## Description

Block quantum code defined on two-dimensional subsystems such that any permutation of the subsystems leaves any codeword invariant.

Dicke states: For \(n\)-qubit block codes, an often used basis for the \(n+1\)-dimensional PI subspace consists of the Dicke states \(|D^n_w\rangle\) -- normalized PI states of \(w\) excitations, i.e., a normalized sum over all binary-string basis elements with \(w\) ones and \(n - w\) zeroes. For example, the single-excitation Dicke state on three qubits is \begin{align} |D_{1}^{3}\rangle=\frac{1}{\sqrt{3}}\left(|001\rangle+|010\rangle+|100\rangle\right)~. \tag*{(1)}\end{align} The \(n+1\)-dimensional PI space can be thought of as a standalone spin-\(n/2\) quantum system, yielding a way to convert between permutation-invatiant qubit codes and \(SU(2)\) spin codes. A single-spin code for the \(SU(2)\) group correcting spherical tensors can be mapped into a PI qubit code with an analogous distance [1][2; Thm. 1].

## Protection

## Decoding

## Parents

## Children

- Binary dihedral PI code
- Combinatorial PI code
- Qudit GNU PI code
- \(((n,1,2))\) Bravyi-Lee-Li-Yoshida PI code
- Clifford spin code — Clifford codes for spins housing representations of \(SU(2)\) yield PI qubit codes with non-trivial distance when the single spin-\(n/2\) is treated as the permutationally invariant subspace of \(n\) qubits via the Dicke-state mapping. The subgroup of gates of a Clifford spin code is implemented transversally via this mapping [1].

## Cousins

- Eigenstate thermalization hypothesis (ETH) code — Several instances of ETH codes contain PI qubit codewords.
- Single-spin code — Single-spin codes are subspaces of a single large spin, which can be either standalone or correspond to the PI subspace of a set of spins via the Dicke state mapping.

## References

- [1]
- S. Omanakuttan and J. A. Gross, “Multispin Clifford codes for angular momentum errors in spin systems”, Physical Review A 108, (2023) arXiv:2304.08611 DOI
- [2]
- E. Kubischta and I. Teixeira, “The Not-So-Secret Fourth Parameter of Quantum Codes”, (2024) arXiv:2310.17652
- [3]
- M. Hagiwara and A. Nakayama, “A Four-Qubits Code that is a Quantum Deletion Error-Correcting Code with the Optimal Length”, (2020) arXiv:2001.08405
- [4]
- A. Nakayama and M. Hagiwara, “Single Quantum Deletion Error-Correcting Codes”, (2020) arXiv:2004.00814
- [5]
- Y. Ouyang and G. K. Brennen, “Finite-round quantum error correction on symmetric quantum sensors”, (2024) arXiv:2212.06285
- [6]
- X. Wang and P. Zanardi, “Simulation of many-body interactions by conditional geometric phases”, Physical Review A 65, (2002) arXiv:quant-ph/0111017 DOI

## Page edit log

- Victor V. Albert (2023-04-18) — most recent

## Cite as:

“PI qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/qubit_permutation_invariant