## Description

Encodes \(K\)-dimensional Hilbert space into a tensor-product or direct sum of factors, with each factor spanned by states of a quantum mechanical spin or, more generally, an irreducible representation of a Lie group.

In the simplest case of a spin half (i.e., a qubit), the canonical states \(|^J_m\rangle\) of a single \(2J+1\)-dimensional factor are labeled by total angular momentum \(J\) (either integer or half-integer) and its \(z\)-axis projection \(m\). There can be multiple factors of the same size, as in the case of atomic or molecular state spaces, and the number of factors can be infinite. In contrast to other qudit codes, spin codes are closely associated with the angular momentum Lie algebra and/or \(SU(2)\), \(SO(3)\), or more general Lie groups.

## Protection

## Parent

## Children

- Qubit code — Spin codes with spin \(\ell=1/2\) correspond to qubit codes.
- Æ code
- Diatomic molecular code
- Magnon code — Magnon codewords are low-energy excited states of the frustration-free Heisenberg-XXX model Hamiltonian [1].
- Twisted \(1\)-group code
- Valence-bond-solid (VBS) code — VBS codewords are eigenstates of the frustration-free VBS Hamiltonian [2,3].
- Single-spin code

## Cousins

- Eigenstate thermalization hypothesis (ETH) code — Relevant many-body systems housing ETH codes include 1D non-interacting spin chains or frustration-free systems such as Motzkin chains and Heisenberg models.
- Movassagh-Ouyang Hamiltonian code — Justesen codes can be used to build a family of \(n\)-qudit Movassagh-Ouyang Hamiltonian spin codes encoding one logical qubit with linear distance. These codes form the ground-state subspace of a frustration-free geometrically local Hamiltonian [4].

## References

- [1]
- M. Gschwendtner et al., “Quantum error-detection at low energies”, Journal of High Energy Physics 2019, (2019) arXiv:1902.02115 DOI
- [2]
- D.-S. Wang et al., “Quasi-exact quantum computation”, Physical Review Research 2, (2020) arXiv:1910.00038 DOI
- [3]
- D.-S. Wang et al., “Theory of quasi-exact fault-tolerant quantum computing and valence-bond-solid codes”, New Journal of Physics 24, 023019 (2022) arXiv:2105.14777 DOI
- [4]
- R. Movassagh and Y. Ouyang, “Constructing quantum codes from any classical code and their embedding in ground space of local Hamiltonians”, (2020) arXiv:2012.01453

## Page edit log

- Thomas Wrona (2022-05-18) — most recent
- Victor V. Albert (2022-02-22)

## Cite as:

“Spin code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/spins_into_spins

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