Molecular code[1]
Description
Approximate quantum code that encodes a finite-dimensional logical space into the Hilbert space of \(L^2\)-normalizable functions on \(SO(3)\), i.e., rotational states of an asymmetric rigid body such as a polyatomic molecule.
Construction is based on nested finite subgroups \(H\subset K \subset SO(3)\). The \(|K|/|H|\)-dimensional logical subspace is spanned by basis states that are equal superpositions of elements of cosets of \(H\) in \(K\). Examples discussed in the original work include cyclic, dihedral, tetrahedral-octahedral, and tetrahedral-icosahedral subgroup embeddings.
Protection
Protects against generalized bit-flip errors \(g\in SO(3)\) that are inside the fundamental domain of \(SO(3)/K\). In the cyclic \(Z_N\subset Z_{dN}\) family, the code corrects sufficiently small rigid-body rotations about any axis and angular-momentum kicks with \(\delta\ell<N/2\). Protection against phase-flip and more general momentum-kick errors is determined by the branching rules of irreps of \(SO(3)\) 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.Each ideal molecular code has a parent Hamiltonian whose ground space is the codespace, and normalizable approximate codewords can be obtained by damping in total angular momentum.Cousins
- Diatomic 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.
- Fiber code— Molecular codes encode quantum information into superpositions of multiple orientations of an asymmetric molecule [1], while fiber codes encode into the fiber associated with a single orientation of certain symmetric molecules [3].
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
- [2]
- E. K. Carlson, “Protecting Molecular Qubits from Noise”, Physics 13, (2020) DOI
- [3]
- V. V. Albert, E. Kubischta, M. Lemeshko, and L. R. Liu, “Quantum theory of molecular orientations”, (2026) arXiv:2403.04572
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