Æ code[1] 

Description

Code defined in a single angular-momentum subspace that is embedded in a larger direct-sum space of different angular momenta, which can arise from combinations of spin, electronic, or rotational, or nuclear angular momenta of an atom or molecule. A code is obtained by solving an over-constrained system of equations, and many solutions can be mapped into existing codes defined on other state spaces.

A simple example of an Æ code is the error-detecting code with codewords \begin{align} \begin{split} |\overline{0}\rangle&=\frac{1}{\sqrt{2}}\left(|J,-m\rangle+|J,m\rangle\right)\\ |\overline{1}\rangle&=|J,0\rangle~, \end{split} \tag*{(1)}\end{align} constructed out of states of total angular momentum \(J\) and its projection \(m\) for any \(J,m\geq 2\). This code detects a single change in \(m\) or \(J\).

Protection

Protects against noise native to atomic and molecular platforms, such as spontaneous emission, stray electromagnetic fields, and Raman scattering. Noise operators arising from these processes, when restricted to the angular momentum degrees of freedom, changes either the total angular momentum or its projection and correspond to matrices whose elements are particular combinations of Clebsch-Gordan coefficients.

Parent

Cousins

  • Diatomic molecular 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.
  • Binomial code — Many well-performing Æ codes can be mapped into shifted versions of binomial codes via the Holstein-Primakoff mapping.
  • GNU permutation-invariant code — Many well-performing Æ codes can be mapped into GNU codes via the Dicke state mapping.
  • Single-spin code — Since Æ codes are defined in a subspace of fixed total angular momentum and protect against errors linear in the momentum generators, they can also be thought of a single-spin codes.

References

[1]
S. P. Jain et al., “Æ codes”, (2023) arXiv:2311.12324
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Zoo Code ID: ae

Cite as:
“Æ code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/ae
BibTeX:
@incollection{eczoo_ae, title={Æ code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/ae} }
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Cite as:

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

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