Diatomic molecular code

Diatomic molecular code[1; Sec. VI]

Description

Approximate quantum code that encodes a qudit in the finite-dimensional Hilbert space of a rigid body with \(SO(2)\) symmetry (e.g., a heteronuclear diatomic molecule). This state space is the space of normalized functions on the two-sphere, consisting of a direct sum of all non-negative integer angular momenta. Ideal codewords may not be normalizable because the space is infinite-dimensional, so approximate versions have to be constructed in practice.

Construction is based on nested subgroups \(H\subset K \subset SO_3\), where \(H,K\) are finite. Codewords consist of orbits of particular position states under \(H\), while some elements of \(K\) can cycle between codewords.

Protection

Protects against sufficiently small shifts in position and small kicks in angular momentum. But unlike the original molecular codes, these codes cannot protect against all product of such shifts and kicks because the underlying state space does not form a group.

Parent

Spin code

Cousins

Molecular code — Molecular (diatomic molecular) codes are constructed using two nested subgroups of \(SO(3)\) on the state space of a particle on \(SO(3)\) (the two-sphere).
Æ code — Diatomic molecular codes are supported on states with various total angular momenta, while the more practical Æ codes are supported on only one subspace of fixed total momentum. The latter codes are thus more practical and more applicable to other spin spaces.

References

[1]: V. V. Albert, J. P. Covey, and J. Preskill, “Robust Encoding of a Qubit in a Molecule”, Physical Review X 10, (2020) arXiv:1911.00099 DOI

Page edit log

Victor V. Albert (2023-11-22) — most recent

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/spins/amo/diatomic_molecular.yml.