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Bosonic quantum Fourier code[1]

Description

Two-mode non-uniform QSC encoding two logical qubits whose projection is onto a copy of an irreducible representation of the single-qubit Pauli group. This code is an extension of the single-logical-qubit code in [2; Eq. (10)], storing an extra logical qubit in the multiplicity space of the Pauli group.

The code admits the following basis of codewords for complex \(\alpha > 0\), up to normalization: \begin{align} \begin{split} |\overline{0,0}\rangle&\propto\left(\left|\alpha\right\rangle -\left|-\alpha\right\rangle \right)\left(\left|i\alpha\right\rangle +\left|-i\alpha\right\rangle \right)\\ |\overline{0,1}\rangle&\propto\left(\left|i\alpha\right\rangle -\left|-i\alpha\right\rangle \right)\left(\left|\alpha\right\rangle +\left|-\alpha\right\rangle \right)\\ |\overline{1,0}\rangle&\propto\left(\left|i\alpha\right\rangle +\left|-i\alpha\right\rangle \right)\left(\left|\alpha\right\rangle -\left|-\alpha\right\rangle \right)\\ |\overline{1,1}\rangle&\propto\left(\left|\alpha\right\rangle +\left|-\alpha\right\rangle \right)\left(\left|i\alpha\right\rangle -\left|-i\alpha\right\rangle \right) \end{split} \tag*{(1)}\end{align}

Gates

The single-qubit Pauli group can be realized via Gaussian rotations [1].Kerr interactions yield some Clifford gates. There is a Hadamard gate, up to a global rotation [1].A logical \(ZZ\)-gate can be performed using squeezing operators and quantum Zeno effect [1].

Decoding

The code is stabilized by the two-mode parity operator and annihilated by the operators \(\hat{a}_1^4 - \alpha^4\) and \(\hat{a}_1^2 \hat{a}_2^2 + \alpha^4\) [1].

Cousin

  • Pauli group-representation QSC— The bosonic quantum Fourier code and the Pauli group-representation QSC are both group-representation codes with \(G\) being the single-qubit Pauli group.

Primary Hierarchy

Parents
The bosonic quantum Fourier code has non-uniform \(\pm 1\) coefficients.
The bosonic quantum Fourier code is a group-representation code with \(G\) being the single-qubit Pauli group.
Bosonic quantum Fourier code

References

[1]
A. Leverrier, “Bosonic quantum Fourier codes”, (2025) arXiv:2505.16618
[2]
A. Denys and A. Leverrier, “Quantum Error-Correcting Codes with a Covariant Encoding”, Physical Review Letters 133, (2024) arXiv:2306.11621 DOI
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Zoo Code ID: fourier_bosonic

Cite as:
“Bosonic quantum Fourier code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/fourier_bosonic
BibTeX:
@incollection{eczoo_fourier_bosonic, title={Bosonic quantum Fourier code}, booktitle={The Error Correction Zoo}, year={2025}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/fourier_bosonic} }
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Cite as:

“Bosonic quantum Fourier code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/fourier_bosonic

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/oscillators/qsc/fourier_bosonic.yml.