High-threshold fault-tolerant quantum computation with the Gottesman-Kitaev-Preskill qubit under noise in an optical setup

To implement fault-tolerant quantum computation (FTQC) with continuous variables, continuous variables need to be digitized using an appropriate code such as the Gottesman--Kitaev--Preskill (GKP) qubit. The scheme introduced in [K. Fukui et. al., Phys. Rev. X 8, 021054 (2018)] has reduced the threshold of a squeezing level required for continuous-variable FTQC to less than 10 dB, assuming noise derived from the GKP qubit itself. In this work, we propose a scheme to improve noise tolerance during the construction of large-scale cluster state used for FTQC with the GKP qubits. In our scheme, a small-scale cluster state is prepared by employing a maximum-likelihood method, the entanglement generation via the Bell measurement, and a probabilistic reliable measurement. Then, a large-scale cluster state is construct from the small-scale cluster states via the encoded Bell measurement. In the numerical calculations, we assume errors derived from the two-mode gate and loss in the homodyne measurement in addition to noise from the GKP qubit itself. The results show that the thresholds of a squeezing level are around 8.1, 9.6, and 12.4 dB for loss in the homodyne measurement 0, 5, and 10 %, respectively. Hence, this work provides a way toward continuous-variable FTQC with a feasible squeezing level.