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Bosonic codes allow the encoding of a logical qubit in a single componentdevice, utilizing the infinitely large Hilbert space of a harmonic oscillator.In particular, the Gottesman-Kitaev-Preskill code has recently beendemonstrated to be correctable well beyond the break-even point of the bestpassive encoding in the same system. Current approaches to quantum errorcorrection (QEC) for this system are based on protocols that use feedback, butthe response is based only on the latest measurement outcome. In our work, weuse the recently proposed Feedback-GRAPE (Gradient Ascent Pulse Engineeringwith Feedback) method to train a recurrent neural network that provides a QECscheme based on memory, responding in a non-Markovian way to the full historyof previous measurement outcomes, optimizing all subsequent unitary operations.This approach significantly outperforms current strategies and paves the wayfor more powerful measurement-based QEC protocols.

Boosting the Gottesman-Kitaev-Preskill quantum error correction with non-Markovian feedback