Hessian QSC[1] 

Description

Quantum spherical code encoding a logical qubit, with each codeword an equal superposition of vertices of a Hessian complex polyhedron. For the unit sphere, the codewords are \begin{align} |\overline{0}\rangle &= \frac{1}{\sqrt{27}}\left( \sum_{\mu,\nu=0}^{2} |0,\omega^{\mu},-\omega^{\nu}\rangle + |-\omega^{\nu},0,\omega^{\mu}\rangle + |\omega^{\mu},-\omega^{\nu},0\rangle \right) \tag*{(1)}\\ |\overline{1}\rangle &= \frac{1}{\sqrt{27}}\left( \sum_{\mu,\nu=0}^{2} |0,-\omega^{\mu},\omega^{\nu}\rangle + |\omega^{\nu},0,-\omega^{\mu}\rangle + |-\omega^{\mu},\omega^{\nu},0\rangle \right)~, \tag*{(2)}\end{align} where \(\omega = e^{\frac{2\pi i}{3}}\).

Figure I: Projection of the double Hessian code constellation with each copy of the Hessian logical constellation marked in a different colour.

Protection

The Hessian QSC is a \(\langle 4, 5, 9 \rangle\) code, i.e. it detects 8 photon losses and protects against 3. The code also detects up to 4 ladder errors (losses or gains). The code resolution \( d_E = 1.0\).

Parent

  • Quantum spherical code (QSC) — The Hessian QSC is an example of a QSC with logical constellation built from the Hessian complex polyhedron.

Cousin

  • Hessian polyhedron code — The Hessian QSC is the quantum generalization of the classical Hessian polyhedron code.

References

[1]
S. P. Jain et al., “Quantum spherical codes”, (2023) arXiv:2302.11593
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Zoo Code ID: hessian_qsc

Cite as:
“Hessian QSC”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/hessian_qsc
BibTeX:
@incollection{eczoo_hessian_qsc, title={Hessian QSC}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/hessian_qsc} }
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Cite as:

“Hessian QSC”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/hessian_qsc

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/oscillators/coherent_state/hessian_qsc/hessian_qsc.yml.