GNU permutation-invariant code[1]
Description
Can be expressed in terms of Dicke states where the logical states are \begin{align} |\overline{\pm}\rangle = \sum_{\ell=0}^{n} \frac{(\pm 1)^\ell}{\sqrt{2^n}} \sqrt{n \choose \ell} |D^m_{g \ell}\rangle~. \end{align} Here, \(m\) is the number of particles used for encoding \(1\) qubit, and \(g, n \leq m\) are arbitrary positive integers. The state \(|D^m_w\rangle\) is a Dicke state -- a normalized permutation-invariant state on \(m\) qubits with \(w\) excitations, i.e., a normalized sum over all basis elements with \(w\) ones and \(m - w\) zeroes.
A qudit extension of such codes, based on a correspondence with binomial codes, exists [2].
Protection
Parent
Cousins
- Bacon-Shor code — Symmetrized versions of the Bacon-Shor codes are GNU codes
- Hamiltonian-based code — GNU codes lie within the ground state of ferromagnetic Heisenberg models without an external magnetic field [3].
- Approximate quantum error-correcting code (AQECC) — GNU codes protect approximately against amplitude damping errors.
- Binomial code — Binomial codes and GNU codes are both related to spin-coherent states, and a qudit generalization can be obtained from qudit binomial codes ([2], Appx. C).
- Quantum repetition code — GNU codewords for \(g=1\) reduce to the phase-flip code.
Zoo code information
References
- [1]
- Y. Ouyang, “Permutation-invariant quantum codes”, Physical Review A 90, (2014). DOI; 1302.3247
- [2]
- V. V. Albert et al., “Performance and structure of single-mode bosonic codes”, Physical Review A 97, (2018). DOI; 1708.05010
- [3]
- Y. Ouyang, “Quantum storage in quantum ferromagnets”, Physical Review B 103, (2021). DOI; 1904.01458
Cite as:
“GNU permutation-invariant code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/gnu_permutation_invariant