Diatomic molecular code[1; Sec. VI]
Description
Approximate quantum code that encodes a qudit in the infinite-dimensional Hilbert space of a rigid body with \(SO(2)\) symmetry, e.g., a heteronuclear diatomic molecule. The physical space is \(L^2(S^2)\), equivalently the orientation space \(SO(3)/SO(2)\) of a linear rotor, 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 rotations about any axis and small kicks in angular momentum. In the simplest cyclic family, angular-momentum kicks with \(\ell<N/2\) are correctable. But unlike molecular codes on \(SO(3)\), these codes cannot in general protect against arbitrary products of such rotations and kicks because the underlying state space \(S^2\) is not a group and rotations on \(S^2\) have fixed points.Cousins
- Molecular code— Molecular codes live on \(SO(3)\) for asymmetric rigid bodies, whereas diatomic molecular codes live on the homogeneous space \(S^2=SO(3)/SO(2)\) for linear rotors.
- Æ code— Diatomic molecular codes are supported on states with various total angular momenta, while Æ codes are supported on only one subspace of fixed total momentum. The latter codes are more practical and applicable to other spin spaces.
Member of code lists
Primary Hierarchy
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