Non-Markovian Feedback for Optimized Quantum Error Correction

Bosonic codes allow the encoding of a logical qubit in a single component device, utilizing the infinitely large Hilbert space of a harmonic oscillator. In particular, the Gottesman-Kitaev-Preskill code has recently been demonstrated to be correctable well beyond the break-even point of the best passive encoding in the same system. Current approaches to quantum error correction (QEC) for this system are based on protocols that use feedback, but the response is based only on the latest measurement outcome. In our work, we use the recently proposed feedback-GRAPE (gradient-ascent pulse engineering with feedback) method to train a recurrent neural network that provides a QEC scheme based on memory, responding in a non-Markovian way to the full history of previous measurement outcomes, optimizing all subsequent unitary operations. This approach significantly outperforms current strategies and paves the way for more powerful measurement-based QEC protocols.