Description
PI code whose codewords can be expressed as superpositions of Dicke states with coefficients are square-roots of the binomial distribution.
In terms of Dicke states, logical codewords for codes encoding a single qubit [1] are \begin{align} |\overline{\pm}\rangle = \sum_{\ell=0}^{m} \frac{(\pm 1)^\ell}{\sqrt{2^m}} \sqrt{m \choose \ell} |D^n_{g \ell}\rangle~. \tag*{(1)}\end{align} Here, \(n\) is the number of particles used for encoding \(1\) qubit, and \(g, m \leq n\) are arbitrary positive integers. Codes with higher logical dimension are developed in Ref. [2]. Each Dicke state in the code can be shifted by adding a shift \(s\) to both \(n\) and \(g\).
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 \(O(n^2)\) [3].
Notes
The degree of entanglement in (non-concatenated) GNU codes scales at most logarithmically in their distance [4; Appx. D].
Parent
- Qudit GNU PI code — Qudit GNU codes encoding logical qubits reduce to GNU codes.
Children
- Four-qubit single-deletion code — The four-qubit single-deletion code is a GNU code for \(g=m=2\) [3].
- \(((9,2,3))\) Ruskai code — The \(((9,2,3))\) Ruskai code is a GNU PI code [1].
- Quantum repetition code — GNU codewords for \(g=1\) reduce to the phase-flip repetition code.
Cousins
- Combinatorial PI code — Combinatorial PI codes \(Q_{g,(m-1)/2,g-1,+}\) are GNU codes for odd \(m\) [5; Prop. 5.4].
- Bacon-Shor code — GNU codes of length \((2t+1)^2\) result from projecting Bacon-Shor codes into the PI qubit subspace [1].
- Frustration-free Hamiltonian code — GNU codes lie within the ground state of ferromagnetic Heisenberg models without an external magnetic field [6].
- Binomial code — Binomial codes and GNU codes related via the Holstein-Primakoff mapping [7] (see also [8]). A qudit generalization of GNU codes can be obtained from qudit binomial codes [9; Appx. C].
- Error-corrected sensing code — GNU codes can be used to sense signals within the PI subspace [10].
- 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 and J. Fitzsimons, “Permutation-invariant codes encoding more than one qubit”, Physical Review A 93, (2016) arXiv:1512.02469 DOI
- [3]
- Y. Ouyang, “Permutation-invariant quantum coding for quantum deletion channels”, 2021 IEEE International Symposium on Information Theory (ISIT) 1499 (2021) arXiv:2102.02494 DOI
- [4]
- S. Bravyi, D. Lee, Z. Li, and B. Yoshida, “How much entanglement is needed for quantum error correction?”, (2024) arXiv:2405.01332
- [5]
- A. Aydin, M. A. Alekseyev, and A. Barg, “A family of permutationally invariant quantum codes”, Quantum 8, 1321 (2024) arXiv:2310.05358 DOI
- [6]
- Y. Ouyang, “Quantum storage in quantum ferromagnets”, Physical Review B 103, (2021) arXiv:1904.01458 DOI
- [7]
- T. Holstein and H. Primakoff, “Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet”, Physical Review 58, 1098 (1940) DOI
- [8]
- C. D. Cushen and R. L. Hudson, “A quantum-mechanical central limit theorem”, Journal of Applied Probability 8, 454 (1971) DOI
- [9]
- V. V. Albert et al., “Performance and structure of single-mode bosonic codes”, Physical Review A 97, (2018) arXiv:1708.05010 DOI
- [10]
- Y. Ouyang and G. K. Brennen, “Finite-round quantum error correction on symmetric quantum sensors”, (2024) 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 PI code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/gnu_permutation_invariant