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
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Zoo Code ID: molecular

Cite as:
“Molecular code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/molecular
BibTeX:
@incollection{eczoo_molecular, title={Molecular code}, booktitle={The Error Correction Zoo}, year={2021}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/molecular} }
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Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/groups/molecular.yml.