Numerically optimized bosonic code[1,2] 


Bosonic Fock-state code obtained from a numerical minimization procedure, e.g., from enforcing error-correction criteria against some number of losses while minimizing average occupation number. Useful single-mode codes can be determined using basic numerical optimization [1,2], semidefinite-program recovery/encoding optimization [3,4], or reinforcement learning [5,6].

The smallest numerically optimized code protecting against a single loss error is the \(\sqrt(17)\) code [1], \begin{align} \begin{split} |\overline{0}\rangle&=\frac{1}{\sqrt{6}}\left(\sqrt{7-\sqrt{17}}|0\rangle+\sqrt{\sqrt{17}-1}|3\rangle\right)\\ |\overline{1}\rangle&=\frac{1}{\sqrt{6}}\left(\sqrt{9-\sqrt{17}}|1\rangle-\sqrt{\sqrt{17}-3}|4\rangle\right)~, \end{split} \tag*{(1)}\end{align} correcting a single loss error. The average occupation number of the codewords is \(\approx 1.6\), which is \(0.4\) photons lower than that of the smallest binomial code with the same level of protection.


Number phase codes protect from a finite number of loss events. However, unlike Fock-state codes, their protection does not stem from a Fock-state spacing.




M. H. Michael et al., “New Class of Quantum Error-Correcting Codes for a Bosonic Mode”, Physical Review X 6, (2016) arXiv:1602.00008 DOI
V. V. Albert et al., “Performance and structure of single-mode bosonic codes”, Physical Review A 97, (2018) arXiv:1708.05010 DOI
K. Noh, V. V. Albert, and L. Jiang, “Quantum Capacity Bounds of Gaussian Thermal Loss Channels and Achievable Rates With Gottesman-Kitaev-Preskill Codes”, IEEE Transactions on Information Theory 65, 2563 (2019) arXiv:1801.07271 DOI
P. Leviant et al., “Quantum capacity and codes for the bosonic loss-dephasing channel”, Quantum 6, 821 (2022) arXiv:2205.00341 DOI
Z. Wang et al., “Automated discovery of autonomous quantum error correction schemes”, (2021) arXiv:2108.02766
Y. Zeng et al., “Approximate Autonomous Quantum Error Correction with Reinforcement Learning”, Physical Review Letters 131, (2023) arXiv:2212.11651 DOI
M. Lin, C. Chamberland, and K. Noh, “Closest Lattice Point Decoding for Multimode Gottesman-Kitaev-Preskill Codes”, PRX Quantum 4, (2023) arXiv:2303.04702 DOI
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Cite as:
“Numerically optimized bosonic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_numopt, title={Numerically optimized bosonic code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Numerically optimized bosonic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.