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~. \tag*{(1)}\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 \(m\) spin-half systems with \(w\) excitations, i.e., a normalized sum over all basis elements with \(w\) ones and \(m - w\) zeroes.
Protection
Depends on the family. One family which is completely symmetrized versions of Bacon-Shor codes (parameterized by \(t\)) protects against arbitrary weight-\(t\) spin errors. Additionally, codes with large enough length \((t+1)(3t+1)+t\) can approximately correct \(t\) spontaneous decay errors.
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 [2].
- Approximate quantum error-correcting code (AQECC) — GNU codes protect approximately against amplitude damping errors.
- Binomial code — Binomial codes and GNU codes related via the Holstein-Primakoff mapping [3] (see also [4]). A qudit generalization of GNU codes can be obtained from qudit binomial codes [5; Appx. C].
- Quantum repetition code — GNU codewords for \(g=1\) reduce to the phase-flip code.
- Error-corrected sensing code — GNU codes can be used to sense signals within the permutation-invariant subspace [6].
References
- [1]
- Y. Ouyang, “Permutation-invariant quantum codes”, Physical Review A 90, (2014) arXiv:1302.3247 DOI
- [2]
- Y. Ouyang, “Quantum storage in quantum ferromagnets”, Physical Review B 103, (2021) arXiv:1904.01458 DOI
- [3]
- T. Holstein and H. Primakoff, “Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet”, Physical Review 58, 1098 (1940) DOI
- [4]
- C. D. Cushen and R. L. Hudson, “A quantum-mechanical central limit theorem”, Journal of Applied Probability 8, 454 (1971) DOI
- [5]
- V. V. Albert et al., “Performance and structure of single-mode bosonic codes”, Physical Review A 97, (2018) arXiv:1708.05010 DOI
- [6]
- Y. Ouyang and G. K. Brennen, “Quantum error correction on symmetric quantum sensors”, (2022) arXiv:2212.06285
Page edit log
- Victor V. Albert (2022-04-26) — most recent
- Victor V. Albert (2021-12-16)
- Benjamin Quiring (2021-12-16)
Cite as:
“GNU permutation-invariant code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/gnu_permutation_invariant