# Molecular code[1]

## Description

Encodes finite-dimensional Hilbert space into the Hilbert space of \(\ell^2\)-normalizable functions on the group \(SO_3\). Construction is based on nested subgroups \(H\subset K \subset SO_3\), where \(H,K\) are finite. The \(|K|/|H|\)-dimensional logical subspace is spanned by basis states that are equal superpositions of elements of cosets of \(H\) in \(K\).

## Protection

Protects against generalized bit-flip errors \(g\in SO_3\) that are inside the fundamental domain of \(G/K\). Protection against phase-flip errors determined by branching rules of irreps of \(G\) into those of \(K\), and further into those of \(H\).

## Notes

Physical space characterizes orientations of a rigid body in 3D, which correspond to rotational states of an asymmetric molecule. See APS Physics Synopsis [2] and Physical Review Journal club discussing molecular applications.

## Parents

## Cousin

- Diatomic 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).

## 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
- [2]
- E. K. Carlson, “Protecting Molecular Qubits from Noise”, Physics 13, (2020) DOI

## Page edit log

- Victor V. Albert (2021-10-29) — most recent

## Cite as:

“Molecular code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/molecular