[Jump to code hierarchy]

2D bosonization code[1,2]

Description

A mapping between a 2D lattice of qubits and a 2D lattice quadratic Hamiltonian of Majorana modes. This family also includes a super-compact fermionic encoding with a qubit-to-fermion ratio of \(1.25\) [2; Table I].

Protection

The original code [1] can be converted via Clifford operations into codes whose distance runs up to \(7\) while preserving the code rate [3].

Encoding

Tensor-network realization [4], extended to periodic boundary conditions and sectors of odd fermionic charge [5].

Cousins

  • Jordan-Wigner transformation code— The 2D bosonization code can be converted into chains of JW-transformed qubits by a linear-depth circuit [2; Fig. 24].
  • Kitaev honeycomb code— Embedding each physical qubit into two fermions via the tetron code is useful for exactly solving the Kitaev honeycomb model Hamiltonian [6] and other qubit Hamiltonians on certain graphs [7,8]. It also allows the logical subspace of the Kitaev honeycomb model to be formulated as a joint eigenspace of certain Majorana operators [9; Sec. 4.1], which admit braiding-based gates due to their non-Abelian statistics and which can be used for topological quantum computation. When done in reverse, this embedding can be thought of as a 2D bosonization fermion-into-qubit encoding by converting to a relabeled square lattice and performing single-qubit rotations [1][2; Sec. IV.B].

References

[1]
Y.-A. Chen, A. Kapustin, and Đ. Radičević, “Exact bosonization in two spatial dimensions and a new class of lattice gauge theories”, Annals of Physics 393, 234 (2018) arXiv:1711.00515 DOI
[2]
Y.-A. Chen and Y. Xu, “Equivalence between Fermion-to-Qubit Mappings in two Spatial Dimensions”, PRX Quantum 4, (2023) arXiv:2201.05153 DOI
[3]
Y.-A. Chen, A. V. Gorshkov, and Y. Xu, “Error-correcting codes for fermionic quantum simulation”, SciPost Physics 16, (2024) arXiv:2210.08411 DOI
[4]
S. K. Shukla, T. D. Ellison, and L. Fidkowski, “Tensor network approach to two-dimensional bosonization”, Physical Review B 101, (2020) arXiv:1909.10552 DOI
[5]
O. O’Brien, L. Lootens, and F. Verstraete, “Local Jordan-Wigner transformations and topological sectors on the torus”, Physical Review B 111, (2025) arXiv:2404.07727 DOI
[6]
A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
[7]
A. Chapman and S. T. Flammia, “Characterization of solvable spin models via graph invariants”, Quantum 4, 278 (2020) arXiv:2003.05465 DOI
[8]
S. J. Elman, A. Chapman, and S. T. Flammia, “Free Fermions Behind the Disguise”, Communications in Mathematical Physics 388, 969 (2021) arXiv:2012.07857 DOI
[9]
A. Roy and D. P. DiVincenzo, “Topological Quantum Computing”, (2017) arXiv:1701.05052
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: 2d_bosonization

Cite as:
“2D bosonization code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/2d_bosonization
BibTeX:
@incollection{eczoo_2d_bosonization, title={2D bosonization code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/2d_bosonization} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/2d_bosonization

Cite as:

“2D bosonization code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/2d_bosonization

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/fermion_into_qubit/bosonization/2d_bosonization.yml.