We propose two schemes to obtain Gottesman-Kitaev-Preskill (GKP) error syndromes by means of linear-optical operations, homodyne measurements, and GKP ancillas. This includes showing that for a concatenation of GKP codes with an [n,k,d] stabilizer code only 2n measurements are needed in order to obtain the complete syndrome information, significantly reducing the number of measurements in comparison to the canonical concatenated measurement scheme and at the same time generalizing linear-optics-based syndrome detections to higher GKP codes. Furthermore, we analyze the possibility of building the required ancilla states from single-mode states and linear optics. We find that for simple GKP codes this is possible, whereas for concatenations with qubit Calderbank-Shor-Steane codes of distance d≥3 it is not. We also consider the canonical concatenated syndrome measurements and propose methods for avoiding crosstalk between ancillas. In addition, we make use of the observation that the concatenation of a GKP code with a stabilizer code forms a lattice in order to see the analog information decoding of such codes from a different perspective allowing for semianalytic calculations of the logical error rates.
Frank Schmidt and Peter van Loock
Phys. Rev. A 105, 042427