Single-spin code 


Permutation invariant qubit code encoding a \(K\)-dimensional Hilbert space into the maximally symmetric subspace or collective spin of \(2\ell\) spin-half systems. This \(2\ell+1\)-dimensional Hilbert space can be thought of as a standalone spin-\(\ell\) quantum system.


Noise models can be categorized as those that cause the state to leave the maximally symmetric subspace and those that do not. The former include single-spin errors akin to qubit Pauli noise. The latter include collective rotations or decays.

Transversal Gates

When the physical Hilbert space is thought of a collective spin, logical gates for spin codes have the form \(U^{\otimes N}\), where \(U\) is a local rotation on the physical system.


  • Spin code
  • Permutation-invariant code — Single-spin codes are subspaces of a single large spin, which can be either standalone or correspond to the permutation-invariant subspace of a set of spins.



  • Quantum spherical code (QSC) — Single-spin codes whose codewords are expressed in terms of discrete sets of spin-coherent states may also be interpreted as QSCs.
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Zoo Code ID: single_spin

Cite as:
“Single-spin code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
  title={Single-spin code},
  booktitle={The Error Correction Zoo},
  editor={Albert, Victor V. and Faist, Philippe},
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Cite as:

“Single-spin code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.