Single-spin code 


An encoding into a monolithic (i.e. non-tensor-product) Hilbert space that houses an irreducible representation of \(SU(2)\) or, more generally, another Lie group. In some cases, this space can be thought of as the permutation invariant subspace of a particular tensor-product space.


Noise models can be categorized as those that cause the state to leave the maximally symmetric subspace and those that do not.

Noise models that do not preserve the PI subspace correspond to models for the tensor-product case. Single spin-half errors do not preserve permutation symmetry and correspond to qubit Pauli noise.

Noise models that preserve the PI subspace are typically relevant to the case of a monolithic spin. They include collective rotations or decays. A continuous-time single-spin noise channel akin to the depolarizing channel is the Landau-Streater channel [1]. A particular error basis of interest consists of the spherical tensors [2].

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.




  • PI qubit code — Single-spin codes are subspaces of a single large spin, which can be either standalone or correspond to the PI subspace of a set of spins via the Dicke state mapping.
  • 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.
  • Æ 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.


L. J. Landau and R. F. Streater, “On Birkhoff’s theorem for doubly stochastic completely positive maps of matrix algebras”, Linear Algebra and its Applications 193, 107 (1993) DOI
S. Omanakuttan and J. A. Gross, “Multispin Clifford codes for angular momentum errors in spin systems”, Physical Review A 108, (2023) arXiv:2304.08611 DOI
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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.
@incollection{eczoo_single_spin, title={Single-spin code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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Cite as:

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