  Bosonic quantum error correction is a viable option for realizing
error-corrected quantum information processing in continuous-variable bosonic
systems. Various single-mode bosonic quantum error-correcting codes such as
cat, binomial, and GKP codes have been implemented experimentally in circuit
QED and trapped ion systems. Moreover, there have been many theoretical
proposals to scale up such single-mode bosonic codes to realize large-scale
fault-tolerant quantum computation. Here, we consider the concatenation of the
single-mode GKP code with the surface code, namely, the surface-GKP code. In
particular, we thoroughly investigate the performance of the surface-GKP code
by assuming realistic GKP states with a finite squeezing and noisy circuit
elements due to photon losses. By using a minimum-weight perfect matching
decoding algorithm on a 3D space-time graph, we show that fault-tolerant
quantum error correction is possible with the surface-GKP code if the squeezing
of the GKP states is higher than 11.2dB in the case where the GKP states are
the only noisy elements. We also show that the squeezing threshold changes to
18:6dB when both the GKP states and circuit elements are comparably noisy. At
this threshold, each circuit component fails with probability 0.69%. Finally,
if the GKP states are noiseless, fault-tolerant quantum error correction with
the surface-GKP code is possible if each circuit element fails with probability
less than 0.81%. We stress that our decoding scheme uses the additional
information from GKP-stabilizer measurements and we provide a simple method to
compute renormalized edge weights of the matching graphs. Furthermore, our
noise model is general as it includes full circuit-level noise.
