automata”, SciPost Physics 4, (2018) arXiv:1711.08196 DOI
- [5]
- J.-S. Xu, K. Sun, Y.-J. Han, C.-F. Li, J. K. Pachos, and G.-C. Guo, “Simulating the exchange of Majorana zero modes with a photonic system”, Nature Communications 7, (2016) arXiv:1411.7751 DOI
- [6]
- K. J. Sung, M. J. Rančić, O. T. Lanes, and N. T. Bronn, “Simulating Majorana zero modes on a noisy quantum processor”, Quantum Science and Technology 8, 025010 (2023) arXiv:2206.00563 DOI
- [7]
- 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
- [8]
- X. Mi et al., “Noise-resilient edge modes on a chain of superconducting qubits”, Science 378, 785 (2022) arXiv:2204.11372 DOI
- [9]
- A. Kitaev and C. Laumann, “Topological phases and quantum computation”, (2009) arXiv:0904.2771
- [10]
- M. Barkeshli, Y.-A. Chen, S.-J. Huang, R. Kobayashi, N. Tantivasadakarn, and G. Zhu, “Codimension-2 defects and higher symmetries in (3+1)D topological phases”, SciPost Physics 14, (2023) arXiv:2208.07367 DOI
- [11]
- 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