A memory-assisted decoder for approximate Gottesman-Kitaev-Preskill codes

We propose a quantum error correction protocol for continuous-variable finite-energy, approximate Gottesman-Kitaev-Preskill (GKP) states undergoing small Gaussian random displacement errors, based on the scheme of Glancy and Knill [Phys. Rev. A 73, 012325 (2006)]. We show that information gained from multiple error-syndrome extractions can be used to enhance the protection of GKP encoded qubits from amplitude damping. Furthermore, we show that the expected total displacement error incurred in multiple rounds of error followed by syndrome extraction is bounded by $2\sqrt{\pi}$. By recompiling the syndrome-extraction circuits, we show that all squeezing operations can be subsumed into auxiliary state preparation, reducing them to beamsplitter transformations and quadrature measurements.