PI qubit code 

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

Permutation invariant qubit codes of distance \(d\) can protect against \(d-1\) deletion errors [3,4], i.e., erasures of subsystems at unknown locations.

Gates

There is a measurement-free code-switching protocol between a qubit stabilizer code and a PI qubit code [5].

Decoding

Schur-Weyl-transform based decoder [6]. Here, one first measures the total angular momentum of consecutive pairs of qubits, and then its projection modulo some spacing. Recovery can be performed by applying geometric phase gates [7] and the quantum Schur transform.

Parents

Children

Cousins

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, “Permutation-Invariant Quantum Codes with Transversal Generalized Phase Gates”, (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, Y. Jing, and G. K. Brennen, “Measurement-free code-switching for low overhead quantum computation using permutation invariant codes”, (2024) arXiv:2411.13142
[6]
Y. Ouyang and G. K. Brennen, “Finite-round quantum error correction on symmetric quantum sensors”, (2024) arXiv:2212.06285
[7]
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

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: qubit_permutation_invariant

Cite as:
“PI qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/qubit_permutation_invariant
BibTeX:
@incollection{eczoo_qubit_permutation_invariant, title={PI qubit code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/qubit_permutation_invariant} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/qubit_permutation_invariant

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/permutation_invariant/qubit_permutation_invariant.yml.