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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.

Encoding

Finite-depth quantum circuits with LOCC for Dicke states [57].

Gates

There is a measurement-free code-switching protocol between a qubit stabilizer code and a PI qubit code [8].

Decoding

Schur-Weyl-transform based decoder [9]. Here, one first measures nested total angular momenta, i.e., that of the first qubit, the first and second, followed by the first, second, and third, etc. Then, for codes with spacing, one measures the projection of the angular momentum modulo the spacing. Recovery can be performed by applying geometric phase gates [10] or the quantum Schur transform.

Cousins

Primary Hierarchy

Parents
PI qubit code
Children
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].

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]
H. Buhrman, M. Folkertsma, B. Loff, and N. M. P. Neumann, “State preparation by shallow circuits using feed forward”, Quantum 8, 1552 (2024) arXiv:2307.14840 DOI
[6]
L. Piroli, G. Styliaris, and J. I. Cirac, “Approximating Many-Body Quantum States with Quantum Circuits and Measurements”, Physical Review Letters 133, (2024) arXiv:2403.07604 DOI
[7]
J. Yu, S. R. Muleady, Y.-X. Wang, N. Schine, A. V. Gorshkov, and A. M. Childs, “Efficient preparation of Dicke states”, (2024) arXiv:2411.03428
[8]
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
[9]
Y. Ouyang and G. K. Brennen, “Finite-round quantum error correction on symmetric quantum sensors”, (2024) arXiv:2212.06285
[10]
X. Wang and P. Zanardi, “Simulation of many-body interactions by conditional geometric phases”, Physical Review A 65, (2002) arXiv:quant-ph/0111017 DOI
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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} }
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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.