GNU permutation-invariant code

GNU permutation-invariant code[1]

Description

Can be expressed in terms of Dicke states whose coefficients are square-roots of the binomial distribution. 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. Each Dicke state in the code can be shifted by adding a shift \(s\) to both \(m\) and \(w\).

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.

Decoding

For a family of shifted gnu codes, decoding can be done using projection, probability amplitude rebalancing, and gate teleportation in time \(O(n^2)\) [2].

Parents

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 related via the Holstein-Primakoff mapping [4] (see also [5]). A qudit generalization of GNU codes can be obtained from qudit binomial codes [6; 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 [7].
Æ code — Many well-performing Æ codes can be mapped into GNU codes via the Dicke state mapping.

References

[1]: Y. Ouyang, “Permutation-invariant quantum codes”, Physical Review A 90, (2014) arXiv:1302.3247 DOI
[2]: Y. Ouyang, “Permutation-invariant quantum coding for quantum deletion channels”, 2021 IEEE International Symposium on Information Theory (ISIT) (2021) arXiv:2102.02494 DOI
[3]: Y. Ouyang, “Quantum storage in quantum ferromagnets”, Physical Review B 103, (2021) arXiv:1904.01458 DOI
[4]: T. Holstein and H. Primakoff, “Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet”, Physical Review 58, 1098 (1940) DOI
[5]: C. D. Cushen and R. L. Hudson, “A quantum-mechanical central limit theorem”, Journal of Applied Probability 8, 454 (1971) DOI
[6]: V. V. Albert et al., “Performance and structure of single-mode bosonic codes”, Physical Review A 97, (2018) arXiv:1708.05010 DOI
[7]: Y. Ouyang and G. K. Brennen, “Quantum error correction on symmetric quantum sensors”, (2023) 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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/spins/transversal/gnu_permutation_invariant.yml.